A Unified Theory of Voting: Directional and Proximity Spatial Models

A Unified Theory of Voting Directional and Proximity Spatial Models

This book addresses the questions: How do voters use their own issue positions and those of candidates to decide how to vote? How do candidates choose policy positions in response to the behavior of voters? Does a voter tend to choose the candidate who most nearly shares the views of the voter or rather a candidate who holds more extreme or intense views but in the same direction as the voter, perhaps because voters discount candidates’ abilities to implement the policies they advocate? The authors develop a unified model that incorporates these and other voter motivations and, using conditional logit and other statistical methods, assess its empirical predictions – for both voter choice and candidate strategy – in the United States, Norway, and France. The analyses show that a combination of motivations involving proximity, direction, discounting, and party identification is compatible with the choices made by voters and with the mildly but not extremely divergent policies that are characteristic responses to these choices in both two-party and multiparty electorates. All of these motivations are necessary to understand the linkage between candidate issue positions and voter preferences. Samuel Merrill III is Professor of Mathematics and Computer Science at Wilkes University, Wilkes-Barre, PA. He received a Ph.D. in mathematics from Yale University. His research interests include mathematical modeling in voting behavior, party strategy, and social choice, as well as medical statistics. He is the author of Making Multicandidate Elections More Democratic (Princeton University Press, 1988) and has published in a number of journals, including the American Political Science Review, the American Journal of Political Science, Public Choice, and the Journal of the American Statistical Association. Bernard Grofman is Professor of Political Science and Social Psychology at the University of California, Irvine. He received a Ph.D. in Political Science from the University of Chicago. His major fields of interest are American politics, comparative election systems, and social choice theory. He has published in a number of journals, including the American Political Science Review, the American Journal of Political Science, and Public Choice, and he has authored or co-edited eleven books, including Information, Participation and Choice (University of Michigan Press, 1995).

A Unified Theory of Voting Directional and Proximity Spatial Models

SAMUEL MERRILL III BERNARD GROFMAN

PUBLISHED BY CAMBRIDGE UNIVERSITY PRESS (VIRTUAL PUBLISHING) FOR AND ON BEHALF OF THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge CB2 IRP 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia http://www.cambridge.org © Cambridge University Press 1999 This edition © Cambridge University Press (Virtual Publishing) 2003 First published in printed format 1999

A catalogue record for the original printed book is available from the British Library and from the Library of Congress Original ISBN 0 521 66222 2 hardback Original ISBN 0 521 66549 3 paperback

ISBN 0 511 01159 8 virtual (eBooks.com Edition)

Contents

List of Tables and Figures Acknowledgments

page

1 Introduction 1.1 How Do Voters Decide? 1.2 Spatial Models 1.3 Overview Part I

Models of Voter Behavior

2 Alternative Models of Issue Voting 2.1 Proximity Models 2.1.1 The Downsian Proximity Model 2.1.2 The Grofman Discounting Model 2.2 Directional Models 2.2.1 The Matthews Directional Model 2.2.2 The Rabinowitz–Macdonald Directional Model 2.3 Comparison of Models 3 A Unified Model of Issue Voting: Proximity, Direction, and Intensity 3.1 Limitations of Pure Models 3.2 The Unified Model 3.3 Relation between the Grofman Discounting Model and the RM Model with Proximity Constraint 3.4 Conclusions v

ix xiii 1 1 4 10 17 19 19 19 22 23 25 29 32 38 38 40 47 50

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Contents

4 Comparing the Empirical Fit of the Directional and Proximity Models for Voter Utility Functions 4.1 Discriminating between Models 4.2 Utility Curves 4.3 Correlation and Regression Analyses of Pure Models of Voter Utility 4.3.1 U.S. Data 4.3.2 Non-U.S. Data 4.4 Discussion 5 Empirical Model Fitting Using the Unified Model: Voter Utility 5.1 Testing the Proximity and Directional Models of Voter Utility within a Nested Statistical Framework 5.2 Correlation Analysis 5.3 Fitting the Unified Model of Voter Utility via Nonlinear Regression 5.4 Parameter Estimates for the Mixed Proximity– RM Model 5.5 Discussion 6 Empirical Fitting of Probabilistic Models of Voter Choice in Two-Party Electorates 6.1 Probabilistic Models 6.2 A Unified Model of Voter Choice 6.3 Fitting the Conditional Logit Model to American NES Data 6.4 Discussion 7 Empirical Fitting of Probabilistic Models of Voter Choice in Multiparty Electorates 7.1 Multiparty Elections 7.2 Mixed Deterministic and Probabilistic Models 7.3 Fitting the Conditional Logit Model to Norwegian Data 7.4 Fitting the Conditional Logit Model to French Data 7.5 Discussion and Conclusions

52 52 53 59 60 64 66 67 67 68 70 74 78 81 81 82 84 89 91 91 92 95 103 105

Contents

Part II

Models of Party or Candidate Behavior and Strategy

8 Equilibrium Strategies for Two-Candidate Directional Spatial Models 8.1 Stable Strategies 8.2 Nash Equilibrium under the Grofman Discounting Model and Constrained Directional Models 8.3 Nash Equilibria under the Matthews Directional Model 8.3.1 Characterization of Condorcet Directional Vectors in Two Dimensions 8.3.2 The Condorcet Vacuum for American and Norwegian Data 8.4 Strategies when Different Models Govern Each Candidate 8.5 Conclusions 9 Long-term Dynamics of Voter Choice and Party Strategy 9.1 Why Is There Limited Polarization and Alternation of Parties? 9.2 Base Dynamic Model under Discounting 9.3 Convergence to Separate Points of Stability for Each Party under the Base Model 9.4 Party Strategy under Discounting 9.5 Modifications of the Model for Asymmetric Parties and Disparate Discount Factors 9.6 Discussion 10 Strategy and Equilibria in Multicandidate Elections 10.1 Multicandidate Equilibria 10.2 A Multidimensional Convergent Equilibrium 10.3 Divergent Equilibria with Partisan Voting and the Effect of a Directional Component 10.4 Regions of Candidate Support for Directional Models for More than Two Candidates 10.5 Discussion and Conclusions

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107 109 109 110 114 115 120 123 127 128 128 131 133 135 138 141 144 144 145 148 151 156

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Contents

11 Strategy under Alternative Multicandidate Voting Procedures 11.1 Alternative Voting Procedures 11.2 Are Centrists or Extremists Favored? 11.3 Simulation Results 11.4 Conclusions Taking Stock of What’s Been Done and What Still Needs to Be Done Future Work

158 158 160 161 162

Postscript

Appendices 3.1 Mixed Proximity–RM Models 4.1 Methodology: Data Analysis 4.2 Methodology: Linear versus Quadratic Utility Functions 4.3 Methodology: Mean versus Voter-specific Placements of Candidates 5.1 The Nature and Magnitude of Projection Effects 5.2 Interpretation of Model Parameters 5.3 The Westholm Adjustment for Interpersonal Comparisons 7.1 Methodology: The Lewis and King Critique 7.2 Methodology: English Translations of Questions from the Norwegian Election Studies 7.3 A Strategic Probabilistic Model of Voter Choice 8.1 Notes on Equilibrium Analysis 8.2 Use of Harmonic Decomposition to Determine Equilibria Glossary of Symbols References Index

164 166 170 170 172 173 174 179 181 181 186 189 189 191 193 195 196 207

List of Tables and Figures

Tables 3.1 Summary of pure models as special cases of the unified model and specification of utility functions page 4.1 Correlation between model predictions and thermometer scores using voter-specific placements of candidates: American NES 1980–96 4.2 Adjusted R-squared values for proximity and RM model predictions of thermometer scores using voter-specific placements of candidates: American NES 1968–92 5.1 Correlation between model predictions and thermometer scores using projection-adjusted placements of candidates: American NES 1980–96 5.2 Parameter estimates for the unified model using projection-adjusted placements of candidates: American NES 1980–96 5.3 Comparative parameter estimates for the unified model using different scoring methods: American NES 1980–96 5.4 Comparative parameter estimates for the RM model with proximity constraint using different scoring methods: American NES 1980–96 6.1 Maximum likelihood estimates for parameters of ix

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List of Tables and Figures

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6.2 6.3 7.1

7.2

7.3

7.4

7.5

9.1 11.1 A.5.1

A.7.1

the proximity–Matthews probabilistic submodel of voter choice: American NES 1980–96 Parameter estimates for RM–Matthews probabilistic submodel of voter choice: American NES 1980–96 Log-likelihoods for nested models of voter choice: American NES 1980–96 Model-predicted vote share for the 1988 French Presidential Survey, based on the issue of immigration Parameter estimates for the unified model of voter choice and its submodels (seven issue dimensions): Norwegian Election Study 1989 Parameter estimates for the unified model of voter choice and its submodels (five issue dimensions): Norwegian Election Study 1993 Parameter estimates for the unified model of voter choice and its submodels (two issue dimensions): Norwegian Election Study 1993 Parameter estimates for the unified model of voter choice and its submodels (four issue dimensions): French Presidential Election Study 1988 Medians, means, and standard deviations for selfand candidate placements: American NES 1980–96 Mean rank of winning candidate by voting system, ranked by nearness to neutral point Standardized regression coefficients for the 1989 Norwegian Election Study for various modeling choices for proximity and RM utilities Parameter estimates for Merrill’s strategic voting model for Norway and Sweden

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Figures 1.1 Directional voting with discounting page 2.1 Downsian spatial model 2.2 Utilities for various configurations for which voter, V, and candidate, C, agree on the horizontal

6 21

List of Tables and Figures

2.3 2.4 2.5 3.1 3.2

3.3 4.1

4.2

4.3 7.1 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 9.1 9.2

dimension (economic policy) but disagree on the vertical dimension (social policy) Utility under the RM directional model Utility curves for pure models for voter in a fixed position Indifference lines under alternative models Indifference lines under directional models Comparison of utility curves for Grofman discounting model and mixed proximity–RM model for voter in a fixed position Equivalence of Grofman discounting model and mixed proximity–RM model Plots of mean thermometer scores for candidates versus liberal–conservative location of the respondent: 1984 American NES Plots of mean thermometer scores for candidates versus liberal–conservative location of the respondent: 1988 American NES Mean thermometer scores, stratified by voter location (L/C): 1984 American NES Scatter plot of Norwegian parties in 1993 with respect to left–right and EEC scales Nash equilibrium under the Grofman discounting model Yolk and pseudo-yolk Indifference line for candidates C* and C and characteristic vector A of the support set of C* Star angles for small electorates Star for 201 voters uniformly distributed on disc Star for 200 voters uniformly distributed on disc Star for 201 voters from the 1984 American NES Star for 201 voters following a tripolar distribution Star for 201 voters from the 1989 Norwegian Election Study Strategy for challenger Base dynamic model: Median voter at 0 Base dynamic model (median voter at 0; d = 0.5):

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List of Tables and Figures

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9.3

9.4 9.5 9.6 10.1 10.2

10.3

10.4

A.4.1 A.7.1

Location of party Right in elections won by Right, for three starting values of the status quo Base dynamic model (median voter at 0; d = 0.5): Movement of the location of the status quo with victories alternating between parties Strategic dynamic model: Median voter at 0 Unbalanced dynamic model: Median voter offset from 0 Unbalanced dynamic model: Movement of the status quo for voter median at 0.25 (d = 0.5) Distribution of support in the Adams probabilistic model Model predictions of party locations versus actual (mean) party placements by model type: Norway 1989 Regions of support for the RM and proximity models: No candidate in the interior of the convex hull of the others Regions of support for the RM and proximity models: One candidate in the interior of the convex hull of the others Utility differences by model Regression of estimated bias in the mixing parameter for simulated data

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Acknowledgments

No work stands on its own. We draw heavily from the basic Downsian proximity voting framework and the huge literature extending Downs’s work, particularly that of Enelow, Hinich, and Munger.1 We were directly inspired by the directional voting ideas of Matthews (1979), which, in turn, can be linked to ideas in social choice in Schofield (1983, 1985); and by the work of Rabinowitz, Macdonald, and their co-authors (e.g., Rabinowitz and Macdonald, 1989) and the literature their work has inspired.2 1

2

Recent important works focusing on proximity spatial models of electoral competition include Enelow and Hinich’s The Spatial Theory of Voting (1984); Ordeshook’s Game Theory and Political Theory (1986); Mueller’s Public Choice II (1989); and Hinich and Munger’s Ideology and the Theory of Political Choice (1994). Enelow and Hinich’s second book (Advances in the Spatial Theory of Voting, 1990) treats multicandidate electoral competition and the effect of policy preferences on the part of candidates, among other topics. While the principal focus of Coughlin’s Probabilistic Voting Theory (1992) is the proximity model, he also derives equilibrium results for candidate strategy under assumptions based on directionality and for a fixed-sum constraint on the available resources to be redistributed to voters. Pierce’s Choosing the Chief: Presidential Elections in France and the United States (1995a), van der Eijk and Franklin’s Choosing Europe? (1996), and Hinich and Munger’s recent book, Analytical Politics (1997) include comparison of the Rabinowitz–Macdonald directional model and the proximity model as one among many topics. However, no previous book has systematically compared directional spatial models with the traditional proximity models or investigated the implications of a unified approach for candidate strategy and equilibrium as we do here. Our bibliography lists 35 papers since 1989 dealing with the Rabinowitz–Macdonald directional model, of which at least 29 have already appeared in print at the time of this writing. There is also a direct link between the 1970s work of the mathematical psychologist Douglas Carroll and the Rabinowitz–Macdonald directional model. The Matthews directional model, on the other hand, has received much less attention in the recent literature on candidate and political party competition and to our knowledge has not been the subject of any empirical analysis prior to the publication of four of our own papers on directional models in the 1990s.

xiii

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Acknowledgments

We particularly thank James Adams, Jay Dow, Steven Brams, Richard Potthoff, and Robert Tuttle for reading portions of the manuscript and for helpful comments and discussions of it. We also thank them – as well as Scott Feld, Gary King, Nicholas Miller, George Rabinowitz, and Anders Westholm – for general discussions or for correspondence about related topics over the years that helped lay the groundwork for the present work. James Adams kindly gave his permission to include a summary of joint work in progress with one of the authors and made numerous and helpful comments about many aspects of the manuscript. At Cambridge University Press, we thank Alex Holzman, Holly Johnson, and Genevieve Scandone. For library assistance, we thank Dorothy Green and Clover Behrend. The data sets used in the study were made available by the Interuniversity Consortium for Political and Social Research (ICPSR) and the Norwegian Social Science Data Services (NSD) and are acknowledged with appreciation. We thank Ola Listhaug and Atle Alvheim for facilitating our use of the NSD data. Bernt Aardal and Henry Valen were the Principal Investigators of the Norwegian Election Studies and the Norwegian Bureau of Statistics collected the data. Roy Pierce was the Principal Investigator of the 1988 French Presidential Election Survey. A portion of the research for this book was done while the first author was a Visiting Scholar in the Department of Biostatistics at the University of Washington. The contributions of the second author to the completion of this manuscript were supported by National Science Foundation Grant No. SBR 446740-21167, Program in Methodology, Measurement, and Statistics (to Bernard Grofman and Anthony Marley). Neither the NSF, the ICPSR, the NSD, the Norwegian Bureau of Statistics, nor the principal investigators of either of the U.S., Norwegian, or French national election studies are responsible for the analyses or interpretations presented here. Chapters 2, 3, and 5 are based in part on ideas in Merrill and Grofman (1997a, b), while Chapter 8 draws on the work of Schofield (1978, 1985), Matthews (1979), Cohen and Matthews (1980), and Merrill, Grofman, and Feld (1999) and Chapter 9 is drawn from Merrill and Grofman (1998b). Chapters 10 and 11 are based in part

Acknowledgments

xv

on the work of Adams (forthcoming a, 1997a, 1999), Merrill (1993), Adams and Merrill (forthcoming a), and related ideas in Feld and Grofman (1991). Figures 3.3, 8.1, 8.4, 8.5, 8.6, and 10.2 and Tables 4.2 and 11.1 are reprinted or adapted with permission from Kluwer Academic Publishers.

CHAPTER 1

Introduction . . . how should a rational voter calculate the expected utility incomes from which he derives his expected party differential? Anthony Downs, An Economic Theory of Democracy (1957: 39)

1.1 How Do Voters Decide? Imagine that you, a voter, must choose between two or more candidates for elective office. How will you decide for whom to vote? Political scientists and others have studied this question from both theoretical and descriptive perspectives. Roughly speaking they have offered what we might think of as three basic types of answers: Answer one is that you will tend to vote for the candidate whose political party you have come to identify with. In classic works such as The American Voter (Campbell, Converse, Miller, and Stokes, 1960), scholars at the University of Michigan and elsewhere have developed this idea into what is often called the Michigan model of party identification. While not everyone will simply vote their party ID, party ID is seen as the best single advance predictor of the vote, with changing circumstances (such as economic conditions) giving rise to short-run and longer-run electoral forces that favor one candidate/party or another and that lead voters to deviate from their historical party identifications. Answer two is that you will tend to vote for the candidate whose policy views are closest to your own. Following Anthony Downs (l957), we may think of both voters and candidates as points in some n-dimensional issue or policy space. A voter’s location in the space represents the voter’s ideal point (a.k.a. bliss point), whose coordinates tell us the position espoused by that voter on each of the issues. A candidate’s location in the space is taken to be an indicator of the 1

2

Introduction

candidate’s platform, i.e., her1 statement about the policy outcomes on each issue she hopes to achieve were she to be elected. Under the basic Downsian (proximity) model, the voter chooses the candidate closest to his own ideal point. Answer three is that you will tend to vote for the candidate who is most likely to change things in a way that will leave you most satisfied. It might appear that this answer is just a rephrasing of the previous model built upon relative proximity, but it is not. In this directional approach to voting it is critical to have a neutral or status quo point from which to judge expected direction of change.A voter may choose that candidate whose direction of movement – which involves both issue salience and policy preference – most resembles his own. Alternatively – if, for example, conservatives have been in power for a long time – a voter who is conservative, but less so than the party in power, may choose a liberal candidate. The latter can be expected to move the status quo back in the voter’s direction in preference to a conservative candidate, who may attempt to move policy even further to the right.2 The search for a better understanding of voting behavior raises a number of questions. How do voters respond to advocates of policy positions more extreme than their own? Do voters always choose the candidate/party who advocates a policy position closest to their own most preferred position or might voters favor a candidate who takes a position somewhat more intense than their own on the principle that “more is better”? Or might a voter support a candidate who advocates moving the status quo beyond the voter’s ideal policy on the grounds that – following the compromises of office – such a candidate might end up implementing what the voter really wants? In other words, do voters discount the claims of candidates who assert that their election will result in large changes from the status quo? How, if at all, do changes in the location of the status quo affect the choices made by voters? Answers to such questions about voting behavior lead to further 1

2

We use the following convention: To refer to candidates, female pronouns are used; to refer to voters, we use male pronouns. A further approach to voter choice – not as systematically developed as the first three – emphasizes candidate characteristics distinct from ideology such as perceived competence or perceived trustworthiness (see, e.g., Enelow and Hinich, 1984).

Introduction

3

inquiries about the optimal response of candidates. What types of strategies should rational candidates adopt in their reaction to the distribution of voter preferences? In particular, should we expect convergent strategies, i.e., should all candidates/parties be expected to take similar stands, or does greater benefit lie in the adoption of divergent (i.e., dispersed) strategies? Should a party’s strategy be more extreme than the positions of its supporters? How are desirable strategies altered as the policy status quo shifts over a sequence of elections? How closely do the optimal strategies predicted by different models resemble those actually used by parties and candidates in real elections? This book is addressed to political scientists and students of political science who seek answers to questions like the ones above. In this book we present a unified model of voting behavior in which voters have not just one motivation but a mixture of motivations. We demonstrate how various models of voting behavior – entailing the combined effects of proximity, direction, intensity, and discounting – can illuminate how parties strategically choose policy positions in response to voters. We offer empirical tests of these models over a wide range of assumptions and for electorates that operate under a variety of voting systems and political cultures. The empirical evidence suggests a picture intermediate between convergence and extreme divergence of party strategies. For example, in the United States, even though most presidential elections are not seen as sharply polarizing, the Democratic and Republican nominees almost inevitably favor different policies, are responsive to different constituencies, and are perceived to hold different positions by voters in the American National Election Studies. Despite claims to the contrary, the positions of the two major American parties are neither identical to one another, nor are they at the farthest possible opposite extremes of the American spectrum of political thought, i.e., there are both centripetal and centrifugal forces at work. Likewise, for most of the major parties in Western European polities – very different electoral systems and often very different political experiences notwithstanding – party positions are distinct but, except for some minor parties, not extremist. A central concern of this book is to try to determine what – if any – models are compatible with the mildly but not extremely divergent

4

Introduction

policy platforms that appear, empirically, to be characteristic of both two-party and multiparty competition. How can we understand this general phenomenon, which we might call moderate extremism? This book develops both traditional and alternative models of voter decision making, comparing and contrasting the utility functions that characterize each model. We place the models in a unified framework and study the implications – for both voter choice and candidate strategy – of how voters make choices. We fit models representing multiple voter motivations to data from national election studies in three nations that represent different political structures – the United States, Norway, and France – and summarize numerous empirical studies in other nations. The data we analyze suggest that a unified approach that combines elements of the traditional proximity model, the various directional and discounting models, and the Michigan model’s emphasis on party identification is necessary to adequately understand the linkage between voter preferences and candidate issue positions. 1.2 Spatial Models A principal concern in this book is the interdependence between the decisions made by parties and those made by voters. In this chapter we introduce several theoretical models of voters’ evaluations of candidates; in subsequent chapters, we describe these models and their implications in greater detail and assess them empirically for two-party elections in the United States and multiparty contests in Europe. The fundamental vehicle for the translation of issue positions to voter choice is the spatial model of issue voting; in spatial modeling we focus on how issue positions of both voters and candidates (or parties) are translated into voter preferences and candidate strategy. In any spatial model of electoral competition, both voters and candidates are located at ideal points in a multidimensional space, each dimension of which represents a substantive issue. For example, the issue dimension of health care might be represented by a scale that ranges from the belief that government should provide universal health care to the opinion that medical expenses should be paid by individuals and private insurance plans. The spatial modeling approach pioneered by Anthony Downs

Introduction

5

(Downs, 1957), and subsequently developed by numerous scholars, permits us to represent the preferences of voters and the strategies of candidates in a structured manner and to develop mathematical models of the relationships among voters, among candidates, and between voters and candidates. We may then ask: What factors influence the value (usually denoted utility in this economics-inspired literature) that voters attach to seeing particular candidates get elected? Once we know the utility placed by the voters on each of the candidates, we can then seek to model voter choice and candidate strategy. The proximity version of the spatial model has dominated the thinking of political scientists about voting behavior. In literature stemming from Downs, closeness or agreement in political views between voters and candidates has played center stage in attempts to understand voter choice, but there are some aspects of politics that it simply does not do well at explaining. Other factors such as: (1) whether a candidate’s direction of movement from the status quo can be expected to bring the new status quo closer to the voter, and (2) the fact that some voters may be much more concerned with outcomes on certain issues than on others (or have greater intensities of preferences for certain issues than for others) may also be of great importance in understanding voter preferences and the candidate strategies these give rise to. Moreover, (3) the issue positions that voters ascribe to particular candidates need not be the positions that a candidate formally espouses (or that are embodied in the platform of the candidate’s party). Voters may discount platforms by taking into account the likelihood that particular policies will actually be implemented. Under the traditional (Downsian) proximity spatial model, a voter’s utility for a candidate is assumed to increase with proximity to his ideal point (i.e., set of policy preferences). In the most basic Downsian model, vote-maximizing party locations for two-party competition converge toward the median voter location3 of the overall electorate. In general, in the Downsian model, voters’ choices and 3

The median voter is the voter with respect to whom 50 percent or more of the electorate is placed either at the voter’s own position or to the left of it and 50 percent or more is placed to the right. The median voter theorem applies to a onedimensional model.

6

Introduction

Figure 1.1 Directional voting with discounting.

preferences exert a centrist influence on the strategy of candidates seeking the voters’ preferences. In contrast, alternative spatial models – that posit links between voter preference and the direction of movement from the status quo (Matthews, 1979), or the intensity with which issue positions are advocated as well as their direction (Rabinowitz and Macdonald, 1989), or the discounting of candidate claims (Grofman, 1985), or party identification (Campbell et al., 1960) – suggest that some or all candidates may benefit by moving outward from the ideological center in the direction of the preferences of particular constituencies. To illustrate the difference between proximity, directional, and party identification models, let us consider a simple two-party example. Let L be the candidate of the left (say, Liberal) party and R the candidate of the right (say, Conservative) party. In a onedimensional model represented by a nine-point scale ranging from -4 on the left to 4 on the right, let the platform of L be -2, and that of R be 3 (see Figure 1.1). Consider a world with three voters: V1 with ideal point at -3, V2 with ideal point at 1, and V3 with ideal point at 4. Let voters V1 and V2 be Liberals and voter V3 be a Conservative party identifier. If party ID alone determines voting, then V1 and V2 will vote Liberal and L will win. If issue proximity determines voting, then V2 and V3 will vote Conservative and R will win. If the simplest directional model holds, then what will happen? Well, we don’t yet have enough information to answer this question. Imagine, however, that the Conservatives have been in power a long time and the policy status quo is at 2. If the Conservative wins, then we expect the status quo to shift in a positive direction (say, from 2 toward 3). Even though voter V2 is closer to R than to L, he may prefer

Introduction

7

L to R, in directional terms, since a victory by R will move policy outcomes (i.e., the status quo) away from V2’s most preferred outcome. In contrast, a victory by L will begin to shift policies down from 2 closer to 1, where V2 is located. Of course, if the Liberals actually succeed in moving policies all the way (or even most of the way) to -2, voter V2 would be quite upset, but if change in any given election period is only incremental, then voter V2 can afford to vote for the Liberal this time and then wait until the next election to see how far the status quo has moved before he decides whether to vote for the Conservative position to which he is actually closer. Here voter V2 is a moderate voter who is casting a kind of protest vote at the status quo having been shifted too far in one direction (in this case, toward the right). This simple model might be compared with American politics in 1992 in which voter V2 represents a conservative Democratic identifier willing to move the conservative status quo (following twelve years of Republican rule) toward the left. Two years later many such voters reversed this leftward movement by selecting a conservative Congress, only to apply the brakes once again in the election of 1996, by helping reelect a Democratic president and a diminished Republican majority in the House. Note that, under the particular type of directional voting defined above, a voter may change his vote without any change in his own ideal point or in the policy platforms offered by the two party candidates.All that will have changed is the location of the status quo.4 Note also that, in this model, we have natural forces restoring policy moderation, as well as forces fostering party competitiveness when the party in power has gone “too far.”5 For example, if the status quo was at 0, say, then the directional model predicts that voter V2 should again vote for the Conservative platform. A great deal of effort by many scholars (e.g., Markus and Converse, 1979; Page and Jones, 1979; Rabinowitz and Macdonald, 1989; Rabinowitz, Macdonald, and Listhaug, 1991; Platt, Poole, and 4

5

This model has some resemblance to the balancing model of Fiorina (1992) or Alesina and Rosenthal (1995) but it is not identical. In particular it does not require strategic calculations on the part of the voter. The voter votes sincerely, but in directional rather than proximity terms. We shall return to this point in Chapter 9. Cf. Stokes (1963) in his famous essay on forces restoring party competition.

8

Introduction

Rosenthal, 1992; Pierce, 1993, 1995a, 1997; Iversen, 1994; Westholm, 1997) including previous work of our own (Merrill, l994, l995; Merrill and Grofman, 1997a) has gone into trying to decide which of these three models (party ID, Downsian proximity, or some form of directional) fits the data best. In this book we take a rather heretical position. Namely, it is our view that all three models provide useful insights and that, as a practical matter, for any single election, it is difficult to determine which of the three pure models fits the data best, since there is a very large communality between the variance explained by each.6 We believe that the supposed incompatibility between the party ID approach and the approaches that focus on issues has been greatly exaggerated. In particular, those with a given party ID also tend to demonstrate issue proximity to the candidate of their party.7 Also, if we look only at a single election, Downsian and directional models and the party ID model tend to give successful predictions for the same set of voters – namely those that are well-anchored to the political system and see policy differences between the parties/candidates. Perhaps even more importantly, the recent work of Frank Wayman (1996) on the only panel data set to allow us to follow U.S. voters over a substantial time period (nearly twenty years) makes it clear that, rather than party ID being largely immutable over a lifetime, eventually party ID is apt to change to reflect previous voting choices when these are relatively consistent over time and inconsistent with previous party ID. One way to integrate both directional and proximity perspectives is to assume, following ideas in Grofman (1985), that some voters discount the policy positions announced by the candidates. In this discounting approach, if a candidate says she will implement a policy at, say, 4 and the status quo is at 2 (see Figure 1.1), then the voter may assume that the change from the status quo to be expected if that candidate is elected may not be 2 units (i.e., 4 - 2), but rather 2d units, where d is a discounting factor less than or equal to 1 and strictly 6

7

For simplicity of exposition we have treated both the Downsian and Michigan models as pure models, although both draw on multiple factors to explain voting behavior. We will elaborate on this point in Chapter 5 and evaluate the importance of “rationalization” and “projection” effects.

Introduction

9

greater than 0. Now, the voter chooses the candidate who he perceives will implement policy closest to himself. If the discounting factor d is 1, this is simply the proximity model; if the discounting factor d is close to 0, this is essentially a directional model, since each voter (except possibly a voter very near the status quo) will then pick the candidate who will move in the direction from the status quo toward the voter’s bliss point. Thus, the discounting model can be thought of as a mixture of proximity and directional voting with the d value indicating how far along the continuum between directional voting and proximity voting we have gone. Consider Figure 1.1 once again. With the status quo at 2, we have just noted that the swing voter V2 (at 1) will vote for R (at 3) if d is 1 (the proximity model), while V2 will vote for L (at -2) if d is near 0 (the directional perspective). A little arithmetic demonstrates that, in this example, V2 will vote for L if and only if d < 2/3; that is, a mildly conservative voter would vote for the liberal if discounting is sufficient. As before, voter V2 might represent a moderately conservative American voter in 1996. Given the strength of the Republican Congress following the election of 1994, the overall status quo was distinctly conservative.Voter V2 – if he sufficiently discounted the platforms of Clinton and the Republicans – might, according to the argument we have just presented, find a swing back in the Democratic direction preferable. Such support (along with other factors such as a healthy economy) helped Clinton win reelection. A similar phenomenon – in the reverse direction – may have occurred in 1980 as discounting of Ronald Reagan’s platform helped him pick up otherwise slightly left-leaning voters. The example we have worked through for Figure 1.1 shows that it can be useful to think about voter choice as a blend of directional and proximity calculations, with the discounting factor d indicating the relative importance of the two factors.8 Thus, we may think of the notion of discounting as one way to integrate directional and proximity factors. 8

Of course, it is possible that a voter may not apply the same discounting factor to both candidates. We shall take this possibility into account when we model voter choice and candidate strategy in Chapters 8 and 9.

10

Introduction

1.3 Overview Our desire to understand optimal candidate and party strategies leads us to investigate the behavior of voters upon which the rationale for these strategies rests. We will see empirically that these strategies are neither totally convergent (i.e., identical) nor totally divergent (i.e., arbitrarily far apart), but rather intermediate. The models of voter behavior that lead theoretically to the empirically observed behavior of parties are likewise intermediate in the sense that they blend aspects of proximity, direction, intensity, and/or discounting.9 In turn, a multicomponented model of voter behavior is supported by the data from a wide range of national election studies. A central concern of this book is to distinguish between directional and proximity models or combinations thereof – both theoretically and empirically. We do this in three different ways. We look at (1) differences in expectations about voter utility functions, (2) differences in expectations about voter choice functions, and (3) differences in expectations about party strategies. For each of these three we present empirical evidence. Much previous work (e.g., Westholm, 1997; Macdonald and Rabinowitz, 1998) has sought to demonstrate that one pure model of voting behavior or another is superior to all others. We argue instead that the various models are complementary rather than competing and that a unified model that reflects multiple motivations is the preferred model to explain both voter choice and party response. It is not just that a more complicated model yields a better statistical fit – that is to be expected. Rather, it is that for a wide variety of national electorates and for different methodological assumptions, the best fitting models are fairly consistently intermediate between the pure models. Perhaps most striking is the evidence from party strategies. Empirically, these strategies reflect behavior that is nearly optimal if parties are responding to voter behavior stemming from a mixture of voter motivations. Such strategies, however, would be far from 9

Despite our efforts in the volume, we are not really ready to distinguish between interpretations of our unified model as being motivated by discounting, or by a mixture of directional and proximity factors, or by a lack of consistency in employing these factors among individual voters, or by heterogeneity among different voters. We shall, however, have a great deal to say about whether voters use a blend of models rather than specific pure models (see Chapters 3–7 and 10).

Introduction

11

optimal if those motivations were characteristic of any one of the pure models. The remainder of this book is divided into two parts: one part dealing with choice seen from the perspective of the voter; and the second dealing with choice seen from the perspective of the party. Part I consists of Chapters 2–7; Part II consists of Chapters 8–11. A crosscutting division is that between two-party/two-candidate competition on the one hand and multiparty and multicandidate competition on the other. Chapters 2–6 in Part I and Chapters 8 and 9 in Part II fall into the former category; Chapter 7 in Part I and Chapters 10 and 11 in Part II fall into the latter category. Each part of the book begins with two theoretical chapters followed by two or more chapters that are primarily empirical. Chapters 2 and 3 introduce the separate spatial models of voting behavior and then bring them together into a unified model. Chapters 4–7 analyze these models empirically. Chapters 8 and 9 investigate stable patterns of party strategies, whereas Chapters 10 and 11 use the unified model to empirically assess these expectations of party decision making. In Chapter 2 we review the family of proximity and directional spatial models, providing detailed descriptions of the utility functions of each and showing how they are related to one another. As noted previously, there is not just a single directional model, there are various directional models. Some can be thought of in terms of direction of a candidate’s position on a dichotomous issue relative to a neutral point on that issue (Rabinowitz and Macdonald, 1989), some in terms of direction of movement along a vector from a neutral point (Matthews, 1979). Others (Grofman, 1985) make explicit use of the notion of direction of movement from a status quo point that changes from election to election. Some emphasize the vote-attracting powers of intense advocacy of issue positions (Rabinowitz and Macdonald, 1989), whereas others (Matthews, 1979) reflect only the relative salience of different issues and not the overall intensity of voter or candidate preferences. We compare these directional models with each other and consider how they differ in their implications from the classical proximity model. Chapter 3 introduces a general unifying model that contains most of the existing models in the literature as special cases. In this chapter,

12

Introduction

we develop a unified model for the shapes of voter utility functions – i.e., for the relationships between voters’ evaluations of candidates and the voters’ spatial locations – that encompasses both directional ideas and the standard proximity model as special cases. This unified model has as polar special cases the Rabinowitz–Macdonald directional model, the Matthews directional model, and the Downsian proximity model. We envision a voter as preferring a candidate who proposes to move policy in the direction desired by the voter. The voter may be responsive to strong advocacy for this direction of movement – whereas less intense advocacy in the same direction may sound weak or may be dismissed as an instance of “me-too.” At the same time, voter attitudes are assumed to be tempered by a concern that the candidate’s policy not be too distant from the voter’s own preferred position. The first of these motivations is one of direction; the second, one of proximity. These motivations are subsumed in the unified model and may induce a voter to support a candidate who proposes to move policy in the direction desired by the voter but is moderately more extreme than the voter himself. The unified model is characterized by two parameters, labeled b and q. The model is economical in that it depends on only two parameters, yet it permits us to distinguish between proximity and directional models on the one hand and between different versions of directional models on the other. The first of the parameters may be interpreted as the relative importance of proximity versus direction in the voter’s calculations, or – following Iversen (1994) – as the degree by which directional voting is constrained by distance between voter and candidate. Alternatively, we may explain the first parameter, b, in terms of discounting and the location of the status quo. From the perspective of a given voter, under appropriate restrictions, b is equivalent to the discount factor d of the Grofman discounting model, that is, a mixed proximity/directional model can be interpreted as a pure discounting model. The second parameter, q, is an indicator of the extent to which the directional component of voter choice corresponds to the Rabinowitz–Macdonald directional model as opposed to the Matthews directional model. Thus, q represents directional intensity,

Introduction

13

i.e., the extent to which utility increases with both voter and candidate distance from the neutral point. Chapters 4–7 develop empirical analyses of voter utility and voter choice. Chapter 4 provides a number of empirical analyses of voter utility – both graphical and numerical – that compare these alternative pure models for a variety of two-party and multiparty electorates in the United States and Europe. In discussing these analyses, we introduce several methodological considerations involved in such empirical testing. Chapter 5 continues the empirical analysis, dealing with the statistical fitting of the unified model of voter utility and of models that are nested within the unified framework. We fit the unified model so as to draw inferences not just about the relative contributions of the various model components but also about the interaction of these components with candidate type (i.e., incumbent versus challenger). Following Markus and Converse (1979), we also seek to measure the importance of rationalization effects that occur as voters attempt to assign locations to candidates that may reflect projection or contrast effects based on the voter’s own preferences for the candidates (cf. Page and Jones, 1979). The key result in this chapter is that all three components – proximity, direction, and intensity – are needed to explain variations in utility and that the mix of these constituents varies with type of candidate, with intensity being a greater factor for challengers than for incumbents. In Chapter 6 we shift from a voter utility to a voter choice perspective and from a deterministic to a probabilistic framework. We argue that voter choice models either avoid or are less sensitive to the many methodological problems that bedevil empirical testing of the voter utility models to which so much attention has been paid in recent literature. For the two-party United States electorate, we estimate parameters of a unified model of voter choice that controls for partisanship of the voter and specifies the probability of voter choice for each party/candidate. The voter choice version of the unified model (a conditional logit model) introduced in Chapter 6 is best suited for multiparty electorates. In Chapter 7, we fit this voter choice model for the multiparty/multicandidate elections of Norway and France. In addition to looking at politics from the voters’ perspective, we

14

Introduction

look at voting from the perspective of parties or candidates. Voter preferences – mediated by party strategies that define the choices open to the voters – ultimately determine public policies via the outcome of elections. Because of this intermediating role of party strategies, we shift our concern in Part II to party and candidate strategy. We wish to understand candidate/party strategies when voting is based on more than just simple issue proximity. The classic inference about candidate response is embodied in the median voter theorem: in a single-stage two-candidate race under majority rule and with only one dimension, both candidates – if rational – are drawn to the location of the median voter. Although this neat conclusion dissolves into instability or multiple potential equilibria for higher dimensions or more than two candidates, extensive recent work has shown that some degree of centrism is often associated with the proximity model.10 What can be said about optimal candidate/party strategies when we integrate directional and proximity ideas? Are configurations of candidate positions that should remain stable over a period of time (termed equilibria) necessarily central or may they be dispersed? What happens if we introduce discounting, partisan voting, or more than two candidates?11 In Chapter 8, instead of looking directly at voter utility and choice, we focus on optimal candidate or party response to the behavior of the voters. Such a response is embodied in the concept of a strategy equilibrium, i.e., a set of candidate/party strategies (expressed as points in a spatial model) from which no candidate/party would have 10

11

The work on centrality of outcomes is usually tied to geometric constructs such as the yolk and the heart (McKelvey, 1986; Feld, Grofman, and Miller, 1988; Miller, Grofman, and Feld, 1990; Schofield and Tovey, 1992; Tovey, 1992; Schofield 1993, 1996) that are known to be centrally located in the space of voter ideal points. See Chapter 8. There are really three different substantive areas of application of spatial models of parties: (1) models of committee voting tied to the literature on social choice and social welfare orderings stemming from Arrow (1962); (2) models of candidate competition, stemming from the work of Downs (1957); and (3) models of coalition formation (e.g., Axelrod, 1970; Grofman, 1982; Laver and Schofield, 1990). A number of the mathematical results in these literatures are equivalent or very similar, e.g., the search for the core of a spatial voting game is essentially equivalent to the search for an equilibrium location of candidates in two-party competition and is closely related to the search for a stable coalition structure. In various chapters we draw on work from the first as well as the second of these literatures.

Introduction

15

an incentive to depart. A strategy equilibrium – if it exists at all – may differ under various models such as proximity voting, directional voting, and the unified model. First we show, for two-party/twocandidate competition in one dimension, that discounting (or equivalently, mixing proximity and directional effects) may give rise to a distinct pair of stable strategies for the two parties, i.e., a divergent equilibrium. The more disparate the discount factors for the two candidates, the greater the divergence. Chapter 8 also considers the directional analog to the basic equilibrium concept in the proximity context, the core, which for majority rule games is also known as the Condorcet winner.12 We refer to this equilibrium concept as the Condorcet directional vector. We summarize some key geometric results that characterize the spatial region of instability, i.e., the region of potential neutral (or status quo) points from which there is no Condorcet directional vector. Next, in Chapter 9, we seek to understand the long-run dynamics of voter choice and party strategies. If voter choice is sensitive to the location of the status quo – which can be expected to change over time – then there should be forces restoring party competition, as one party takes the status quo “too far” in a given direction. We will show that, in contrast to a pure proximity model, the discounting model – and hence a model that combines directional and proximity aspects – implies either alternation in power between the parties or a pattern of repeated wins by one party, punctuated with single victories by the opposing party.13 On the theoretical side, we show how to extend our unified approach to party strategy from two-candidate/two-party contests to multicandidate and/or multiparty competition. On the empirical side we show in Chapter 10 that a mixed proximity and directional model 12

13

A Condorcet winner is a candidate who is preferred over each of the others by a majority of the electorate, i.e., could beat (or tie) each of the others in two-way contests. In further work, we hope to link the spatial literature on party competition more closely to the literature on party realignment. Lubell (1952) has hypothesized that, generally, in the United States, one party has been the sun and the other the moon, i.e., that we have long periods of one-party dominance with the major party the driving force behind policy change. Our analysis, thus, provides an alternative to that of Riker’s work on realignment (Riker, 1962, 1982), which holds that a long period of one-party dominance eventually ends when a majority coalition becomes too difficult to maintain.

16

Introduction

is by far the best predictor of the actual locations assumed by parties in the multiparty electorates of Norway and France, which are indicative of many European polities. Thus, empirical fitting provides strong support for a model of voter choice that combines both proximity and directional aspects. On the theoretical side, for three or more candidates, we investigate equilibria for directional models and the influence of partisan preference on equilibria for a spectrum of models. We see that introduction of a probabilistic element into a model of voter choice can lead to a convergent equilibrium under proximity assumptions. Moreover, if party ID is added as an explanatory variable, the equilibrium may remain, but one in which the parties maintain distinct strategies. Also, as a directional component is entered into the model, these optimal party locations spread further and further apart. Chapter 11 extends our analysis to types of voting rules where voters must do more than identify their single most preferred candidate, e.g., the Borda count, the single-transferable vote (STV), plurality with runoff, and approval voting. What stands out is that the preference for a mixed model over any pure one, which we have demonstrated for the standard plurality system of voting, extends to alternative voting rules as well. The postscript reflects on what we have done and what is left to be done. We reiterate our findings that – regardless of whether evidence is drawn from voter utility, voter choice, or party strategies – a mixed model dominates any pure model. Thus, even if a reader objects to any particular methodology, the combined evidence is overwhelming. We also reiterate the other major finding presented in this book; namely, that moderate divergence in party strategies is both to be expected and empirically observed. Thus, for both voter utility and voter choice, the unified model predicts that voters will use a mix of proximity and direction, whereas for party strategies, the unified model predicts and we empirically observe moderate divergence. Finally, we consider topics that must be left largely for future research, including issue salience, voter heterogeneity, ambiguity of or uncertainty about candidate positions, persuasion, and distinctions between issue space and ideological space.

PART I

Models of Voter Behavior

CHAPTER 2

Alternative Models of Issue Voting . . . [the voter] knows that no party will be able to do everything that it says it will do. Hence, he cannot merely compare platforms; instead he must estimate in his own mind what the parties would actually do were they in power. Anthony Downs, An Economic Theory of Democracy (1957: 39)

2.1 Proximity Models 2.1.1 The Downsian Proximity Model To establish a baseline, we describe in some detail the classic Downsian proximity model. As in all spatial models, both voters and candidates are represented by points in a multidimensional space that reflect their opinions on issues; each issue corresponds to one dimension of the space. In the simplest case, there is only one issue (say, economic policy) and, hence, one dimension. At this level, the spatial model is no more than a formalization of the familiar left–right or liberal-to-conservative scale, represented by a line. By convention, numerical values on this scale increase from left to right. Other things being equal, it thus seems plausible that, say, a liberal voter on economic policy will prefer a candidate who is liberal on that issue to either a middle-of-the-road or conservative candidate and will prefer a moderate over a conservative. This suggests that the voter’s evaluation, or utility, may be based on proximity in the spatial model, with highest utilities reserved for candidates located close by on the policy scale. This is the basic idea in the Downsian proximity model. If we introduce a second issue (say, social policy), it too can be represented by a liberal–conservative scale and the two issues can be combined into a two-dimensional spatial model, represented by the plane. We will associate location along the horizontal axis with position on one issue (say, economic policy) and location along the verti19

20

Part I Models of Voter Behavior

cal axis with position on another issue (say, social policy). Values on this second scale increase from the bottom to the top. It is convenient, although not necessary for the proximity model, to center both scales at 0, so that negative values refer to leftist views and positive values to rightist views. Figure 2.1 depicts a voter, V, who is strongly conservative on economic policy (located at +3) but only slightly conservative on social policy (located at +1). We denote the voter’s overall position by the ordered pair, (3, 1), where the first coordinate, 3, denotes the voter’s position on economic policy and the second coordinate, 1, denotes his position on social policy. He must choose between two candidates. The first, C1, agrees with him on economic policy (although she is slightly less conservative) but is slightly liberal on social policy, and is placed, let us say, at (2, -1). The second candidate, C2, agrees with the voter on social policy, but is slightly liberal on economic policy, and is placed at (-1, 1). The basic Downsian principle suggests that the voter will prefer the closer of the two candidates, C1, who is a little more than two units from the voter, whereas C2 is 4 units away (see Figure 2.1). For two dimensions, we may generalize this situation by denoting the voter’s spatial location by the ordered pair V = (v1, v2), where v1 is the voter’s position on the first (economic-policy) scale and v2 is the voter’s position on the second (social-policy) scale. Thus, V is a vector that can be interpreted either as a point whose coordinates are v1 and v2 or as an arrow emanating from the origin and ending at this point. Similarly, the spatial location of a typical candidate, C, is denoted by the vector C = (c1, c2). The (Euclidean) distance between V and C is given (according to the Pythagorean rule) as 2

(v1 - c1 ) + (v2 - c2 )

2

For example, if the candidate is C1 in Figure 2.1, the distance between V and C is (3 - 2) 2 + (1 - (-1)) 2 = 5 = 2.24 . In general, the Downsian proximity model (the traditional spatial model) specifies that utility – i.e., a voter’s quantitative evaluation of a candidate – is a declining function of policy distance from voter to candidate. Thus, a voter’s utility is greatest for a candidate holding identical positions with the voter on all issues and drops off as the

Alternative Models of Issue Voting

21

Figure 2.1 Downsian spatial model.

candidate’s spatial position recedes from that of the voter. The voter is assumed to prefer candidates for whom he has higher utility. As do many other researchers, we use the quadratic proximity utility function, under which utility declines with the square of the distance between voter and candidate. This function is defined by 2

[

U (V, C) = - (v1 - c 2 ) + . . . + (vn - cn )

2

]

or more compactly as n

2

U (V, C) = -Â (vi - ci ) = - V - C

2

i =1

where V = (v1, . . . , vn) and C = (c1, . . . , cn) denote the vectors of voter and candidate locations in n-dimensional issue space, vi and ci are the voter and candidate locations on the ith issue dimension, i = 1, . . . , n, and U(V, C) denotes the utility of voter V for candidate C.1 Note that a negative sign is used in defining this utility function, so that utility declines with distance. 1

For any vector X, ||X|| denotes Euclidean length or distance, i.e., X =

n

Âx i =1

2 i

. Thus,

||V - C|| is the Euclidean distance between V and C; ||V - C||2 is squared Euclidean distance. Under a linear proximity function, utility declines in proportion to the distance itself, i.e., U(V, C) = -||V - C||. See Appendix 4.2 for a comparison of these metrics.

22

Part I Models of Voter Behavior

2.1.2 The Grofman Discounting Model Grofman (1985) proposed two modifications of the standard proximity model: (1) accounting for the location of a status quo point and (2) explicit discounting of candidate positions. In fact, discounting of party or candidate claims not only is common sense but rests on Downs’s own words. Referring to a rational voter, Downs (1957: 39) writes: . . . [the voter] knows that no party will be able to do everything that it says it will do. Hence, he cannot merely compare platforms; instead he must estimate in his own mind what the parties would actually do were they in power.

Grofman argued that voters are likely to discount candidate claims and that – in judging candidate policies – voters are likely to use location relative to the voter’s perception of current policies, i.e., relative to the location of the status quo. Grofman focused on the magnitude of expected shifts from the status quo, after discounting.2 As we will see in Chapter 9, this property of the discounting model – for a twoparty system in which the parties have policy motivations – generates convergence of party positions to separate locations and predicts frequent alternation of electoral success. Discounting candidate claims, with the nature of the discount depending upon the location of the status quo, can explain how voter choice may change while both candidate and voter ideal points remain the same. For example, if the status quo shifts to the left, a moderately conservative voter may shift from support of a moderately liberal candidate to support of a strongly conservative one, as the expected (limited) achievement of the latter in shifting the status quo rightward leads to an outcome more in line with the voter’s own goals. Under the Grofman modification of the proximity model, spatial locations represent ideal positions in the Downsian sense. Grofman assumes that voters believe that a candidate will not actually move the status quo to the platform location (ideal point) proposed by the candidate but only part way in the direction of that platform point. If we assume for simplicity that all voters agree on a common discounting factor for each candidate, then they believe that the candi2

In Chapter 3 we shall show how Grofman’s notion of discounting, which was developed as a modification of the standard proximity model, can also be incorporated into a directional approach.

Alternative Models of Issue Voting

23

date will implement policies at a location intermediate between the status quo and the candidate’s ideal point, in proportion to the discounting factor. A voter’s utility for a candidate is the same as in the Downsian proximity model except that the discounted position of the candidate is used in place of her ideal position. We will refer to a proximity model with both a status quo point and discounting factors as the Grofman discounting model. 2.2 Directional Models There are two basic spatial models based on directionality that can be seen as alternatives to the standard (Downsian) proximity model. The first compares the direction of policy movement desired by a voter – from a status quo point or a policy neutral point – with the direction of policy movement espoused by a candidate. Direction of policy movement entails both desired or proposed policy changes on each of several issues as well as relative salience to the voter or candidate among these issues. The second accounts not only for direction but also for the overall intensity of the voter’s preferences on the issues as well as the overall intensity with which each contender advocates issue positions. Both types of directional models give rise to very different implications for the shapes of voters’ utility functions and for voters’ choice of candidates than does the proximity model. Both also can be shown to imply strategies on the part of candidates that may be centrifugal in nature (or at least not centripetal) and that may thus contrast strikingly with those implied by a pure proximity model.3 Directional models that provide alternatives to the more usual proximity approach go back at least to Carroll (1972) and Reynolds (1974). Carroll (see also Carroll, 1980; De Soete and Carroll, 1983) provided a utility function for psychological scaling, referring to it as the “vector” or “wandering vector” model and contrasting it with the proximity model – a model he referred to as the “Coombsian unfolding model” after Coombs (l964). De Soete and Carroll (l983) provided 3

As we shall see in Chapter 6, the effects of party ID may also contribute to the ideological dispersal of vote-maximizing parties. Likewise, incumbency advantage (Feld and Grofman, 1991) or an advantage on valence issues (Macdonald and Rabinowitz, 1998) may induce a less-advantaged opponent to move away from the position of the median voter.

24

Part I Models of Voter Behavior

applications of the wandering vector model (and other multidimensional spatial models) to a wide variety of data, including data on political party policy positions in Italy and Sweden.4 A second line of alternative modeling – also arising from the psychological literature – began with Fishbein (1963), who presented what he labeled the “ab” model. Fishbein argued that a person’s attitude toward an object is a function of his belief, b, about the object and his evaluation (own opinion), a, about the object. The model postulates that attitude is the product of belief and evaluation, i.e., ab. In general, a person has not one but several beliefs and evaluations about (different aspects of) an object, so that attitude is expressed as a sum of products of the form ab. These ideas suggested to Reynolds (1974) how to aggregate a voter’s beliefs about a candidate’s issue positions and the voter’s evaluation5 (own opinion or position) on the issues. For each issue, Reynolds placed both belief and evaluation on scales using positive and negative values. For example, +3 might represent a strongly conservative position and +1 a slightly conservative position, whereas -3 would represent a strongly liberal position, etc. Reynolds predicted voter attitudes toward candidates by summing, over issues, the product of the voter’s belief about a candidate’s position on each issue and the voter’s own evaluation on the issue. Using American National Election Study (NES) data, Reynolds obtained substantial correlations between attitude toward candidates and his predictive measure based on beliefs and evaluations. Independently, Rabinowitz (1973, 1978) and Rabinowitz and Macdonald (1989) developed a spatial theory of political competition between candidates or parties based on directional intensities of candidates and voters over dichotomous issues. Their utility function used a scalar product (or dot product) mathematically equivalent to the utility functions of Carroll and of Reynolds, but one in which the constituent factors represent intensities rather than beliefs and evaluations. Rabinowitz and his co-authors have written a series of papers comparing predictions of their directional model, both for voter utility 4

5

The reader is referred to Carroll and De Soete (l991) for a useful general survey of the mathematical psychology literature on multidimensional scaling. Later we shall use “evaluation” in a different sense, to refer to a voter’s overall utility for a candidate.

Alternative Models of Issue Voting

25

and for the strategic behavior of candidates or parties, with those of the proximity model. We will discuss their findings in subsequent chapters. Matthews (l979) proposed that candidates be associated with directional vectors from a neutral or status quo point representing the state of current policies rather than with policy locations.6 His model assumes that voters choose among candidates in terms of direction of preferred movement alone rather than in terms of overall intensity (or policy distance). Matthews argued that voters may perceive candidates as only a marginal distance from the status quo because candidate mobility is restricted for political reasons or because voters may discount the capacity of a winning candidate to move the status quo fully in accord with campaign promises. Accordingly, the direction of movement may be more important than the distance of movement. Second, due to the imperfect flow of information, candidates may succeed only in conveying to voters the directions in which they would shift the status quo rather than the exact positions to which they might move. Candidates may also be uncertain of voter response and, hence, move conservatively within the familiar neighborhood of the status quo, giving signals of direction rather than position. 2.2.1 The Matthews Directional Model Like the Grofman discounting model, the Matthews directional model postulates a neutral or status quo point. According to our interpretation of the model, voters and candidates are positioned at ideal points in space as in the proximity model, but utility reflects only the direction and not the overall intensity of voter and candidate positions. With this in mind, the utility function of the Matthews directional model depends only on the angle between the voter and candidate location vectors that emanate from the neutral point.7 This utility is a declining function of that angle, varying between +1 when voter and candidate agree completely in direction (and the angle is 0 6

7

For related ideas, see also Plott (1967), Sloss (1973), Schofield (1978), and Coughlin and Nitzan (1981b). Note that – in our rendition of the Matthews model – by assigning to each voter and candidate a location anywhere in policy space, we are modifying Matthews’s original idea that the directions of preferred movement by voters and candidates be restricted to points on the circle of radius 1 centered at the neutral point.

26

Part I Models of Voter Behavior

degrees) to -1 for complete disagreement (when the angle is 180 degrees). Taking the neutral point to be the origin, 0, we define the Matthews utility function as the cosine of the angle between voter and candidate vectors (relative to the neutral point), or equivalently by the formula U (V, C) =

V ◊C = cos q V C

n

where V ◊C = Â vi ci = v1c1 + . . . + vn cn is the scalar or dot product of i =1

the vectors V and C representing voter and candidate, respectively, and q is the angle between V and C. ||V|| and ||C|| are the Euclidean lengths of the vectors V and C, respectively, i.e., the distances from the origin to the points represented by V and C.8 In a one-dimensional model, there are only two directions (right and left). With the exception of degenerate cases, the Matthews utility is either +1 or -1 according to whether the voter agrees or does not agree with the candidate on the single issue. In a twocandidate race, the candidate with a majority on her side of the neutral point wins; if both are on the same side, they tie. The intensity with which the position is taken by either voter or candidate makes no difference. For two or more dimensions, the relative intensity with which a voter (or candidate) takes positions on different issues does matter (because, as we will see, it affects the angle), although the overall or across-the-board intensity does not. We illustrate this distinction for the case of two dimensions. Suppose there are two issues, where, say, the horizontal dimension represents economic policy and the vertical dimension represents social policy. Consider a voter, V, and a candidate, C, who agree on one issue (say, economic policy), but not on the other (social policy). First, suppose that both voter and candidate have strong, conservative positions on economics, at 4 on the scale with 0 as the neutral point. The voter is moderately conservative on social policy, located at +1, but the candidate is moderately liberal, located at -1 (see Figure 2.2a). The voter and candidate agree on the issue that both feel strongest 8

If either V or C is 0, the utility is defined to be 0.

Alternative Models of Issue Voting

27

Figure 2.2 Utilities for various configurations for which voter, V, and candidate, C, agree on the horizontal dimension (economic policy) but disagree on the vertical dimension (social policy).

about and the Matthews utility reflects this by assigning a high value of 0.88 (the cosine of the relatively small angle of 28 degrees between the voter and candidate vectors). Note that if the voter (or the candidate) changed his intensity on both issues simultaneously by a factor of two (or any other factor), Matthews utility would be unchanged. This occurs because such a change affects only the length of the voter

28

Part I Models of Voter Behavior

Figure 2.2 (cont.)

(or candidate) vector, not its direction. Hence, the angle would be unchanged and overall intensity has no effect. Now suppose the candidate moderates on economic policy, to location +1 (Figure 2.2b), while other positions are unchanged. The angle between voter and candidate widens and Matthews utility drops to 0.51, reflecting the lack of agreement in intensity between voter and candidate on economic policy, although both still favor conservative policy. Next, suppose the candidate moves to a strongly liberal position (-4) on social policy, so that the candidate is now at location (1, -4) (Figure 2.2c). The angle between the vectors is now 90 degrees and Matthews utility drops further to 0.00, reflecting the lack of agreement in intensity on both issues. Other relationships give the same Matthews utility; see, for example, Figure 2.2d. If we now move the voter to a more conservative position on social policy – to location (1, 4) – Matthews utility becomes negative with a

Alternative Models of Issue Voting

29

value of -0.51 (see Figure 2.2e), reflecting the fact that the voter now disagrees with the candidate on the issue he feels strongest about (the same would happen if it were the candidate who took an extreme position on social policy). Finally, if both voter and candidate take strong but opposite positions on social policy (see Figure 2.2f), Matthews utility drops to -0.88, reflecting disagreement on the issue on which both have strong feelings. Note that in all six configurations, voter and candidate agree on economic policy but disagree on social policy. The widely varying values of Matthews utility that obtain are a measure of the richness of this model and its capacity to account for the relative intensities with which a voter (or candidate) holds positions on different issues. It is remarkable that this occurs even though utility is not affected by across-the-board increases in intensity.9 2.2.2 The Rabinowitz–Macdonald Directional Model The directional model of electoral competition proposed by Rabinowitz and Macdonald (1989) was termed by the originators as the directional spatial model. We shall call it the RM directional model or simply the RM model to distinguish it from the directional model of Matthews. This model postulates that voter utilities are determined by both intensity and communality of direction of their own and candidates’ positions. The RM model stands in contrast to the traditional proximity spatial model, in which voter utilities decrease monotonically with distance from a candidate’s position, and the Matthews model in which voter utility depends on direction alone and not overall intensity. In fact, as we shall see in Chapter 3, the RM utility defines a composite model, of which the Matthews directional function represents the pure directional component. The RM directional theory assumes that most voters have a diffuse preference for a certain direction on an issue but vary in the intensity with which they hold that preference, i.e., in the degree to which that 9

This feature is particularly helpful in empirical analysis based on survey research because – unlike the other utility functions we consider – computed values of the Matthews utility from such data are unaffected by voter-specific dilation or contraction of the issue scales.

30

Part I Models of Voter Behavior

Figure 2.3 Utility under the RM directional model.

issue is important to them (Rabinowitz and Macdonald, 1989). Likewise, candidates are assumed to vary in the intensity with which they advocate one or the other of two possible positions on each dichotomous issue. These assumptions are no different from those made in our interpretation of the Matthews, model, and – just as for the latter model – voter preference in the RM model depends on the interaction of voter and candidate intensities. For the RM model, however, utility not only reflects the relative salience of different issues to voters and candidates but also varies as the overall intensity of voter or candidate increases. Accordingly, under the RM model, the value on each issue scale (each scale is centered at zero) is interpreted as the intensity with which the voter or candidate holds one or the other position on a dichotomous issue. Thus, on a single issue, a voter or candidate is placed at a value on a scale from, say, -3 to +3, where the sign of this value represents direction while the absolute value represents intensity (see Figure 2.3). For example, a voter placed at -2 holds a view on the negative (leftist) side with intensity equal to 2 (where intensity is on a scale from 0 to 3). The neutral coordinate, 0, represents no interest in the issue. The voter’s utility for the candidate is assumed to be proportional to both the voter’s intensity of interest and concern about the issue and to the intensity with which the candidate presents the issue. Thus, the voter’s utility is the product of his coordinate and the candidate’s coordinate and is positive if the coordinates are of the same sign and negative otherwise. In the example above, suppose two candidates place themselves at 1 and 3, respectively, and that a voter is located at 2. Utilities under the RM model for these candidates would be 1 ¥ 2 = 2 and 3 ¥ 2 = 6, respectively. Thus, the second candidate, who is interpreted to be more intense on this issue, would be more attractive to the voter, even though both candidates are equidistant from the voter and both are in the same direction (from the

Alternative Models of Issue Voting

31

neutral point) as the voter. By contrast, for either the Matthews or proximity models, the voter would evaluate these two candidates equally. Under the RM model, the voter’s overall utility for a candidate is the sum of these products over issues, i.e., by definition, the scalar or dot product of the vectors representing the positions of the voter and the candidate, respectively. Thus the Rabinowitz–Macdonald (RM) utility function is defined by10 n

U (V, C) = V ◊C = Â vi ci i =1

where, again, V · C is the scalar or dot product of the vectors V and C representing the voter and candidate, respectively. In one dimension, the RM utility is simply the product of the candidate and voter locations, each computed relative to the neutral point. If candidate A is to the left of candidate B, all voters to the left of the neutral point will vote for A and all to the right of the neutral point will vote for B. Note that this occurs regardless of whether the candidates are on opposite sides or both are on the same side of the neutral point. Suppose, for example, that the neutral point represents indifference between liberal opinion on the left and conservative opinion on the right. In a race between two conservatives, the more extreme conservative will take all the conservative votes (i.e., from voters to the right of the neutral point) while the less extreme conservative takes all the liberal vote. If, in addition, a liberal candidate enters the race, the less extreme conservative will lose all her votes to the new candidate. In fact, the theory implies that the former candidate would not even vote for herself.11 Implications such as these suggest that the pure RM model may be unlikely to faithfully represent voter behavior. Yet, as we shall see in Chapter 3, the intensity and directional aspects highlighted in the RM model may play significant roles as components of a multicomponented model. 10

11

Thus, the RM utility function is the same as the Matthews utility, but not normalized. In contrast, the Matthews utility is normalized by the lengths of the voter and candidate vectors. In Chapter 10, we give a necessary and sufficient condition that a candidate not vote for herself under the RM model.

32

Part I Models of Voter Behavior

If, under the RM model, there are only two candidates – but with one on the left and one on the right – each takes the entire vote on his side of the neutral point and movement toward the extreme or toward the center (without crisscrossing) makes no difference in vote share. In this case there is no incentive to move toward the center. Indeed, in a race with three or more candidates, the interior candidates would have strong incentives to leapfrog more extreme opponents. This would in turn create new interior candidates, who would also leapfrog, causing instability and continued outward movement. These and other features of the RM model suggest that, standing alone, it will not do a good job in accounting for observed patterns of party behavior (in particular for moderate levels of divergence).12 2.3 Comparison of Models We can compare models in three ways: (1) by the shape of voter utility, (2) by voter choice, and (3) by predictions about party strategies (convergence or divergence). Here, in this section of the book, we will focus on differences in voter utility and in voter choice among the models. As can be seen in Figure 2.4 – which plots utility as a function of candidate position for a voter in fixed location – the proximity utility curve peaks for a candidate located at the voter position whereas the utility curve for the Rabinowitz–Macdonald (RM) model is unbounded, continuing to increase as the candidate position becomes more extreme (more intense) than that of the voter. The slope of the RM utility curve, furthermore, depends on the intensity of the voter – that of the proximity curve does not. The Matthews utility curve, on the other hand, is level except for a sharp jump at the neutral point, N. Perhaps the simplest way to characterize the effect of a particular utility function is to determine – in a case in which voters choose between two candidates, A and B – what geometrical regions correspond to (1) voters who would have higher utility for candidate A and hence would select A, (2) voters who would have higher utility for B and would select B, and (3) voters who have equal utility for each and hence would be indifferent about the choice. In a one-dimensional 12

In Chapter 3 we consider modifications of the RM model to address these problems.

Alternative Models of Issue Voting

33

Figure 2.4 Utility curves for pure models for voter in a fixed position.

model using the proximity utility, the midpoint between A and B divides the line into two half-lines, one consisting of voters who favor A and the other of voters favoring B. We refer to this critical point as the point of indifference; a voter at this point should be indifferent between the two candidates. For a two-dimensional proximity model, the set of voters with equal utility for candidates A and B lies on a line that is the perpendicular bisector of the line segment joining A and B (see Figure 2.5a).This line is called the line of indifference and it separates the plane into two halfplanes, each representing the region of support for one of the candidates. Under the Grofman discounting model, the line of indifference between two candidates is the perpendicular bisector of the line segment connecting the discounted positions of the candidates. For both the Matthews and RM directional models, the neutral point in a one-dimensional model is an indifference point.13 Under the 13

If both candidates are on the same side of the neutral point, both have equal utility under the Matthews utility for all voters so that, in fact, all points are indifference points.

34

Part I Models of Voter Behavior

a

b

c Figure 2.5 Indifference lines under alternative models. a. Proximity model: given two candidates, A and B, the indifference line is — perpendicular bisector of AB. b. Matthews directional model: The indifference line bisects angle ANB. — c. RM directional model: The indifference line is perpendicular to AB and passes through the neutral point, N.

RM utility, the more conservative candidate takes all the votes to the right of the neutral point while the more liberal candidate takes all to the left, regardless of whether both candidates are liberal or both are conservative.

Alternative Models of Issue Voting

35

Clearer differences between the RM and Matthews models emerge, however, in two (or more) dimensions when we compare the position of indifference lines. According to the Matthews utility, a voter making a choice between two candidates prefers that candidate whose angular distance from the voter is smaller and is indifferent between the candidates if the angular distance is the same. Accordingly, in two or more dimensions, the line of indifference between two candidates for the Matthews model passes through the neutral point and bisects the angle between the candidate vectors (see Figure 2.5b). For the RM model the indifference line between two candidates also passes through the neutral point but must be perpendicular to the line segment joining the candidates (see Figure 2.5c) rather than bisect the angle between them as in the Matthews model.14 Perspective can be gained on the proximity, Matthews, and RM models by noting that if all voters and candidates are located at points on the circle of unit radius centered at the neutral point, then utility under all three models is equivalent.15 Only insofar as proximity and RM utility functions take into account overall intensity do they differ from the Matthews utility and from each other. If voter and candidate are equidistant from the neutral point (but not at distance 1), both proximity and RM utility increase by a factor of ||V||2 relative to the Matthews utility. Hence, from the point of view of a single voter, utility distinctions between candidates are still equivalent for all three models. The models diverge when voter and candidates are not at the same distance from the neutral point. We have seen, for example, that a moderate voter ranks a moderate and an extreme candidate who are both on his side of the neutral point in opposite ways under the proximity and RM utilities.

14

15

In the Matthews model, the indifference line is perpendicular to the segment joining the normalized locations of the candidates, i.e., the locations after all candidates not at the neutral point are expanded or contracted so that their distances from the neutral point are unity. This segment is a chord of the unit circle and the indifference line bisects the angle at N. If ||V|| = ||C|| = 1, then -||V - C||2 = -(||V||2 + ||C||2 - 2||V|| ||C|| cos q) = 2(cos q - 1), so that proximity utility is a positive linear transformation of the Matthews utility, which in turn is equal to RM utility when V and C are on the unit circle, as can be seen by inspecting the latter two utility functions.

36

Part I Models of Voter Behavior

It is instructive to return to the configurations depicted in Figure 2.2 and compare the utilities assigned by the proximity and the two directional models and given under each subplot in the figure. All three models give high utility to the configuration in plot a and low utility in plot f. The Matthews and RM utilities agree in plots c and d because the angle is 90 degrees in each case. But the proximity model assigns widely varying utilities to these latter two situations, reflecting the greater lengths of the vectors in plot c. In general, if both voter and candidate vectors (relative to a neutral point) are dilated by some factor, proximity utility is multiplied by that factor squared; in other words, proximity utility is quite sensitive to acrossthe-board changes in scale. The RM utility is also affected the same way, as can be seen by changing the lengths of either the voter or candidate vectors in plot b or e. For both the proximity and RM models, it may seem difficult to justify such large sensitivities as voter and candidate take more or less extreme stands. On the other hand, it seems unwarranted to ignore them altogether, as does the Matthews utility. In Chapter 3 we develop a unified model of voter utility that brings together the features of the various separate models. In Chapter 4, we assess each of the three pure models – whose properties we have described in this chapter – in terms of their empirical fit to the shape of voter utility functions. The unified model is fit to data in Chapter 5. In Chapters 6 and 7 we extend the unified model to voter choice as opposed to voter utility and assess its empirical fit to several data sets. The unified model establishes a linkage between issue information, voting behavior, and candidate strategies that incorporates all the influences specified by the models described above. Much attention in the literature has been given to assessing the relative merits of pure models. We do not claim to have a definitive statement regarding whether one pure model is in some sense superior to another and seriously doubt whether a clear answer is attainable with the limitations of methodology and data gathering.16 But we believe that we demonstrate a more important empirical finding – that proximity, directional, 16

Lewis and King (1998) support this view, based on statistical issues of model identifiability. We will discuss their methodological criticisms in Appendix 7.1.

Alternative Models of Issue Voting

37

and intensity components ALL play significant roles in voter choice and hence in the positioning of candidates and parties. Moreover, we also show that these three determinants of voter utility and voter choice appear to vary in their importance over different candidates and voters, with preferences for incumbents (as known commodities) best predicted by the proximity and Matthews models.

CHAPTER 3

A Unified Model of Issue Voting: Proximity, Direction, and Intensity . . . spatial [i.e., proximity] and directional theories may not be incompatible but instead may complement one another in explaining patterns of voting behavior. Torben Iversen, “Political Leadership and Representation in West European Democracies” (1994: 49)

3.1 Limitations of Pure Models In Chapter 2, we introduced two pure models of voting behavior: the Downsian proximity model and the Matthews directional model. We saw that the former could be modified by discounting (Grofman), the latter by taking overall voter or candidate intensity into account (Rabinowitz and Macdonald). Even with these emendations, each of these models singles out one (or in some cases two) aspects of voter decision making. We expect each to be of limited usefulness by itself in explaining voter behavior, and empirical studies bear out this expectation (see Chapters 4–7). In this chapter, we specify a unified model intended to incorporate the features of all the various pure models. In order that utility not increase without bound as candidates recede from the neutral point, Rabinowitz and Macdonald (1989) incorporate into their model a penalty for extremism, on the grounds that voters will tend to find unacceptable a candidate or party whose stands are too “far-out.” To this end, they define a circle or region of acceptability, centered at the neutral point, beyond which a candidate suffers a loss of utility for all voters.1 Although Rabinowitz and Macdonald consider the “region of acceptability” 1

In three dimensions, this circle is replaced by a sphere, and in general, by a hypersphere. For simplicity, we retain two-dimensional terminology, referring to circles and lines.

38

A Unified Model of Issue Voting

39

to be part of the definition of their directional model, they provide no decision rule for specifying which parties fall into the region. We agree that there may well be parties/candidates whose locations are treated as so extreme that distances calculated to them are not meaningful. For example, Damgaard (1969) notes that in multiparty systems it has often been the case that communist parties or right-wing monarchical parties are considered uncoalitionsfähig, i.e., so extreme as to be unacceptable as possible coalition partners for the other parties. Nevertheless, the notion of a single circle of acceptability appears inadequate as it ignores the fact that assessment of extremeness depends heavily on the voter’s own position. Iversen (1994) argues that the constraint implicit in Rabinowitz and Macdonald’s idea of a circle of acceptability can better be modeled by a function idiosyncratic to each voter. Certainly, the perception of unacceptability due to extremism is not the same for each voter. For example, in American politics, a strong conservative may consider a liberal Democrat unacceptable, but certainly a liberal Democrat would not. At the same time, the sharp edge of Rabinowitz and Macdonald’s region of acceptability is hard to justify, and the radius of the circle would need to be estimated from the data to achieve a testable model. Iversen suggests subtracting from each voter’s RM utility function a quantity that grows slowly while the candidate is near the fixed voter but more rapidly as the candidate recedes. Such a quantity can be specified by a quadratic function of the distance between voter and candidate. This yields what we term the RM model with proximity constraint. The quadratic utility function permits the directional effects to dominate over short distances whereas the proximity effects dominate over greater distances where issue differences may be perceived as extreme. Iversen (1994) argues that the directional or scalar product component of utility (which is proportional to candidate intensity) reflects policy leadership, whereas the proximity term (which drops off with the candidate’s distance from the voter’s ideal point) reflects a candidate’s or party’s representational role. Mathematically the RM model with proximity constraint constitutes a linear mixture of the RM directional model and the (quadratic)

40

Part I Models of Voter Behavior

proximity model.2 In fact, such a mixed model could also be given a more direct rationale. It seems reasonable to believe that the utilities of voters may be influenced both by similarity between their spatial position and that of a candidate and by the intensities with which they hold their respective positions. This is particularly true for the type of survey data used to test alternative models. Voter response, such as that elicited from questions in the American NES surveys, may reflect either or both of these influences. A typical question, such as item number 746 in the 1984 American NES Codebook, is phrased in a way that blurs the line between these motivations: Some people feel the government in Washington should see to it that every person has a job and a good standard of living. Others think the government should just let each person get ahead on his own. Where would you place yourself on this scale, or haven’t you thought much about this?

A voter’s reported position on this issue may be either an attempt for the voter to locate himself on a policy axis of governmental intervention or a reflection of the salience of the issue, or both. Thus a centrist placement may reflect preference for moderate government intervention, lack of interest in the issue, or some combination of these attitudes (Poole and Rosenthal, 1984a). An extreme position represents preference for a policy at one end of the policy spectrum, but may also represent strong interest in the issue. Thus, according to the information gathered in surveys such as the American National Election Studies, a voter’s utility for a party may reflect a mixture of these two types of information, that is, a mixture of the proximity and directional utility functions. Alternatively, a mixed model, if interpreted as incorporation of a proximity component into a directional model, builds into the latter model a penalty for extremism (or excessive intensity), rather than imposing it via an ad hoc region of acceptability. 3.2 The Unified Model The models considered so far have a number of implausible implications, e.g., the RM model implies an unwarranted advantage for 2

Although points in these spatial models represent different entities conceptually – an ideal policy point in a proximity model and a combination of intensity and direction in the RM model – their mathematical representation is the same so that both pure models may be embedded in the same space mathematically.

A Unified Model of Issue Voting

41

extreme candidates and has the peculiar feature that sometimes candidates may not vote for themselves (see Chapter 10); the onedimensional Matthews model implies indeterminacy when both candidates lie on the same side of the neutral point. Similarly, the most basic proximity model does not appear to predict the prevalence of divergence of candidate positions observed in real campaign strategies3 (see, e.g., Bullock and Brady, l983; Poole and Rosenthal, l984b; Grofman, Griffin, and Glazer, l990), whereas the RM model appears to overpredict it. These inadequacies of the various pure models provide additional motivation for the development of a more general model. We have indicated that it is reasonable to combine directional, intensity, and proximity ideas in a single model because all may influence voter behavior. In addition, voters may use proximity to evaluate some candidates (e.g., incumbents whose locations can be reasonably well identified in terms of their previous policies), but use direction of likely change to evaluate other candidates whose policy positions may not be so precisely identifiable (e.g., challengers without a track record). It is important to recognize that mixing directional and proximity ideas is not mixing apples and oranges, since these pure models can be shown to be arrayable on a single continuum, and thus distinguishable by a single parameter. To draw together these perspectives, Merrill and Grofman (1997a) introduce a unified model specified by two parameters. The first defines a proximity versus directionality continuum anchored by the pure proximity (Downsian) model at one end and a directional–intensity (RM or Matthews) model at the other. The second parameter defines an intensity continuum anchored by the Matthews model at one end and the RM model at the other. We describe the intensity component first. The RM utility is the product of a pure directional utility (the Matthews utility) and a pure intensity factor (the product of the lengths of the location vectors).4 Both the RM and Matthews directional models can be embedded in a common one-parameter model whose utility function is the product 3

4

R. Potthoff (personal communication) has suggested, however, that candidates might assume divergent strategies in an effort to move the median voter position rather than to adopt it. Also there are institutional features, such as primaries, that can help account for party divergence (Aranson and Ordeshook, 1972; Coleman, 1972; Owen and Grofman, 1995). See also Chapters 9 and 10. Denoting the voter’s spatial location by the vector V and the candidate’s location by

42

Part I Models of Voter Behavior

of the directional Matthews utility and an intensity factor determined by an intensity parameter, q, which varies between 0 and 1. The resulting utility function will be referred to as the damped directional utility function.5 The first factor is directional whereas the second reflects only intensity. If q = 0, overall intensity has no bearing on utility and the damped model is the Matthews directional model. If q = 1, intensity is fully combined with direction and the RM model is defined. Relative to the RM model, the damped directional utility – for q strictly between zero and one – dampens the increase in the utility of candidates as they become more extreme. To get a better sense of the meaning of intermediate values of q, note that under the Matthews model the voter applies the scalar product utility as if all candidates were located on the unit circle, one unit from the neutral point. Under the damped directional model with intermediate values of q, the voter treats candidates as if they were — placed at ||C||q, e.g., Î||C||, if q = 1/2. Suppose there are two candidates in a two-dimensional spatial model, the first at (4, 0) and the second at (0, 1) (see Figure 3.1). Under the Matthews utility, the first candidate can be normalized to (1, 0) and the indifference line between them is at 45 degrees above the horizontal; under the RM utility, this line is at 76.0 degrees. Voters between these two lines have opposite preferences under the two models, reflecting the voters’ reactions to the first candidate’s intensity. For an intermediate value of q, say, q = 1/2, the first candidate is treated as being located at (2, 0) and the indifference line is at 63.4 degrees, approximately halfway between the indifference lines for the pure models. Accordingly, as a rough approximation, we can interpret the vector C, the RM utility function can be expressed as a product of the Matthews utility function and a pure intensity component as follows: Ê V◊C U ( V, C) = V ◊ C = Á Ë V C 5

ˆ ˜( V C ) ¯

In a one-dimensional model, V and B are scalars. The damped directional utility function is defined by Ê V◊C ˆ q q -1 U ( V, C) = Á ˜ ([ V C ] ) = V ◊ C = [ V C ] Ë V C ¯ if C π 0 and V π 0, and 0 otherwise, where the exponent q is the intensity parameter.

A Unified Model of Issue Voting

43

Figure 3.1 Indifference lines under directional models.

q as a measure of distance between the models; values of q substantially below 0.5 suggest a model more like that of Matthews; those well above 0.5 suggest utilities more like those of Rabinowitz and Macdonald. We next turn to an explication of the proximity versus directional factor. As we have seen, Iversen (1994) suggests idiosyncratically adjusting the scalar product that defines the RM utility function by subtracting a quantity proportional to the square of the distance between voter and candidate, i.e., by subtracting the (quadratic) proximity utility. This yields the RM model with proximity constraint or mixed proximity–RM model, which Iversen refers to as the representation policy leadership model. The RM model with proximity constraint is defined by the utility function6 U (V, C) = 2(1 - b)V ◊ C - b V - C

2

The model is specified by the value of the proportionality constant, b, which indicates the strength of the constraint. Note that if b = 0, there 6

n

As before, V ◊ C = Â vi ci denotes the scalar product of the vectors V and C; ||V - C||2 i =1

denotes the square of the Euclidean distance between V and C.

44

Part I Models of Voter Behavior

is no constraint, i.e., the proximity component of the voter utility function drops out. By simultaneously multiplying the directional component by the value 2(1 - b), we anchor the other end of the proximity–directional scale with the value, b = 1, i.e., when b = 1, the composite model becomes the pure proximity model. Mathematically, this composite model is equivalent to the mixed RM and proximity model introduced by Rabinowitz and Macdonald (1989) with mixing parameter b. As we have seen, under the proximity model, for a fixed voter, utility peaks for a candidate at the ideal point of the voter. By contrast, under the RM model, utility increases (or decreases) without bound as the candidate recedes from the neutral point, i.e., increases in issue intensity. The shape of the utility curve for an RM model with proximity constraint is intermediate, peaking at a point more remote from the neutral point than that for a pure proximity model. Because, for a fixed voter and for the status quo at the neutral point, utilities under the RM model with proximity constraint and the Grofman discounting model differ only by a constant (see Section 3.3), peaking is the same for the two models. The similarity of utilities between the Grofman discounting model and the RM model with proximity constraint, each contrasted with the plots for pure proximity and pure RM utilities, is illustrated in Figure 3.2, where voter utility is plotted as a function of spatial position of the candidate. For a fixed voter position, utility under the directional model with proximity constraint is maximized for a candidate in the same direction as the voter but more extreme by the factor 1/b. Thus, this model accounts for candidate behavior more extreme than that of proximity but less than that of the RM model.7 We summarize the interpretation of the parameter b in the RM model with proximity constraint. When b = 0 we have the pure RM directional model; when b = 1 we have the pure Downsian proximity model. For any specified model, the indifference line consists of those points representing voters who are indifferent between the two can7

Iversen (1994) found, for example, that – for a large number of European parties – the mean position of a party’s officials (assumed to represent true party position) tends to be about twice as extreme as the mean for the party’s supporters. This would suggest that the factor 1/b equals 2, i.e., that b is in the vicinity of 0.5, if voters, on the average, choose parties of maximum utility. We will elaborate on this and other empirical questions in Chapter 5.

A Unified Model of Issue Voting

45

Figure 3.2 Comparison of utility curves for Grofman discounting model and mixed proximity–RM model for voter in a fixed position.

didates. For a mixed proximity–RM model with mixing parameter b between 0 and 1, the indifference line is intermediate between the locations of the (parallel) indifference lines of the two pure models in proportion to the value of b. Thus, values of b > 1/2 imply that voter choice is closer to the proximity model than to the RM directional model; values of b < 1/2 imply that it is closer to the directional model. Pulling together these two strands of generalization, we have a twoparameter unified model defined by a utility function that incorporates both the damped directional model and the constraint of the proximity model. The unified model is defined by the utility function8 U (V, C) = 2(1 - b)

V◊C V C

q

[ V C ] -b V-C

2

where b is a mixing parameter between directional and proximity components, and q is an intensity parameter. If q = 1, the unified model 8

Here, again, V · C denotes the scalar product of the vectors V and C.

46

Part I Models of Voter Behavior

Table 3.1. Summary of pure models as special cases of the unified model and specification of utility functions Model

b

q

d

U(V, C)

Pure proximity (Downs)

1

NA

1

- V - C = -Â (vi - ci )

Pure direction (Matthews)

0

0

NA

Direction plus intensity (RM)

0

1

NA

Proximity plus discounting (Grofman)

1

n

2

2

i =1

V◊C = cos q V C n

Unified model Basic version: Discounting version:

V ◊ C = Â vi ci i =1

NA

2

n

d<1

- V - dC = - Â (vi - dci )

2

i =1

a

a

NA

2( 1 - b )

V◊ C q [ V C ] -b V -C V C

a

a

d<1

2( 1 - b )

V◊ C q [ V C ] - b V - dC V C

2

2

a

Not restricted. Source: Adapted from Merrill and Grofman (1997a: Table 1).

is equivalent to the RM model with proximity constraint. If, furthermore, b = 0, the model defines the RM directional model. The combination of b = 0 and q = 0 specifies the Matthews directional model. If b = 1, a pure proximity model is defined. A pure intensity model, without a directional component, makes no sense behaviorally and is not a special case of the unified model. Finally, if the status quo point is the same as the neutral point, we might incorporate the Grofman discounting model into a threeparameter unified model by discounting the candidate position (i.e., multiplying it by a factor d) in the proximity term of the unified model.9 Because the parameters b and d are not independent, 9

The three-parameter unified model is obtained by replacing the proximity term ||V - C||2 in the two-parameter unified model by ||V - dC||2, where d is the common discounting factor for both candidates, i.e., U ( V, C) = 2(1 - b )

V◊C q [ V C ] - b V - dC V C

2

A Unified Model of Issue Voting

47

however, we will focus on the two-parameter version of the unified model and incorporate the Grofman discounting model into our unified framework in a different way (see Section 3.3). To better understand the relationships among the three parameters b, q, and d and the various models, the chart in Table 3.1 may be useful. Thus, voter choice or voter utility can be modeled as representing proximity alone, direction and intensity alone, direction with proximity constraint, or, alternatively, in terms of proximity to discounted positions. The effect of each of these factors on voter utility can be modeled as special cases of a single utility function. In Section 3.3 we provide an alternative unified perspective for these seemingly disparate models in terms of what we call “shadow” positions. We demonstrate that voter choice in a variety of spatial models including directional components can be viewed as proximitybased choices in which the candidate locations are replaced by shadows, where shadow locations are defined by a simple transformation. Thus voters choose that candidate whose shadow is nearer. In particular, we will show that voter choice is identical for the Grofman discounting model (with common discounting factor d) on the one hand and the RM model with proximity constraint (with mixing parameter b) on the other, provided that b = d and the status quo is located at the neutral point. This approach, based on the work of Merrill and Grofman (1998a), unifies our understanding of what otherwise appear to be disparate spatial models. 3.3 Relation between the Grofman Discounting Model and the RM Model with Proximity Constraint The pure proximity model and the pure scalar product model give rise to quite different expectations as to the nature of voter utility functions. Nevertheless, there are areas of significant coincidence between the utility functions of mixed directional–proximity models and the Grofman discounted version of the proximity model. Suppose that the status quo point of the Grofman model and the neutral point of a mixed proximity–RM (directional with proximity constraint) model are the same. Assume further that each candidate has the same discount factor, d, and that the mixing parameter of the mixed model is denoted by b.

48

Part I Models of Voter Behavior

For a fixed voter, if b is equal to d, the utility functions for the two models are equivalent in the sense that one is a (positive) linear transformation of the other.10 Thus, utility for the Grofman discounting model can be obtained from that for the mixed proximity–RM model by multiplying by a positive constant (b) and then subtracting a constant. In particular, any voter indifferent between two candidates under one model will likewise be indifferent under the other. This observation can also be seen geometrically, as illustrated in two dimensions in Figure 3.3, where with a suitable change of coordinates we take the status quo point to be the origin, so that the discounted position of A is given by A9 = dA. For two candidates, A and B, the indifference line in the undiscounted Downsian model is the perpendicular bisector of the segment joining A and B. In the discounted version, it is the perpendicular bisector of the segment joining A9 and B9 (see Figure 3.3). By elementary geometry, these indifference lines are parallel. As d moves from 0 to 1, the discounted indifference line moves proportionately from the origin to the perpendicular bisector of the segment joining A and B. This indifference line is identical to that for the RM model with proximity constraint with mixing parameter b = d, because both are perpendicular to the segment joining A and B and located at a distance proportional to b (= d) from the neutral point to the bisector of this segment. This follows because – under the RM model with proximity constraint – a voter V is indifferent between candidates A and B if V lies on the perpendicular bisector of the segment connecting A9 and B9, where A9 = bA and B9 = bB (see Figure 3.3).11 In other 10

To see this, denote by UPRM the utility under the mixed proximity–RM model and by UG utility under the Grofman discounting model. Then: 2

UG = - V - bC = -( V - bC) ◊ ( V - bC) = - V ◊ V + 2 bV ◊C - b 2C◊C 2

2

2

- V + b 2 V + b[-b V + 2 V ◊C - b C = bU PRM - (1 - b ) V

11

2

]

2

Equality of the expression in brackets and UPRM follows after expanding the squared distance as a scalar product in the defining equation for the RM model with proximity constraint. Voter V is indifferent between candidates A and B if and only if È V - b (A + B) ˘ ◊ [A - B] = 0 ÎÍ 2 ˚˙

A Unified Model of Issue Voting

49

Figure 3.3 Equivalence of Grofman discounting model and mixed proximity–RM model. (Sources: Adapted from Merrill and Grofman, 1998a: Figure 1.)

words, voter choice is as if the candidate positions were shrunken (or dilated) to their “shadows,” A9 and B9, and voters behaved as under a proximity model, selecting not the nearer candidate but the candidate whose shadow is nearer. Thus, if there is a single discount factor, both voter utility and voter choice for the Grofman variant of the proximity model are equivalent to that for a mix of the Downsian proximity model and the RM model.12 We summarize in the following proposition. Proposition 3.1. Indifference curves are identical and utility functions for a fixed voter are equivalent between the Grofman

12

i.e., if V lies on the hyperplane perpendicular to A - B passing through b (A + B) , 2 or equivalently, if V lies on the perpendicular bisector of the segment connecting A9 and B9. This general point is noted in passing in Rabinowitz, Macdonald, and Listhaug (l993) at footnote 3, but without any formal mathematical development of the exact link.

50

Part I Models of Voter Behavior

discounting model (with common discounting factor d) and the directional model with proximity constraint (with mixing parameter b) provided that b = d and the status quo is located at the neutral point. Thus, under the assumptions given, these two models are indistinguishable on the basis of voter choice or on the basis of utility curves for a fixed voter. If the status quo point is the same as the neutral point, N, we may illustrate (see Figure 3.3) the indifference lines defined by the proximity model (the perpendicular bisector of the line connecting A and B); the Matthews directional model (the bisector of the angle, ANB); the RM directional model (the perpendicular through N to the line connecting A and B); and the Grofman discounting model with d = 0.5 (the line halfway between the lines for the proximity and RM models and parallel to each). The last-named line is identical to the indifference line for the mixed proximity–RM model with b = 0.5. As can be seen in Figure 3.3, voters with ideal points that lie in the angles defined by the RM and Matthews indifference lines are predicted to vote differently by the two models. Voters between the parallel indifference lines for the RM, Grofman discounting, and proximity models vote differently between one or more pairs of these models. 3.4 Conclusions We have shown how to embed four representations of voter utility – the Rabinowitz–Macdonald directional–intensity model, the Matthews pure directional model, the standard Downsian proximity model, and the Grofman discounting proximity model – into a common geometric framework. Two or three parameters capture the differences among the various models and allow us to specify a very general class of hybrid models. Empirically, the unified approach will permit us to test the full set of models – both pure and hybrid forms – simultaneously rather than piecemeal. This advantage is analogous to that offered by analysis of variance, in which one tests several variables simultaneously, with each serving as a control for the others, as opposed to the piecemeal approach of applying separate t-tests. Of

A Unified Model of Issue Voting

51

course, for some comparisons, statistical tests are inadequate to distinguish alternative models, e.g., between the Grofman discounting model and the constrained RM model. In the process of developing our unified model, we have separated the directional and intensity components of the Rabinowitz– Macdonald directional model, with the Matthews model best thought of as embodying a pure directional utility. The modeling of voting behavior as reflecting some combination of proximity, directionality, and intensity effects, as well as the incorporation of discounting into a unified spatial framework, offers, we believe, a significant theoretical synthesis of the existing literature on formal models of voting. We have also seen that, under a number of apparently disparate spatial models, voter choice – and to some extent voter utility – can be viewed as selection according to proximity to shadow candidates. These shadow positions are obtained by a simple projection of the actual or declared positions of the candidates.Analytically, the shadow positions can be obtained by simple multiplication by a constant (i.e., the projection is a pure contraction or dilation). For the RM model with proximity constraint, this constant is just the mixing parameter, b; for the Grofman discounting model with status quo at the neutral point, the constant is the common discounting factor, d. In each case, the multiplication constant is normally less than one, so that the transformed positions represent a contraction. The indifference line between the two candidates is the perpendicular bisector of the segment joining the transformed or shadow candidates. Thus it typically lies in a position between that for a pure proximity model and a pure RM model.

CHAPTER 4

Comparing the Empirical Fit of the Directional and Proximity Models for Voter Utility Functions All politics . . . are based on the indifference of the majority. James Reston, The New York Times (June 12, 1968)

4.1 Discriminating between Models We may assess a voter’s evaluation of a candidate through either (1) voter choice, i.e., whom the voter votes for, or (2) voter utility, i.e., the voter’s quantitative evaluation of the candidate on some scale. Whom one votes for may be the ultimate question, but utility – which relates to enthusiasm for (or against) a candidate – has important secondary effects. For example, if the value attached to getting one’s preferred candidate elected is high enough, the citizen may be motivated not merely to vote but also to participate in politics as an opinion leader, activist, or financial contributor. Moreover, such participatory and contributory activities on the part of one voter may well influence how others vote. Discrimination between competing models of voter choice may appear more difficult than discriminating among competing models of voter utility, because focusing only on the either–or decision of voter choice makes use of only some of the information available to us about the shape of voter preferences. On the other hand, as we shall see in Chapters 6 and 7, statistical testing of voter choice models may avoid certain methodological difficulties that plague attempts to test models of voter utility. Models that make clearly different predictions about the shape of voter utility functions may have much greater predictive overlap in terms of voter choice (especially for only two alternatives); however, as we will see in Chapter 10, different models may make very different predictions about the spatial configurations of candidates or parties as they respond to voter evaluations. 52

Comparing the Empirical Fit

53

In this and the following chapter, we will focus on voter utility. In the present chapter, we limit our model assessment to pure models of voter utility. In Chapter 5 we will fit the unified model. In Chapters 6 and 7, we will extend our unified model to voter choice (as opposed to voter utility) and incorporate probabilistic and partisan components.1 In this chapter, we will look at data from national election studies in the United States, Belgium, Britain, France, Germany, Norway, and Sweden to test the competing predictions of proximity and directional models. 4.2 Utility Curves The analysis of utility curves and the fitting of models can be operationalized based on respondent self-placement and candidate placement on issue scales and by thermometer feeling scores (interpreted as utilities). For example, in recent American National Election Studies (NES), respondents in a national random sample were asked to provide self-placement and placements of candidates on seven-point scales for a variety of issues. We will look at data from NES studies for five presidential election years from 1980 to 1996.2 The data from these scales can be given a spatial interpretation, with each issue corresponding to one dimension in a multidimensional issue space. Each 1–7 scale has been centered at 0 by subtracting 4, so that each ranges from -3 on the left to +3 on the right. The methodology used in employing these scales is discussed in detail in Appendix 4.1. Similar scales are used in the national election studies discussed below for six European electorates. Descriptions of these studies will be introduced as needed. Ideally the contrasting predictions of the proximity and directional models can best be seen by plotting – for a fixed voter – voter utility as a function of spatial positions of the candidates in a one1 2

In Chapter 10 we compare models in terms of their predictions about party strategy. Eight issues were available in 1980, 1984, and in 1988, five in 1992, and eleven in 1996. Responses were obtained on seven-point scales in all five NES studies for liberal-toconservative placement (L/C), government services, defense spending, and government responsibility for jobs. Responses were also obtained in one or more of these studies for U.S. attitude toward Russia, government effort for blacks (called minority aid in 1980), inflation/unemployment, women’s role/rights, government effort for women, U.S. involvement in Central America, national health care, abortion, reduction of crime, jobs versus the environment, and environmental regulation.

54

Part I Models of Voter Behavior

dimensional model (as in Figure 2.4). Thus we might hope to distinguish between the proximity and directional models based on the shapes of the utility curves.3 We will first, however, consider mean candidate thermometer scores as the location of the voter changes. Although fraught with all of the problems caused by making use of interpersonal comparisons, this method will nonetheless give us some suggestion of the shape of utility curves and will be easier to interpret because of limited information available in the American NES about candidate locations. For example, Figure 4.1 provides plots for six real or potential candidates surveyed in the 1984 NES (Jesse Jackson, Edward Kennedy, Walter Mondale, Gary Hart, Gerald Ford, and Ronald Reagan), giving mean thermometer scores versus the voter’s self-placement on the liberal–conservative (L/C) scale. Median placements (over respondents) for the L/C scale are available only for the two nominees and are indicated on the horizontal axis: Mondale (-1.26) and Reagan (1.90).4 The utility curves for the two nominees continue to increase beyond these median candidate positions, as would be predicted by the RM model. On the other hand, neither maintains the same slope, which is a violation of what would be predicted by the RM model; in fact, the curve for Mondale peaks at -2 on the L/C scale. Although, unfortunately, as noted previously, median voter placements in the NES are available only for Reagan and Mondale, peak utilities for Hart and Ford at moderate locations (the utility curve for Hart peaks at -2 and -1, whereas that for Ford is essentially level between 1 and 3) fit their moderate images – though both curves probably peak beyond where their median placements might have been. Similarly, the steady increase (to the left) for Kennedy and Jackson fit their more extreme images. Kennedy’s curve does not level off until -2 to -3, whereas the curve for Jackson appears to continue to increase to the limit of the scale. The facts that the peaks of candidate 3

4

As we saw in Chapter 3, for a mixed proximity–RM model, utility peaks at a point more distant than the voter’s ideal point. For the Grofman discounting model with status quo at the neutral point and common discount factor, the location of the peak point is likewise more distant than the voter’s ideal point. Discrimination between these two models is difficult. We use median as opposed to mean placements, because the former is a more robust estimator of central tendency for skewed data and because it is more comparable to the location of the median voter, a position that is relevant to equilibrium analysis (see Chapter 8).

Comparing the Empirical Fit

55

Figure 4.1 Plots of mean thermometer scores for candidates versus liberal–conservative location of the respondent: 1984 American NES. Note: Arrows denote median placements of candidates Mondale (M) and Reagan (R).

utility occur at different points for different candidates and that these peaks tend to be near the (presumed) ideological locations of the candidates tend to support the proximity model, but the varying steepness of the curves (least for Hart and Ford and greatest for Kennedy and Reagan) provides support for the RM directional model. In sum, the mixed evidence, especially the occurrence of peaking at points beyond the median locations of the respective candidates, supports a mixed proximity–RM model. Comparable plots for Michael Dukakis, Jesse Jackson, George Bush, and Ronald Reagan in the 1988 American NES (see Figure 4.2) suggest a similar but less clear-cut story. Dukakis peaks at -2; Bush and Reagan both peak at 2 to 3 on the L/C scale. Only Jackson appears to increase to the limit of the scale. Again, this pattern of peaking beyond one’s median placement tends to support a mixed model.

56

Part I Models of Voter Behavior

Figure 4.2 Plots of mean thermometer scores for candidates versus liberal–conservative location of the respondent: 1988 American NES. Note: Arrows denote median placements of candidates.

Under the proximity model, utility is expressed as a loss function and is thus largest when that loss is zero, i.e., when the voter and candidate have the same location. Hence, the utility of voters who locate at the same point as the candidate should be the same for all candidates, namely 0, independently of the position of the candidate. For convenience, we refer to this utility as peak utility. Figure 4.1 suggests at least qualitatively that in 1984 peak utility is lower for the moderate candidates, Hart and Ford, than for the others. In 1988, however, little or no such effect is observed (see Figure 4.2).5 Likewise, the prox5

As a corollary, under the proximity model, peak utility should not depend on the voters’ placements of a single candidate. Stratifying the data from the 1984 NES by candidate placement, Merrill (1995) finds that for both Reagan and Mondale, peak utilities increase monotonically as the candidate position recedes from the neutral point, as one might expect under the RM model.

Comparing the Empirical Fit

57

imity model predicts equality of slopes of the utility curves for different candidates. Yet the gradual slopes of the two moderates (Hart and Ford) in the 1984 NES (see Figure 4.1), contrasted with the steep slopes of Kennedy and Reagan, contradict this expectation. In summary, many pure model expectations of the proximity and RM models are not fulfilled and, in each case, the divergence is in the direction of the other spatial model. Rabinowitz, Macdonald, and Listhaug (1993) analyze several British elections from 1979 to 1992, presenting plots of mean party evaluation versus voter location, analogous to our Figures 4.1 and 4.2. Although these authors argue that these plots support the RM model, the plots for the last two elections (1987 and 1992) for which better voter placement data are available show considerable leveling off toward the right for the Conservative party and toward the left for Labor. These plots appear more in tune with a mixed model rather than a pure RM model.The authors observe, however, that the almostlevel plots for the Alliance are what is to be expected from the RM model for a party near the 0-point, because all voters should be indifferent to such a party and hence give it the same issue-based utility. Granberg and Gilljam (1997), however, study six Swedish national election studies and fail to find support for the hypotheses suggested by the RM model that centrists have no real party preferences and that voters slightly on one side of the neutral point prefer a much more intense party on the same side over a party on that side closer to their own position. The plots so far are all subject to the pitfalls of interpersonal comparisons. To reduce this problem, we stratify the data by respondent self-placement on the L/C scale. Placing the 1984 candidates in a plausible left-to-right order (Jackson, Kennedy, Mondale, Hart, Ford, Reagan), we plot mean thermometer scores over candidates – one plot for each possible location of voter self-placement on the L/C scale (see Figure 4.3). According to the RM model, we would expect the scores to decrease monotonically for each group of voters to the left of center and to increase monotonically for each group to the right of center. This expectation appears to be borne out for the latter groups but not the former. The flatness of the plot for voters located at the neutral point bears out the expectations of the RM model, since such voters should have the same (zero) utility for all candidates. Thus, the

Figure 4.3 Mean thermometer scores, stratified by voter location (L/C): 1984 American NES.

Comparing the Empirical Fit

59

zero location (4 on the traditional 1–7 scale) appears to be used by voters who are indifferent rather than by voters who prefer a moderate policy position. The patterns of the plots are confounded, however, by nonspatial intercandidate comparisons. For example, the evaluations of candidates were no doubt affected by variables such as racial prejudice or incumbency not directly measured by the NES policy scales. The fact, however, that voters with L/C = -1 provide peak evaluations to Hart, while those at -2 and -3 give their highest ratings to Mondale and Kennedy, respectively, are difficult to explain by arguments other than spatial proximity.6 The plots in Figure 4.3 are suggested by similar plots – stratified by voter location – in Westholm (1997: Fig. 4) for the 1989 Norwegian Election Study in which mean thermometer scores for seven major parties are plotted against mean party placement on a left–right scale. Westholm finds that for nine out of ten voter locations, the peak utility occurs at the voter location, offering much clearer evidence for the proximity model for these Norwegian data than appears from the American NES data analyzed here.7 4.3. Correlation and Regression Analyses of Pure Models of Voter Utility Turning now from graphical to analytical methods, we focus first on the predictive power of competing pure models for thermometer scores. To this end, we compare the regression (correlation) coefficients of alternative models using voter-specific placements of candidates by the voters. In their work, Rabinowitz and Macdonald use mean candidate placements over voters, rather than voter-specific placements. For example, Rabinowitz and Macdonald (1989) obtain a consistent 6

7

In the 1988 American NES, candidate-placement data permit us to plot mean thermometer scores (see Figure 4.2) directly against median placements of candidates (from left to right: Jackson, Dukakis, Bush, and Reagan). Voters located at -1 or -2 give peak scores to Dukakis (unlike those at -3, who give Jackson equivalent scores), as might be expected from the proximity model. Median placements of Bush and Reagan are, however, too similar to permit meaningful inference. This may be because Westholm is looking at data on parties in a well-established multiparty system rather than at candidates in a two-party system several of whom are from the same party.

60

Part I Models of Voter Behavior

preference for the RM model over the proximity model by comparing R-squared values for separate regressions of thermometer scores on model predictions for the American NES for four elections during the period 1972–84.8 Rabinowitz, Macdonald, and Listhaug (1991, 1993) obtain similar results for several recent elections in Norway and Britain, respectively, as do Aarts, Macdonald, and Rabinowitz (1996) for the Netherlands. Most researchers other than Rabinowitz and Macdonald and their collaborators agree that voter-specific (idiosyncratic) placements of candidates better reflect the information used by voters in decision making than do mean placements, i.e., that use of voter-specific placement reduces measurement error. We find this argument convincing and focus on evidence based on voter-specific placements. The main arguments in the debate over mean versus voter-specific placements of candidates are outlined in Appendix 4.3, along with a discussion of the empirical fit of various models obtained by the researchers who have compared the fit of these models when mean issue location is used with that when individual voter placement is used. When mean placements are used, the empirical fit of the proximity model for voter utility is substantially reduced compared to the situation when voter-specific placements are used. For most data sets, this use of mean placements is the primary reason the results reported by Rabinowitz and Macdonald strongly favor the RM model whereas the results of others do not.9 4.3.1 U.S. Data Using voter-specific placements of candidate positions in the five American NES studies 1980–96 (see Table 4.1), we do not find large differences among the mean correlations between model predictions and thermometer scores for the three separate pure models – proximity, RM, and Matthews.10 Nevertheless, correlations for the 8

9

10

They report R-squared values controlling for party identification, race, and region, and also with no controls. The latter R-squares are simply the squares of Pearson correlation coefficients. The disparity between results for mean and for voter-specific placement of candidates largely disappears for voter choice as opposed to voter utility models because voter choice models suppress the interpersonal comparison of utilities (see Chapters 6 and 7). For further discussion see Merrill and Grofman (1997a).

Table 4.1. Correlation between model predictions and thermometer scores using voter-specific placements of candidates: American NES 1980–96 1980 (n = 429)

1984 (n = 1,095)

1988 (n = 932)

1992 (n = 901)

1996 (n = 891)

Means

Model

Carter

Reagan

Reagan

Mondale

Bush

Dukakis

Bush

Clinton

Clinton

Dole

a

b

All

Proximity (quadratic) Rabinowitz– Macdonald Matthews directional

.542

.651

.733

.595

.632

.620

.646

.648

.702

.588

.651

.620

.636

.552

.693

.714

.642

.633

.641

.624

.629

.740

.624

.653

.646

.649

.585

.713

.728

.625

.677

.669

.638

.609

.768

.650

.679

.653

.666

a b

Incumbents. Challengers.

62

Part I Models of Voter Behavior

Matthews model are significantly higher than those for either the proximity or RM models ( p = 0.02 and 0.03, respectively), although the RM and proximity values are not significantly different ( p = 0.15).11 Using voter-specific candidate placement, Krämer and Rattinger (1997) regress thermometer scores separately on (quadratic) proximity and RM model predictions (while also controlling for party identification, race, and region) for the American NES for the seven presidential election years during 1968–92 and compare the adjusted R-squared values obtained (see Table 4.2). Under subjective location of candidate issue positions – which Krämer and Rattinger, like most authors, argue is more plausibly linked to individual voting decisions – correlations for the proximity model are generally higher than those for the RM model (except for Ford and Carter in 1976 and Mondale in 1984, with a tie for Reagan in 1980), but the differences they find, like ours, are not statistically significant ( p = 0.12).12 Our interest is, however, less with which pure model is the better predictor than it is with whether predictive differences in the power of the competing pure models vary with the type of candidate. When candidates in the 1980–96 American NES are broken down into two types, incumbents and challengers, somewhat different patterns begin to appear.13 The advantage of the Matthews model over the RM model appears greater among the incumbents (mean model difference in correlation = 0.026) than among challengers (0.007), although the significance level for the comparison t-test is only 0.16. By contrast, the mean correlation for the RM model exceeds that for proximity for challengers by 0.026, but for incumbents, by only 0.002. 11

12

13

The p-values are based on paired t-tests that control for candidate identity. These and similar statistical tests assume, implicitly, that the elections observed are a sample of a larger population. If a linear, rather than quadratic, Euclidean metric is used for the proximity model, correlations increase fairly uniformly over candidates with an average increase of 0.032. The mean correlation for the linear proximity model is significantly greater than that for the RM model ( p = 0.04) but not significantly greater than that for the Matthews model ( p = 0.87). Substantively, correlations for the proximity model are comparable to those for the directional models. Because Krämer and Rattinger use control variables and different rules for admissible cases than we do, their comparative results for the proximity and RM models differ somewhat from ours for specific elections. The basic conclusion that the two models are generally of similar predictive strength, however, is the same in both studies. Incumbent is defined to mean representative of the incumbent party.

Comparing the Empirical Fit

63

Table 4.2. Adjusted R-squared values for proximity and RM model predictions of thermometer scores using voter-specific placements of candidates: American NES 1968–92 Model

Incumbent

Challenger

1968 Proximity model RM model Difference

37.4 35.6 1.8

28.9 28.1 .8

8.5 7.5 1.0

1972 Proximity model RM model Difference

39.9 39.1 .8

41.1 39.7 1.4

-1.2 -.6 -.6

1976 Proximity model RM model Difference

33.0 34.3 -1.3

32.4 35.0 -2.6

.6 -.7 1.3

1980 Proximity model RM model Difference

36.3 34.9 1.4

34.8 34.8 .0

1.5 .1 1.4

1984 Proximity model RM model Difference

54.7 52.3 2.4

44.6 46.0 -1.4

10.1 6.3 3.8

1988 Proximity model RM model Difference

46.3 44.5 1.8

41.8 41.1 .7

4.5 3.4 1.1

1992 Proximity model RM model Difference

41.0 38.0 3.0

43.8 42.9 .9

-2.8 -4.9 2.1

Difference

Source: Adapted from Krämer and Rattinger (1997: Table 5).

Again, however, the difference is not statistically significant. Krämer and Rattinger do not compare incumbents and challengers. Based on their data, however, the mean difference between proximity and RM values of adjusted R-squared is significantly higher for incumbents than for challengers ( p = 0.03) In fact this difference is greater in all of the seven elections except 1972, as can be seen in Table 4.2.

64

Part I Models of Voter Behavior

In Chapter 5 we shall return to the possibility of a difference between incumbents and challengers as to which model of utility best fits. 4.3.2 Non-U.S. Data One of the most careful comparisons of the proximity and RM models is that of Maddens (1996) concerning the 1991 parliamentary elections in Flanders. Maddens addresses the argument that the typical bipolar scale used for issue variables in national election studies confounds intensity with extremity. His measure of party intensity is based on a separate survey of party candidates, which requested not only issue positions but also which issues they would emphasize in the campaign. These issue positions were used in the proximity analysis; a scale constructed from the responses concerning emphasis was used for intensity. This procedure also avoids the use of mean voter placements of parties. Voter self-placements and intensities were, however, obtained from a single 0–10 scale from a representative sample of 2,691 Flemish voters. Maddens performs separate logistic regressions (with three control variables) for each of the six larger Belgian parties, obtaining a substantially higher coefficient for the proximity model for the Christian Democrats, a substantially higher coefficient for the RM model for the Vlaams Blok, and about equal coefficients for the other four parties. The Christian Democratic party was the largest member of the government coalition, whereas the Vlaams Blok was a far-right party not part of the government. This contrast tends to support the distinction we have made between incumbent and challenging candidates. Maddens also performs separate regressions for each issue, finding considerable variation of issue impact across parties and a generally larger issue effect when a party stresses that issue. Both of these observations are expectations of the RM model. At the same time some issues affect the parties irrespective of intensity, as might be expected under a proximity model. Thus, again, we find support for a mixed model. In addition to their study of the American electorate, Krämer and Rattinger (1997) compare R-squared values for up to five German parties for each of the years 1990, 1991, and 1992. Using a much larger set of control variables than in their American study, they obtain – for

Comparing the Empirical Fit

65

both mean and voter-specific candidate placement – a slight advantage for the proximity model in West Germany and a slight advantage for the RM model in East Germany.14 Their extensive set of control variables, however, appears to leave little for any of the issue-based models to explain. Pierce (1993, 1995a, 1997) compares the proximity and RM models for the 1988 French Presidential Survey and the 1988 American NES. Overall, Pierce argues that the proximity model (using voter-specific placements) best predicts a plausible distribution of candidate choice. Pierce (1993) finds that the linear proximity model is generally superior to the RM model but Pierce (1997) finds that the quadratic proximity model is only sometimes superior. This contrast is in accord with our results (see note 11, above) and those of Merrill (1995) for American NES data. Gilljam (1997) compares the correlation between feeling thermometer scores and the predictions of the proximity and RM models for four electoral studies in Sweden (five parties in 1979, seven in 1982 and 1988, and eight in 1991) and Norway (seven parties in 1989). For the Swedish data, Gilljam (using voter-specific placement) finds the proximity model superior in more cases than the RM model. For the Norwegian data, he finds a slight preference for the RM model (the mean difference between the correlation coefficients for the two models is 0.026). Van der Eijk and Franklin (1996: 347) compare the predictive ability of the proximity and RM utility functions for scores of voter preference as part of a larger study of party choice. They find the variance explained by the proximity function higher than that for the RM function for all 14 political systems of the European Union that they investigate. Overall, we would conclude that the studies of Merrill, Merrill and Grofman, Maddens, Krämer and Rattinger, Pierce, and Gilljam – which cover five European electorates in addition to that of the United States – demonstrate no strong, consistent preference for either the proximity or the RM model.15 If anything, these studies suggest that a mixed model may be preferred, and they provide some 14

15

Values of N vary from 689 to 1,889 for West Germany and 285 to 565 for East Germany. Van der Eijk and Franklin’s study is considerably more favorable to the proximity model.

66

Part I Models of Voter Behavior

evidence that the best pure model may depend upon whether the candidate whose utility is being evaluated is a challenger or an incumbent. However, all of these authors who compare voter-specific and mean placements find that use of the former favors the proximity model and argue that this use more realistically represents the information voters are likely to employ in making voting decisions. 4.4 Discussion The analyses in this chapter provide evidence that both the pure proximity and the pure RM spatial models fall short of accounting for the shape and position of individual utility curves of voters and the variation of utility over voters. Graphically, we have seen that utility over voters appears to peak at a point more extreme than that of the candidate – an expectation intermediate between that predicted by the pure proximity and pure RM models. The shape of the utility curve for a typical fixed voter – which involves no interpersonal comparison of utility – likewise appears intermediate between the pure model expectations. Analytically, a number of researchers – working with a variety of national electorates – find strong associations between candidate evaluations and utility as predicted from either the proximity or the RM model. These associations are usually comparable in strength. Our own analysis suggests a similar and possibly higher degree of association for the Matthews directional model. In Chapters 5–7, we assess the unified model and its components for data from the United States and several European electorates. In the process, we address a number of methodological issues. For example, none of the studies we have summarized in this chapter attempts to measure the potential to bias the results in favor of the proximity model that may result from projection or rationalization in subjective candidate placements. In Chapter 5, we treat the problem of rationalization–projection effects in detail, introduce a modification in the proximity component of the unified model to adjust for them, and reanalyze empirical data under a form of the unified model that has been adjusted for rationalization–projection.

CHAPTER 5

Empirical Model Fitting Using the Unified Model: Voter Utility The best lack all conviction, while the worst are full of passionate intensity. William Butler Yeats, “The Second Coming”

5.1 Testing the Proximity and Directional Models of Voter Utility within a Nested Statistical Framework Theoretically, we have shown that proximity, directional, and intensity components can all be embedded in a two-parameter unified model of the shape of voter utility functions, of which the traditional proximity spatial model, the Rabinowitz–Macdonald (RM) model, and the Matthews directional model are all special cases. In developing this unified model we showed that the RM directional model, despite its name, is best viewed as a mixed model with both directional and intensity components, whereas the Matthews model best represents a pure directional component. To provide a preliminary test of this unified model, we look at the nature of voter utility functions for major candidates for the U.S. presidency during the period l980–96 by fitting the unified model developed in Chapter 3 to American NES data. In Chapters 6 and 7 we develop and test a unified model of voter choice for these and other data. Not unexpectedly, by testing nested submodels, we conclude that, in most applications, pure models must be rejected in favor of a model incorporating two or more of the proximity, directional, intensity, or discounting components. Also – like Enelow, Endersby, and Munger (l993) – we provide evidence that different models of utility functions best explain voter utility for different types of candidates. Our empirical analysis shows that in American politics the intensity component of the unified model for voter utility functions is significant for pres67

68

Part I Models of Voter Behavior

idential challengers but not for incumbents. The intuition here is a rather simple one: Incumbent locations can be reasonably precisely located, whereas challengers’ positions are seen primarily in terms of intensity and direction of change.1 The desire of voters for greater intensity on the part of the challenger may be interpreted as follows. A voter’s ideal challenger may be one who espouses more intense or extreme views than the voter – but just extreme enough that the discounted position of the candidate agrees with the ideal location of the voter. Enelow, Endersby, and Munger (l993) suggest that such a candidate may need to “overshoot the mark,” advocating stronger policies than voters actually want in order to offset the expected discount. Alternatively, one may argue with Rabinowitz and Macdonald (1989) that a voter may directly prefer a candidate whose intensity on issues exceeds his own. Such a voter first seeks a candidate who is on the same side of the issues; the more intensely she advocates that position the better. 5.2 Correlation Analysis In Chapter 3, we argued that better estimates of utility are obtained by using voter-specific placement of candidates instead of mean placement. Voter-specific placement, however, might bias model estimates – in favor of the proximity model – because of the likelihood of projection, namely, the tendency of respondents to place favored candidates near themselves (see, e.g., Macdonald and Rabinowitz, 1997). The effect of projection can be reduced using a regression technique, which we describe in Appendix 5.1. Our results below indicate that estimated values of model parameters are similar whether or not we adjust for projection; in other words, the effect of projection on these values is minimal. We first reassess Pearson correlation coefficients for the five American NES studies during 1980–96, previously considered with unadjusted data in Table 4.1. Table 5.1 shows that after adjustment for projection the mean correlations for the three separate models – 1

As we shall see in Chapter 8, the implications of the incumbent–challenger dichotomy are significant for candidate strategy. The incumbent is led to seek the center, whereas the challenger is either forced by vote-maximizing concerns to stake out more extreme positions or at least is not disadvantaged by doing so.

Table 5.1. Correlation between model predictions and thermometer scores using projection-adjusted placements of candidates: American NES 1980–96 1980 (n = 430)

1984 (n = 1,091)

1988 (n = 930)

1992 (n = 901)

1996 (n = 892)

Means

Model

Carter

Reagan

Reagan

Mondale

Bush

Dukakis

Bush

Clinton

Clinton

Dole

a

b

All

Proximity (quadratic) Rabinowitz– Macdonald Matthews directional RM with proximity constraint Unified model

.543

.653

.736

.598

.476

.534

.560

.471

.478

.496

.559

.550

.554

.552

.693

.717

.643

.438

.542

.524

.491

.564

.498

.559

.573

.566

.585

.711

.730

.627

.473

.565

.551

.492

.586

.529

.585

.585

.585

.569

.706

.749

.655

.494

.572

.568

.508

.566

.531

.589

.594

.592

.605

.726

.773

.668

.510

.590

.583

.519

.590

.545

.581

.583

.611

a b

Incumbents. Challengers.

70

Part I Models of Voter Behavior

proximity, RM, and Matthews – still do not differ greatly overall.2 As with the situation for unadjusted data considered in Chapter 4, the mean correlation for projection-adjusted utilities (using paired, two-tailed t-tests that control for candidate identity) is significantly higher for the Matthews directional model than that for either the proximity model ( p = 0.02) or the RM model ( p = 0.005). They are not significantly higher for the RM model than for the proximity model ( p = 0.36). However, the best-fitting unified model (see below) yields higher correlations than each pure model ( p ≥ 0.001 in each case).3 The contrast between correlations for incumbents and challengers is now more apparent. (As in Chapter 4, incumbent refers to the candidate of the incumbent party.) Correlations for the Matthews model are generally higher than those for the RM model but especially among incumbents. Correlations for the RM model are generally lower than those for the proximity model for incumbents (with the notable exception of Clinton in 1996) but higher among challengers. These distinctions between challengers and incumbents – based on correlations alone – are, however, not statistically significant because of the outlier (Clinton in 1996). 5.3 Fitting the Unified Model of Voter Utility via Nonlinear Regression We are now prepared to test the full unified model and its nested submodels using nonlinear regression relating thermometer scores to voter issue positions. Through maximum likelihood estimation (see Dixon, 1990: 921–58), we can find the best fitting value of the intensity parameter q that is used to distinguish between the RM and Matthews directional models and of the mixing parameter b that specifies the mix of proximity and directional components. We propose to test several hypotheses involving the two-parameter unified model. Suppose we observe a value of b between 0 and 1.4 2

3 4

The projection adjustment is based on the full set of issues; Merrill and Grofman (1997a) estimate the adjustment parameter from the liberal–conservative scale alone, obtaining substantively the same relationships between correlations. An RM model with proximity constraint yields intermediate correlations. Note that mathematically b is not restrained to lie in the 0 to 1 range. Values outside that range may occur owing to measurement error or to factors not taken into account

Empirical Model Fitting using the Unified Model

71

Rejection of the hypothesis b = 0 would imply a proximity component; rejection of b = 1, a directional component. Rejection of both would imply support for a mixed directional and proximity model. If q > 0, a significant intensity component would be implied by rejection of q = 0. If one or more of these hypotheses is rejected, we wish to use the size of the parameter estimates in an exploratory sense to assess the extent of mixing of the various pure models and to investigate possible incumbency effects. Table 5.2 provides estimates for the intensity parameter q and the mixing parameter b for the unified model based on the five American NES Studies conducted in 1980–96. The values in Table 5.2 are adjusted for projection; unadjusted values are similar, as can be seen in Table 5.3. In the unified model with parameters b and q, the small values of q for the five incumbents (only for Clinton in 1996 is the hypothesis that q = 0 rejected at the 0.05 level) support the Matthews model with little need for an intensity component. Since the hypotheses b = 0 and b = 1 are also rejected for the three Republican incumbents, the utility functions of these incumbents appear primarily as Matthews directional with proximity constraint, i.e., as a mixed model with two components. For neither of the two Democratic incumbents can the hypotheses b = 0 be rejected, so that a directional model may be more appropriate. For three of the five challengers, however, the hypothesis q = 0 is rejected ( p < 0.05); for the other two the values of q are more than one standard error above 0, suggesting that – for challengers – both pure directional and pure intensity factors play a role, again constrained by a proximity term, i.e., the model has three components. The mean estimate of q for challengers is significantly higher (0.31) than that for incumbents (0.06), with a p-value for the comparison test of 0.01.5 Further interpretation of q and b is given in Appendix 5.2. Direction and intensity appear to better predict voter utilities for challenging candidates. Voters are more likely to have only a vague,

5

in the model. Alternatively, values of b greater than 1 may occur because voters – under the discounting interpretation – “discount” the position of a candidate’s position by inflating it under the belief that she is not as moderate as claimed. Mean values for q continue to be significantly greater (p = 0.03) for challengers than for incumbents when controlling for candidate’s party (in an analysis of variance).

Table 5.2. Parameter estimates for the unified model using projection-adjusted placements of candidates: American NES 1980–96 1980 (n = 430) Parameter

Carter

1984 (n = 1,074) Reagan

Reagan

Unified model (projection adjusted placements) q -.14 .28** .05 (.19) (.11) (.10)

b

.10 (.03)

.07* (.03)

.17** (.02)

Standardized regression coefficients by component: Proximity .26 .16 .41 Directional .39 .58 .40 Note: Standard errors are in parentheses. * Significantly different from 0 at the .05 level. ** Significantly different from 0 at the .01 level.

1988 (n = 927)

1992 (n = 901)

1996 (n = 892)

Mondale

Bush

Dukakis

Bush

Clinton

Clinton

Dole

.44** (.09)

.13 (.19)

.22 (.12)

.04 (.19)

.38* (.19)

.24* (.12)

.22 (.15)

.12** (.03)

.22** (.05)

.14** (.04)

.23** (.05)

.18** (.06)

.02 (.03)

.14** (.05)

.20 .50

.27 .28

.23 .29

.32 .29

.18 .36

.03 .57

.19 .38

Table 5.3. Comparative parameter estimates for the unified model under different scoring methods: American NES 1980–96 1980 (n = 430) Carter Estimates for the intensity Candidate placement Voter specific Projection adjusted Mean

1984 (n = 1,091) Reagan

Reagan

1988 (n = 913)

1992 (n = 890)

1996 (n = 892)

Mondale

Bush

Dukakis

Bush

Clinton

Clinton

Dole

.44 .44 .49

.12 .13 .30

.14 .22 .13

.06 .04 .67

.40 .38 .71

.16 .24 .40

.19 .22 .21

.12 .12 .07

.10 .22 .07

.09 .14 .05

.20 .23 .03

.33 .18 .16

.06 .02 .10

.06 .14 .01

parameter q in the unified model -.13 -.14 -.03

.26 .28 .72

.05 .05 .54

Estimates for the intensity parameter b in the unified model Candidate placement Voter specific .10 .07 .17 Projection adjusted .10 .07 .17 Mean .13 .11 .10

Source: Adapted in part from Merrill and Grofman (1997a: Table 4).

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Part I Models of Voter Behavior

directional idea of their stance on issues, whereas incumbent positions are often uncomfortably cast in stone. Thus, intensity is more important in shaping voter utility for challengers than for an incumbent administration or governing party. In summary, the nonlinear regression analysis supports and extends the results suggested by the correlation analysis. On the one hand, for challengers, the intensity aspect of the RM model augments the Matthews directional model, whereas, for incumbents, intensity plays little or no role, especially for Republican candidates. On the other hand, the proximity constraint is in nearly all cases an essential component of voter utility, especially for incumbents. 5.4 Parameter Estimates for the Mixed Proximity–RM Model Previous research has considered the mixed proximity–RM model without a parameter for intensity (effectively, q = 1). Under this assumption, our (projection-adjusted) estimates for the mixing parameter b range from 0.12 to 0.64 (see Table 5.4) with both b = 0 and b = 1 rejected in all cases (one-sided p < 0.05).6 This implies that the marginal addition of either the proximity or directional component at this stage is statistically significant. Since b can be interpreted in this model as the proportional position of the indifference line between its position under RM and under proximity (see Chapter 3), we may infer support for a model intermediate between these two. In this reduced model, the mean estimate of the mixing parameter b for challengers (0.36) appears slightly lower than that for incumbents (0.45), although the difference is not statistically significance [the comparison is statistically significant in the data set (unadjusted for projection) considered by Krämer and Rattinger (1997), which we discussed in Chapter 4). An analysis of variance shows that the 6

Without projection adjustment, estimates of b range from 0.23 to 0.57 (see Table 5.4) with again both pure models rejected in all cases at the 0.05 level. Estimates were also obtained (without projection adjustment) with party ID as a control variable (the NES variable for party ID varies from 0 for a self-identified strong Democrat to 6 for a self-identified strong Republican). These values range from 0.23 to 0.74, and are slightly (but significantly) higher than those obtained without the control variable (the mean value of b with the control variable is 0.45; without it, the mean is 0.38). See Chapters 6, 7, and 10, in which the control variable, party ID, is an important constituent of the model for voter choice as opposed to voter utility.

Table 5.4. Comparative parameter estimates for the RM model with proximity constraint under different scoring methods: American NES 1980–96 1980 (n = 430) Carter

1984 (n = 1,095) Reagan

Reagan

1988 (n = 932) Mondale

Estimates for mixing parameter b in the RM model with proximity restraint Candidate placement Voter specific .37 .28 .52 .25 Projection adjusted .37 .29 .53 .25 Mean .24 .16 .13 .09 Source: Adapted in part from Merrill and Grofman (1997a: Table 4).

1992 (n = 901)

1996 (n = 892)

Bush

Dukakis

Bush

Clinton

Clinton

Dole

.44 .59 .13

.34 .42 .13

.57 .64 .04

.55 .38 .20

.23 .12 .19

.26 .45 .09

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Part I Models of Voter Behavior

mean for b is significantly higher (p = 0.04) for Republican than for Democratic candidates and, furthermore, that this effect is close to significantly greater (p = 0.06) for incumbents than for challengers, after controlling for the party of the candidate. In general, however, this one-parameter model, which lumps together intensity and direction, fails to delineate the incumbency effect with the clarity apparent in the two-parameter unified model. Table 5.4 also presents estimates for b based on voter-specific and mean placements of candidates. Estimates of b using mean placements are much lower – i.e., more favorable to the RM theory – than those based on voter-specific placement. (Recall that a pure RM model corresponds to b = 0, whereas b = 1 defines a pure proximity model.) Using mean placements, Rabinowitz and Macdonald (1989) present model ratios (equal to 1/b in our notation) for several NES data sets through 1984 and conclude a strong preference for the RM model. Converted to betas, their estimates are 0.12 for Reagan and 0.10 for Mondale in 1984 and are similar to ours; their values of 0.38 for Carter and 0.09 for Reagan in 1980 are somewhat different due to different numbers of cases used. Similarly, Aarts, Macdonald, and Rabinowitz (1996) use mean placements to analyze the Dutch elections of 1971, 1986, and 1994, grouping all parties together but keeping issues separate. Their model ratios – converted to betas and averaged over the several issues – are 0.05, 0.46, and 0.38, for the three elections, respectively. The latter two values support a mixed model, not a pure RM model, despite the fact that mean party placements are used. This conclusion is obscured when inspecting Rabinowitz and Macdonald’s reciprocal version of the mixing parameter, which does not vary linearly with the location of the indifference plane. Dow (1998a) also evaluates the proximity and RM models by estimating the mixing parameter b for the RM model with proximity constraint for the four American NES data sets for 1980–92. Dow includes nonvoters, voters for third-party candidates, and respondents who placed the Democrat to the right of the Republican. His less restrictive criteria increase the number of cases by a factor of two or more. Furthermore, Dow controls for race, party ID, and region. Like Krämer and Rattinger, he argues that voter-specific placement is better grounded in spatial theory and – using voter-specific placements – obtains a mean estimate over the eight candidates of 0.51 for

Empirical Model Fitting using the Unified Model

77

b (compared to 0.14 using mean placements). Dow also recomputes the estimates for an “informed” electorate – defined as those respondents who could place themselves and the candidates on all issue scales – obtaining the same mean estimate of b as before. Thus, Dow’s results, like ours, support a mixed model – not the superiority of one pure model over another – and suggest that the parameter estimate does not depend on voter sophistication. Iversen (1994) approaches the problem of assessing a mixed proximity–RM model by a different technique. Reasoning that in a mixed proximity–RM model with mixing parameter b, the voter with maximum utility for a fixed candidate with spatial location C should be located not at C but further out at the point C/b, he concludes that the ideal location for this candidate is at C/b.7 For example, if b = 0.5, this ideal location is about twice as extreme as the location expected under a pure proximity model, where candidate and (mean) voter positions should be the same. To test this expectation, he compares the mean location of a party’s supporters with the mean placement of that party by an elite consisting of party officials, presumably able to place the party more accurately than the mass public. His analysis applies to six European electorates (Belgium, Britain, Denmark, France, West Germany, and the Netherlands). Using data from the European Political Party Middle Elite study of delegates to the European party conference in 1979 for party placement and Eurobarometer data for the location of supporters, he regresses mean party placement on mean supporter position. Iversen obtains a slope coefficient of 1.9, suggesting a mixing parameter equal to the reciprocal of 1.9, or about 0.5, which would specify a mixed model with roughly equal proximity and RM components. In summary, both unadjusted and projection-adjusted parameter estimates as well as Iversen’s alternative approach suggest that issueoriented voting behavior has both proximity and directional components of comparable strength. Estimates for q are more supportive of pure direction (Matthews) rather than direction with intensity (RM), at least for incumbents, although challengers may engender a significant intensity component. 7

This need not be the equilibrium location for C, if indeed an equilibrium exists (see Chapters 8 and 10).

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Part I Models of Voter Behavior

Projection-adjusted estimates for b in the directional model with proximity constraint and for b and q in the unified model are generally quite similar to the unadjusted estimates, but are quite different from the estimates based on mean placement. Consequently, our major inference from our projection-adjustment analysis is that projection-adjusted estimates are substantially similar to simple voterspecific estimates. Thus, the latter may be used without undue bias and may be preferred to the more complex projection-adjusted estimates because of their simplicity. In the remaining chapters of this book we will report only the voter-specific estimates without making a projection correction. 5.5 Discussion Fitting the unified model to American NES data from the five presidential election years from 1980 to 1996 suggests that voter utilities have both directional and proximity components and that intensity plays a role for challengers but not necessarily for incumbents. A mixture of the proximity and directional models enhances the fit with empirical data. A simple combination of voter motivations in which directional utilities are tempered by distance provides a rationale for why voters might behave the way the empirical fit to the data suggests. A voter looks for a candidate or party who is moving more or less in his direction on issues of importance to him – that is, who is moving out a similar ray from the neutral point, as is predicted by the (Matthews) directional theory. As voter and candidate recede from the center (along their respective rays), however, their distance apart increases. This greater distance reduces the voter’s utility (as predicted by the proximity component), reflecting the fact that differences matter more to a voter and a candidate who have strong opinions (and are far from the center) than to ones who have more moderate opinions (and are closer to the center). Alternatively, discounting of candidate positions may account for behavior on the part of voters similar to that expected under a mixed proximity and directional model. Although both proximity and directional components contribute in explaining the variation of thermometer scores, Platt, Poole, and Rosenthal (1992) have suggested that uninformed voters may be more

Empirical Model Fitting using the Unified Model

79

influenced by direction, whereas informed voters may be more likely to respond to issue proximity. In general, it is possible that some voters evaluate candidates according to the proximity model, whereas others use a directional measure. In other words, heterogeneity among voters is an alternative explanation to individual mixed utilities.8 Merrill (1995), in studying the 1984 NES, found a modest trend toward the proximity model with greater levels of information and education. Projection-adjusted estimates of b, however, show no consistent relation with these same measures of sophistication. Macdonald, Rabinowitz, and Listhaug (1995), using mean placements and electorates in the United States and Norway, also find little relation between sophistication and the relative predictive strength of the proximity and RM models. Dow (1998a), using a different measure of sophistication, also found no relation with estimates of b. Still, it is possible that an alternative method of subgrouping would reveal heterogeneity for utility functions. We shall address the Lewis and King (1998) critique of the possibility of statistically distinguishing between the proximity and RM utilities in Chapter 7, in the context of models of voter choice.9 We 8

9

See Morris and Rabinowitz (1997), who investigate the implications of a heterogeneous electorate, finding that vote-maximizing candidates should adopt positions that are intermediate between the more extreme ones implied by the RM model and the more centrist ones implied by proximity theory. Lewis and King (1998) point out that researchers on each side of the proximity– directional debate – typified by Macdonald, Rabinowitz, and Listhaug (1998a) and Westholm (1997) – make assumptions that have side effects that favor one model or another. Lewis and King generalize the mixed proximity–RM utility by regressing thermometer scores on the expression

a - bv||V||2 - bc||C||2 - b2||2V · C||2 This model is equivalent to the mixed model we study except that in the Lewis and King model the coefficients of ||V||2 and ||C||2 are allowed to vary separately. Utility under the Lewis and King model (with bv = bc) is a positive linear transformation of utility under our mixed model with b = bv/(bv + b2). Lewis and King conclude that – with the mean placements used by Macdonald et al. – bc is inestimable as long as candidate-specific intercepts are used (Macdonald et al., 1998a, observe that differences of utilities lead to an inestimable parameter b). Thus, in the Lewis and King formulation with intercepts, the proximity requirement that bv = b2 is a substantive restriction, whereas the RM requirement that bc = bv = 0 has little effect because bc is inestimable and empirically bv is close to zero. This imbalance in restrictions favors the RM model. Under Westholm’s assumptions, however, the restriction on bv is partially removed (by subtracting means from both thermometer scores and predicted utilities), thus permitting a better fit for the proximity model.

80

Part I Models of Voter Behavior

shall show there that the bias in estimating the mixing parameter incurred when party-specific intercepts are omitted is of manageable size, so that estimates of b (and inferences from them) are still useful. A mixed model would appear not only to reflect the tendency of voters to prefer candidates with issue positions similar to their own but also to take account of the intensities with which voters and candidates hold these positions. The analysis provides evidence supportive of the view that the same model or the same combination of models, however, may not be best at explaining both voter evaluations of challengers and voter evaluations of incumbents. Either with or without adjustment for projection, we have seen that utility for American presidential incumbents may be described by a twocomponent model representing issue proximity and issue direction. Hypotheses of no proximity effect or no RM effect are both rejected in most cases. The hypothesis of no intensity effect is, however, generally rejected for challengers but not for incumbents. Thus, for challengers, a three-component model incorporating also issue intensity on the part of voters and candidates better represents the empirically observed shape of utility functions. Last, we must remain aware that knowing which model best helps us explain the shape of voter utility functions is not the same as knowing which model best helps explain voter choice. Although voter utilities differ substantially over models, regions in which voter choice differs may be small in elections with a pair of candidates placed roughly symmetrically with respect to the neutral point, as in most U.S. presidential elections.10 We turn to the assessment of voter choice in Chapters 6 and 7, both introducing a probabilistic element into the model and controlling for party identification. This more complex model is most informative in a multiparty electorate. 10

Despite similarities in voter choice between models, the incentives for candidate strategy engendered by these models may be profoundly different, as we shall see in Chapter 10.

CHAPTER 6

Empirical Fitting of Probabilistic Models of Voter Choice in Two-Party Electorates [We add] to our theory that behavior is a resultant of psychological forces our conception of identification with party . . . Angus Campbell et al., The American Voter (1960: 9)

6.1 Probabilistic Models The models considered previously have all been deterministic and issue based, i.e., each voter was assumed certain to select that candidate whose issue-oriented utility was highest. Yet voters use criteria other than issues to choose candidates. The probability, furthermore, that even an issue-oriented voter will select the candidate of higher utility is certainly less than unity if utilities do not differ greatly. Hinich and Munger state (1994: 167): As researchers, we might reasonably divide the set of determinants for a given individual’s vote into those that we, the researchers, can identify and measure in some consistent fashion and those that are idiosyncratic (hence, for us, unpredictable) and specific to each voter.

From the perspective of a candidate or party, such idiosyncratic factors for any given voter are unknown (i.e., are unpredictable) but their probability distributions can be known. Accordingly, they may be modeled as random variables. From a methodological perspective, as the number of candidates or parties increases, it is easier to rely on voter choice to distinguish between models so that we no longer have to depend on the shapes of voter utility functions. Arbitrary assignment of voters to the single candidate (party) of highest utility, however, becomes increasingly misleading. More realistically, in light of unmeasured variables, voters should be assigned to parties on a probabilistic basis. According to what we have described as the Michigan school, party identification is a primary determinant of voter choice, but one that 81

82

Part I Models of Voter Behavior

we have ignored so far in our study of voter utility. Yet a party ID model is not inimical to the spatial or issue-oriented approach that we have pursued in previous chapters. On the contrary, a synthesis of both motivations is appropriate. Our primary purpose in this chapter is to move in these directions. We extend the modeling to include non-issue effects by adding a probabilistic (or stochastic) component to the issue-oriented utility already defined and a measured non-issue effect such as party ID. The probabilistic component is voter-specific and unrelated to issue preferences on the set of issues incorporated into the model. This term incorporates the “exogenous” factors that affect voters’ decisions. We may model these factors as if they were random. The specific extension of probabilistic modeling to our unified model of voter choice is new, but it draws on a long history of probabilistic modeling initiated by Hinich, Ledyard, and Ordeshook (1972, 1973) and Hinich (1977) and extended by Coughlin and Nitzan (1981a), Enelow and Hinich (1982, 1984, 1989, 1990), Erikson and Romero (1990), Coughlin (1992), Merrill (1994), Alvarez and Nagler (1995, 1996), Adams (forthcoming a, b; 1997b; 1999), and others. Inclusion in a probabilistic model of a measured, nonspatial component (such as party ID or candidate competence) was pioneered by Melvin Hinich (see, e.g., Hinich and Pollard, 1981) and carried on by others such as Erikson and Romero (1990). In this chapter we apply the model to two-candidate elections but note certain methodological problems that apply in the two-candidate case. In Chapter 7, we fit this model to multiparty electorates. There has been much controversy (Page and Jones, 1979) stemming from the difficulty in separating partisanship and policy evaluation in determining vote choice. However that methodological controversy may be resolved, we argue that its resolution is largely irrelevant to our analysis. We use our model not in an attempt to show that one effect – party ID or policy location – is more important in determining vote choice but rather to predict the probability of vote choice and thus to evaluate the components of the spatial part of the model. 6.2 A Unified Model of Voter Choice To develop a probabilistic model of voter choice – that is, one that predicts the probabilities that a voter choose each party – we add to

Probabilistic Models in Two-Party Electorates

83

the utility of each voter for each candidate an independent, unmeasured random component that follows a specified distribution. First, denote by sij a spatial-model based deterministic utility of voter Vj for candidate Ci. The value of sij might be specified by the quadratic proximity utility, the RM utility, or the unified model utility, etc., of Chapter 4. Similarly, denote by tij any other measured, nonspatial utility that might, for example, represent party identification. Accordingly, we model the probabilistic component Xij by identically distributed random variables that follow an extreme value distribution.1 This distribution specifies a conditional logit analysis (Train, 1986; Alvarez and Nagler, 1998). These random components are assumed independent over voters and candidates. Thus, the overall utility of voter Vj for candidate Ci is given by the sum of a spatial component, sij, multiplied by a salience coefficient, A; the measured deterministic utility, tij, multiplied by a salience coefficient, B; and the probabilistic component, Xij. Utility is thus defined by the equation:2 U (Vj , Ci ) = Asij + Bt ij + X ij The spatial-model deterministic component might represent a pure directional, pure proximity, or mixed utility. As we have seen, a pure RM directional model predicts that parties tend to perform best when assuming extreme positions; by contrast, under the proximity model for two-candidate races, in one dimension there is a strong centrist incentive – an incentive that becomes indeterminate for multicandidate races. For the multicandidate case, we shall show in Chapter 10 that the pure RM model predicts that only parties positioned at the corners of the polyhedral region defined by the set of party locations receive votes, so that, generally, parties that take more extreme (intense) positions compared to nearby parties garner more support. Much of the motivation for seeking to discriminate between the 1

2

It would appear simplest to model the probabilistic component of the model by normal distributions because of their ease of interpretation. However, the substantially similar extreme value distribution leads to much cleaner expressions for the choice probabilities, i.e., the probabilities predicted by the model that a voter will choose each of the candidates. Note that an individual’s utility for a party is based on the attributes of the party as well as those of the individual. This is characteristic of conditional logit as opposed to what is known as multinomial logit (see Alvarez and Nagler, 1998), although both models are multinomial in nature.

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proximity and directional models lies in these contrasting implications for party strategy and the consequent distribution of the spatial positions of parties, with implications for the levels of party support. Enelow, Endersby, and Munger (l993) fit a model that combines non-issue-based candidate-specific components with discounting of the position of the challenger but no discounting of the position of the incumbent for U.S. presidential data. They find that their hybrid model offers a substantially better fit than either the pure discounting approach or a model with no discounting. Recalling that voter choice under the Grofman discounting model and the mixed proximity–directional model can be equivalent,3 we see that this finding is in line with our result (see Chapter 5) that a directional component is more significant for challengers than for incumbents. 6.3 Fitting the Conditional Logit Model to American NES Data We introduce the use of a probabilistic model in the simple case of a two-candidate race, with data from the American NES. We take sij to be the utility of the (deterministic) unified model (see Chapter 3)4 and tij to represent the partisanship of voter Vj relative to that of candidate Ci, i.e., the distance between the party identification of the voter and the candidate on the 0–6 party identification scale of the NES. This latter value is replaced by its negative so that it declines with greater distance between voter and candidate.5 The Democratic candidate’s partisanship is assumed to be 0, the Republican’s, 6. This makes sense because a strong party-identifying voter (located at 0 or 6) would be expected to exhibit the strongest partisanship effect. Let dij = Asij + Btij denote the deterministic component of utility of 3

4 5

From the point of view of an individual voter (see Section 4.3), utility under the Grofman discounting model and under a mixed proximity–directional model are equivalent if the status quo is identified with the neutral point, so that these two models can be distinguished at best only by using interpersonal comparisons of utilities. Vj ◊ Ci q 2 Thus sij = 2(1 - b ) [ Vj Ci ] - b Vj - Ci Vj Ci The partisanship component is thus defined by tij = -|pVj - pCi| where pVj and pCi are the partisanship of the voter and candidate, respectively, on the 0–6 party identification scale of the NES.

Probabilistic Models in Two-Party Electorates

85

voter Vj for candidate Ci. Under conditional logit analysis (see, e.g., Rivers, 1988; or Adams, 1999), it follows that the probability, pij, that voter Vj vote for candidate Ci is proportional to the exponential of the dij.6 The parameters of this and its nested submodels can be estimated by maximum likelihood estimation.7 There are likely, however, to be candidate-specific (not voterspecific) utility terms. For example, the incumbent’s utility may be augmented because of a favorable economy, and either candidate may be favored for personal qualities. In a two-candidate contest, this introduces one additional parameter or intercept term, which we denote by C (C represents the difference between candidate-specific parameters of the two-candidates, C1 and C2). For simplicity, consider the one-dimensional case. In a deterministic model, the indifference point for proximity is the midpoint between the candidates; for the RM model, it is the neutral point. Intermediate cases are specified by the parameter b. The Matthews directional indifference point is also at the neutral point. Under a probabilistic model, the peak probability gradient will be near the corresponding indifference point. Because the candidate-specific parameter C also shifts the peak gradient along the spatial axis, if the intercept term C is omitted, the estimate for b is biased. For example, in 1984, Mondale was closer to the neutral point than Reagan, but the economy was good and for that (and other) reasons, Reagan won easily. Thus the empirical indifference point (and peak gradient) was to the left of center, overshooting the neutral point (relative to the midpoint between candidates) as in the following diagram: Mondale

6

I

N

Midpoint

The probabilities pij, are given by p ij = exp(dij )

Reagan

2

 exp(d

kj

). Note that the random

k =1

7

variables specifying the probabilistic components of utility do not appear in the expression for the probabilities of voter choice. This is done by maximizing the expression

’p

1j

x1j

(1 - p 1j )

1- x1j

j

where x1j = 1 if voter j votes for candidate 1 and 0 otherwise.

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Part I Models of Voter Behavior

where I is the indifference point, and N is the neutral point. Without the parameter C in the model, we get a negative or 0 estimate for b, suggesting (spuriously) an ultra-RM model. Thus, we model this candidate-specific effect by adding a parameter C to just the first candidate’s utility U (Vj , Ci ) = Asij + Bt ij + C + X ij

(6.1)

in the unified model of voter choice. The introduction of an intercept term, however, leads to a methodological problem. Although the utility functions for the (quadratic) proximity and RM models are quite different, the utility differences between the two candidates are essentially the same for both models, differing only by a constant.8 It is for this reason that we analyzed in Chapter 5 utility functions for individual candidates rather than the difference between candidates, although the latter may be intuitively more appealing. For a two-candidate contest, the probability of voter choice in model (6.1) is the logit of the utility difference, so we cannot expect to distinguish between the RM and proximity models in twocandidate contests using model (6.1).9 The difference in utility functions between candidates for the Matthews model – with its sharp jump at the neutral point – is quite distinct from the corresponding differences for either the RM or proximity functions. Accordingly, we may meaningfully estimate parameters for two submodels of the unified model. The first combines Matthews and proximity components (and is specified by the unified model with q = 0); the second combines RM and Matthews aspects (specified by the unified model with b = 0 and previously designated as the damped directional model in Chapter 4).Tables 6.1 and 6.2 present maximum likelihood estimates for the parameters, A, B, and b or q, as well as the candidate-specific parameter C for these submodels.10 8

9

10

For the (quadratic) proximity model, the difference in utility between candidates C1 and C2 is given by 2V · (C1 - C2) + ||C2||2 - ||C1||2, which differs from the RM utility function only by the constant, ||C2||2 - ||C1||2 (see Appendix 4.2, note 4). Thus our ability to estimate, say, the mixing parameter b, in such a model depends on the difference between voter-specific and mean placements, not on the difference in shapes of utility functions. Alternatively, we could shift to another metric, such as the linear Euclidean metric, but then distinction between the alternative models would rest – somewhat dubiously – on differences between metrics. Since the analysis in Chapter 5 suggests that projection adjustment may be omitted without substantial bias in model estimation, we use simple voter-specific placements

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Table 6.1. Maximum likelihood estimates for parameters of the proximity–Matthews probabilistic submodel of voter choice: American NES 1980–96 Year 1980 (n 1984 (n 1988 (n 1992 (n 1996 (n

= 428) = 1,074) = 910) = 890) = 891)

A

B

b

C

.80 (.19) .73 (.12) .50 (.12) .23 (.11) .87 (.19)

.37 (.06) .35 (.04) .32 (.03) .42 (.04) .39 (.05)

.19 (.14) .41 (.12) .41 (.19) 1.22 (.74) .48 (.21)

.40 (.22) 1.77 (.18) .43 (.12) -.03 (.14) -1.01 (.18)

Note: Standard errors are in parentheses.

Table 6.2. Parameter estimates for RM–Matthews probabilistic submodel of voter choice: American NES 1980–96 Year 1980 (n 1984 (n 1988 (n 1992 (n 1996 (n

= 428) = 1,074) = 910) = 890) = 891)

A

B

q

C

1.29 (.29) .99 (.13) .73 (.16) .53 (.12) 1.27 (.26)

.40 (.06) .38 (.04) .33 (.03) .47 (.04) .40 (.05)

-.21 (.21) .24 (.12) .22 (.18) .33 (.21) .21 (.19)

.42 (.22) 1.39 (.15) .46 (.12) -.06 (.13) -.89 (.17)

Note: Standard errors are in parentheses.

We may compare nested models that differ by a single parameter, such as b or q, by a likelihood ratio test. The difference in log-likelihoods multiplied by the quantity -2 follows a chi-square distribution with one degree of freedom. Accordingly, the log-likelihoods must differ by at least 1.92 to be significantly different at the 0.05 level and by 3.32 for the 0.01 level. Similar conclusions can be drawn by comparing the parameter estimates with their asymptotic standard errors. For the proximity–Matthews submodel (Table 6.1), the estimates for b lie between 0.19 and 0.48, with the exception of that for 1992 of candidates without projection adjustment throughout Chapters 6 and 7. Note also that the Westholm adjustment for interpersonal bias (see Appendix 5.3) is irrelevant in a model of voter choice because such a model suppresses interpersonal comparisons of utility.

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Part I Models of Voter Behavior

Table 6.3. Log-likelihoods for nested models of voter choice: American NES 1980–96 Year

Proximity

RM

Matthews

-90.8**

-99.7**

-89.3**

1984 (n = 1,074)

-226.6**

-253.7**

-243.0**

1988 (n = 910)

-241.5*

-255.7**

1992 (n = 890)

-185.3

1996 (n = 891)

-137.8*

1980 (n = 428)

Matt.–prox.

Matt.–RM

Unified

-88.8**

-85.1

-222.5

-240.8**

-221.1

-249.5**

-239.8

-248.9**

-238.5

-206.0**

-204.2**

-185.2

-202.2**

-185.2

-152.2**

-146.3**

-136.4

-145.7**

-135.6

-87.5*

* Log-likelihood significantly lower than that of the unified model at the 0.05 level. ** Log-likelihood significantly lower than that of the unified model at the 0.01 level.

(b = 1.22). The hypothesis b = 0 is rejected at the 0.05 level for each election except that of 1980, whereas the hypothesis b = 1 is rejected for all but 1992, suggesting a model with both proximity and directional components. For the RM–Matthews submodel (Table 6.2), the values of q vary from 0.21 to 0.33, except for 1980 (q = -0.21). Although the hypothesis that q = 0 is rejected for only one election (1984), the overall results suggest that q is intermediate between 0 (a pure Matthews model) and 1 (a pure RM model), but closer to the Matthews version. Log-likelihoods are presented in Table 6.3. The log-likelihoods for the proximity–Matthews submodel is significantly lower11 than that for the full unified model only for one election (1980), whereas, by contrast, the log-likelihood for the RM–Matthews submodel is significantly lower for all five elections. Thus, the log-likelihood analy11

The probabilistic voter choice model was also fitted with the partisanship term omitted. This was done for two sample elections: 1984 in which the candidate-specific constant C was large and 1992 in which it was near zero. Whereas log-likelihoods are substantially lower (as might be expected following the omission of a significant term), parameter estimates are approximately the same as in the model with the partisanship term. Estimates for b rise by about 0.10 while those for q drop by about the same amount.

Probabilistic Models in Two-Party Electorates

89

sis suggests that the intensity aspect of the RM model is not needed, whereas both the Matthews and proximity components are. Comparing next the proximity–Matthews combination with each of its pure submodels, we see that the log-likelihood of the former is significantly higher than that for the pure proximity model for 1980 and 1984 and significantly higher than that for the Matthews submodel for all elections except 1980. We conclude that both proximity and Matthews directional components are necessary and that together they provide essentially the full explanatory power of the unified model for these data. 6.4 Discussion A priori, it might appear that proximity is more likely to be a good predictor of voter choice, whereas intensity might be more likely to affect the nature of voter utility functions. Thus, we might think that the intensity aspects of the RM directional model would be more prominent in fitting observed voter utility functions than would proximity aspects, but the reverse might be true in predicting voter choices. In comparison with the results in Chapter 5, the estimates obtained in this chapter suggest that there is some trend in this direction. Overall, however, fitting the voter choice model to the twocandidate American experience gives us no reason to question the empirical results from fitting the voter utility model obtained in Chapter 5, which gave support for a mixed proximity and directional model and with preference for the Matthews rather than the RM version of the directional component. In the voter choice model, the intensity aspect of the RM utility is not supported. Analysis of a probabilistic model of voter choice – with or without the effect of partisan proclivity – indicates that both proximity and Matthews directional components are significant but that the addition of an intensity component (à la the RM model) does not significantly increase the model fit. For American elections with two candidates placed at roughly equal distances from the neutral point on most issues, the indifference lines between the candidates do not differ much between models. In fact, if the candidates were exactly equidistant and on opposite sides of the neutral point on all issues, these indifference lines would all

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Part I Models of Voter Behavior

pass through the neutral point and be identical. In multicandidate elections, however, there are many indifference lines between the pairs of candidates – several of whom may be on the same side of the neutral point – providing a much richer opportunity to distinguish between models of voter choice. It is in this setting that the voter choice model is of primary value, and it is this setting to which we turn in the next chapter.

CHAPTER 7

Empirical Fitting of Probabilistic Models of Voter Choice in Multiparty Electorates The political parties created democracy and . . . modern democracy is unthinkable save in terms of the parties. E. E. Schattschneider, Party Government (1942: 1)

7.1 Multiparty Elections In this chapter we fit the conditional logit model of voter choice developed in Chapter 6 – a model that involves both probabilistic voting and the effects of party identification – in the setting in which it is most appropriate: multiparty contests. For this we turn first to the multiparty parliamentary elections of Norway and Sweden, in which voting is for parties, seat assignment is by proportional representation, and party allegiance is relatively strong. Second, we study a presidential election in multiparty France, where voting is for candidates, not parties, and in which the electorate is characterized by more fluid party identification. Data are taken from the Norwegian Election Studies of 1985, 1989, and 1993, the Swedish Election Study of 1979, and the French Presidential Election Survey of 1988. Parameter estimation is by maximum likelihood, as outlined in Chapter 6. In a multiparty election, deterministic models of voter choice, which assign all voters at a given spatial position to only one of the several parties, fit the data so poorly that discrimination between proximity and directional realizations of these models is unconvincing. By introducing into the model an unmeasured non-issue variable in the form of a probabilistic component, as well as incorporating a measure of partisan proclivity, we can obtain an adequate fit to the response data. Under these conditions, we are in a much better position to attempt to discriminate among alternative spatial models. 91

92

Part I Models of Voter Behavior

7.2 Mixed Deterministic and Probabilistic Models We have seen empirically (Chapters 4, 5, and 6) that a mixture of directional and proximity models may provide a more realistic description of voting behavior than any pure model alone. Here we will refer to the unified model of voter utility (see Chapter 3) as the deterministic unified model. Strict adherence to preference based on a deterministic utility function, however, yields misleading estimates of the distribution of party support by spatial position. In particular, when we observe actual data, regions of support overlap markedly among parties, a phenomenon not predicted by a deterministic model that assigns all voters in any spatial position to a single party (as done by Rabinowitz, Macdonald, and Listhaug, 1991, in their study of the Norwegian electorate for the years 1969, 1973, 1981, and 1985). Moreover, the RM model predicts that spatially intermediate or “centrist” candidates would receive few votes. Yet in France, for example, spatially intermediate or “centrist” candidates like Raymond Barre and even Jacques Chirac (when compared to extremist JeanMarie Le Pen) do quite well. Thus, Pierce (1995a, Ch. 5, especially Table 5.4; 1997, Table 3) argues that deterministic predictions of voter choice under the RM model are unreasonable for multiparty France. Prediction errors of these types are eased somewhat if the RM model is replaced by a proximity model, but even here centrist candidates tend to be squeezed from both left and right and, even under the proximity model, are predicted to receive an unrealistically small vote share. For the 1988 French Presidential Survey, Table 7.1 – which is adapted from Pierce (1995a) – provides for each model the distribution of first preferences on the salient issue of immigration. Columns 1 and 3 of the table omit voters for whom the model predicts ties for first preferences; columns 2 and 4 apportion these voters among the predicted first preferences. Pierce argues that many voters with tied first preferences are indifferent on the choice and thus should be omitted. Whether they are or not, it is apparent that deterministic models underestimate the support observed for spatially intermediate candidates. Voter decision making in a multiparty electorate is typically not independent of the historical context. Voters develop allegiances to particular parties over time that influence their decisions, as do issue

Probabilistic Models in Multiparty Electorates

93

Table 7.1. Model-predicted vote share for the 1988 French Presidential Survey, based on the issue of immigration (in percent) (n = 1,013) Proximity model

RM model

Preferred candidate

(1) Ties omitted

(2) Ties apportioned

(3) Ties omitted

(4) Ties apportioned

Mitterrand Barre Chirac Le Pen

26.8 7.7 9.9 14.1

33.1 19.6 21.3 16.6

29.1 0.5 2.2 27.6

32.4 4.5 6.0 29.1

Source: Adapted from Pierce (1995a: Table 5.4), with permission, the University of Michigan Press. Meaningless four-way ties and missing data have been omitted in all columns.

positions in a particular election. To capture this influence for Scandinavian elections – in which a voter votes for a party, not a candidate – we have eschewed what may appear the most obvious item on the typical questionnaire, namely, current party affiliation. For a respondent who reports any affiliation, the answer to this question is normally identical to the party for whom the voter voted; many voters may interpret the two questions as seeking the same information.1 Instead, we specify party ID as the party for whom the respondent voted in the immediately previous election.2 This historical variable 1

2

That this is true more broadly is suggested by Rose and McAllister (1990: 156), who conclude that “In Britain . . . questions about party identification and voting are but two ways of measuring the same thing.” We thank J. Adams for pointing out this reference. In the Norwegian Election Studies of 1989 and 1993, 60.0 and 54.5 percent of respondents, respectively, reported affiliation with the party for whom they voted. However, most of the others reported no affiliation at all (only 6.6 and 5.4 percent, respectively, reported affiliation with a party different from the one for whom they voted). In 1989, 62.1 percent recalled voting for the same party in the previous election; in 1993, the percent was 61.1. Most of the others reported actually switching parties (24.5 percent of all respondents in 1989 and 28.0 percent in 1993) as opposed to not voting or not recalling how they voted. The similarity between the transition vote matrix relating recalled vote in 1985 with actual vote in 1989 in the present study with the corresponding matrix for a Norwegian panel study conducted by Aardal (1990) of respondents interviewed in 1985 and again in 1989 indicates that recalled vote may be reasonably accurate as an indicator of party ID at least in the aggregate.

94

Part I Models of Voter Behavior

reports an actual deed (although, of course, the report could be incorrect) and plays the role of a control variable; the spatial-model variable is left to explain why the voter might vote differently in the current election. In our analysis of the French presidential election of 1988, however, we do use current party affiliation for party ID. This is because French voters vote for a presidential candidate rather than for a party and because historically French party allegiance has been relatively weak and hence is not expected to overwhelm voters’ choices of candidates. Accordingly, we incorporate one control variable (party ID) along with a probabilistic component (representing unmeasured, non-issue variables) into each spatial model. Following the discussion in Chapter 6, we employ a conditional logit analysis.3 The random component of utility of voter Vj for candidate Ci is denoted by Xij. The deterministic component of utility, dij, is the sum of a spatial-based utility, sij, and a measured, non-issue utility, tij, so that overall utility is defined by U(Vj, Cj) = Asij + Btij + Xij. We take the spatial-based utility to be that of the deterministic unified model (or of one of its nested components). The non-issue utility is based on the voter’s proclivity to vote for a party (party ID) estimated either by the respondent’s vote in the immediately previous election or by current party affiliation. Specifically, tij is 1 if voter Vj identifies with party Ci; otherwise it is 0. The probabilistic unified model is specified by these utilities. The probability pij that party Ci is preferred to all other parties by voter Vj can be expressed as a function of the deterministic utilities (see Chapter 6). To assess the conditional logit analysis, we need to address a methodological question. One of the principal observations of the Lewis and King (1998) paper is that if party-specific intercepts are included in a model, it is unidentified so that the mixing parameter(s) 3

The use of identically distributed random errors, although unrestrictive in the twocandidate case, is restrictive in the multiparty setting. Multinomial probit (MNP), which admits correlated error terms, does not appear feasible for the seven-party Norwegian electorate, which constitutes our primary application of the conditional logit analysis. To our knowledge, all previous applications of MNP to voting behavior involve three or five alternatives (see Alvarez and Nagler, 1995, 1998; Dow, 1997; Schofield, Martin, Quinn, and Whitford, 1998). Furthermore, Dow (1998b) points out that MNP estimates may be unstable and argues that any advantages of MNP are marginal when analyzing a fixed set of candidates or parties. Quinn and Martin (1998) draw similar conclusions.

Probabilistic Models in Multiparty Electorates

95

cannot be estimated. If we exclude intercepts from the model, estimates of the mixing parameter(s) can be made, but now they are biased. We can still ask, however, whether we can get a handle on the size of the bias that is to be expected for the type of data that we are likely to fit. Using simulation analysis (see Appendix 7.1), we show that this bias is relatively small, so that the estimates are still useful. Accordingly, we include no party-specific intercepts in the model. 7.3 Fitting the Conditional Logit Model to Norwegian Data We focus first on the Norwegian election studies of 1989 and 1993 conducted by the Norwegian Social Science Data Services (NSD). Each of these election studies asked respondents to place themselves and the parties on scales from 1 to 10 for each of several issues (seven in 1989 and five in 1993). Both studies include a left–right question (which tends to assess the economic dimension) and questions on environmental and immigration policy. The 1989 study also assesses policy in the areas of agriculture, health care, alcohol, and crime; the 1993 survey includes questions about unemployment and entry into the European Economic Community. Seven major parties in Norway are considered, from left to right: Socialist, Labor, Liberal, Center, Christian, Conservative, and Progress. For each study, respondents were restricted to those who preferred one of the seven major parties and placed themselves and each major party on the 1–10 left–right scale. Model utilities were normalized by dividing by the number of usable issues for each respondent (see Appendix 3.1). English language translations of the wording of typical questions are given in Appendix 7.2. Out of 2,196 respondents in 1989, 1,565 were usable; in 1993, 1,446 out of 2,194 were usable. In 1985, only a left–right scale was available; 1,706 of 2,180 respondents were usable. For Norway in 1989 and 1993, Tables 7.2 and 7.3 present maximum likelihood estimates for the model parameters, b, q, or d, as well as the salience parameters, A and B, for the spatial and nonspatial measured components of utility, respectively. Estimates for the full model as well as for nested submodels are provided. For both election years, inclusion of either a proximity or an RM component in the party ID model substantially raises the log-

96

Part I Models of Voter Behavior

Table 7.2. Parameter estimates for the unified model of voter choice and its submodels (seven issue dimensions): Norwegian Election Study 1989 (n = 1,565) Model

A

B

q

b

Log-likelihood

d

Estimates from voter-specific placements of candidates: Proximity–RM

.24 (.01)

— —

.75 (.03)

— —

— —

-2,140.2

— —

2.76 (.06)

— —

— —

— —

-1,883.1

Proximity–party ID

.17 (.01)

2.23 (.07)

1 —

— —

—

-1,545.6

RM–party ID

.14 (.01)

2.48 (.07)

0 —

— —

— —

-1,540.1

Proximity–RM– party ID

.18 (.01)

2.30 (.07)

.51 (.05)

1 —

— —

-1,499.3

Proximity–Matt.– party ID

.89 (.08)

2.22 (.07)

.08 (.02)

0

—

-1,502.2

Grofman discounting

.34 (.03)

2.29 (.07)

— —

— —

.54 (.05)

-1,497.3

Unified model

.44 (.09)

2.28 (.07)

.15 (.05)

.45 (.11)

— —

-1,491.8

Party ID

Estimates from mean placements of candidates: Proximity–RM– party ID

.15 (.01)

2.42 (.07)

.56 (.06)

1 —

— —

-1,609.3

Unified model

.09 (.01)

2.42 (.07)

.41 (.07)

.16 (.34)

— —

-1,603.6

Note: Parameter values in italics are set by design.

likelihood, as does the introduction of a party ID term (i.e., a term that accounts for recalled party vote) into a purely spatial model. In turn, while controlling for a party ID effect, the mixed proximity–RM model raises the log-likelihood considerably over either a pure proximity or pure RM model in 1989. In 1993, however, the corresponding log-likelihood for the mixed proximity–RM model is

Probabilistic Models in Multiparty Electorates

97

Table 7.3. Parameter estimates for the unified model of voter choice and its submodels (five issue dimensions): Norwegian Election Study 1993 (n = 1,446) Model

A

B

q

b

Log-likelihood

d

Estimates from voter-specific placements of candidates: Proximity–RM

.14 (.01)

— —

.79 (.05)

— —

— —

-2,259.9

— —

2.60 (.06)

— —

— —

— —

-1,823.9

Proximity–party ID

.11 (.01)

2.38 (.07)

1 —

— —

—

-1,553.9

RM–party ID

.09 (.01)

2.37 (.06)

0 —

— —

— —

-1,634.9

Proximity–RM– party ID

.11 (.01)

2.39 (.07)

1.06 (.08)

— —

— —

-1,553.6

Proximity–Matt.– party ID

.44 (.06)

2.36 (.07)

.17 (.04)

0

—

-1,537.5

Grofman discounting

.10 (.01)

2.40 (.07)

— —

— —

1.19 (.09)

-1,538.8

Unified model

.87 (.14)

2.37 (.07)

.09 (.02)

-.54 (.14)

— —

-1,529.5

Party ID

Note: Parameter values in italics are set by design.

insignificantly different from that of the proximity version.4 This contrast between the years is reflected in the estimates for the mixing parameter, b. This estimate is 0.51 in 1989, suggesting contributions of both proximity and RM effects.5 In 1993, the estimate is 1.06, indicating no reason to reject a pure proximity model. 4

5

For nested models that differ by a single parameter, log-likelihoods must differ by at least 1.92 to be significantly different at the 0.05 level. The estimates for b for the voter utility model of Chapter 5 range from 0.02 (for Labor) to 0.37 (for the Liberal party). The estimate for b in the voter choice model is higher as expected, because it does not depend on interpersonal comparisons of utilities and thus avoids a bias against the proximity model (see also Appendix 5.3 on the Westholm adjustment). An alternative model was also run for the 1989

98

Part I Models of Voter Behavior

Fitting the full unified model of voter choice permits us to assess the intensity factor, q.We might anticipate that this would be a difficult factor to estimate using a model of voter choice, which by nature focuses on choice rather than intensity of utility, and indeed the results are problematic. For 1989, we obtain an estimate of 0.45 for q (with an increase of 10.4 in log-likelihood over the value for no intensity), suggesting intensity as a moderate factor. In 1993, a two-dimensional model including only the key issues of left–right (a proxy for the economic dimension) and opinion on the EEC (a key issue in 1993) yields an estimate of 0.09 for q and 0.13 for b, suggesting a Matthews model with a proximity constraint. Parameter estimates for this two-dimensional model are given in Table 7.4.6 The theory developed in Section 4.3 indicates that, if the neutral point and the status quo point are assumed the same, the discounting factor d of the Grofman discounting model and the mixing parameter b of the mixed proximity–RM model should be the same. In fact, estimates for these two parameters match very closely for the Norwegian data for both 1989 and 1993 (see Tables 7.2–7.4). For the earlier year, d and b are 0.54 and 0.51, respectively; for the later year, they are 1.19 and 1.06, respectively, for the five-dimensional model and 0.80 and 0.68, respectively, for the two-dimensional version. We cannot even in principle expect to distinguish between these two models using voter choice (as opposed to voter utility).7 Note,

6

7

Norwegian data using separate salience parameters for each issue. Four issues were used (left–right, agricultural policy, alcohol policy, and immigration policy), each of which is known to be of particular importance to at least one party. The resulting salience parameter for left–right is about four to five times higher than that of each of the other issues, but the estimate of 0.51 for the mixing parameter b is the same as in the model above. For 1993, the estimate for q in the model with five issue dimensions is -0.54 (with an increase of 8.0 in log-likelihood), a peculiar value that is difficult to interpret because of the negative sign. It may be that models fit with issue-specific salience parameters would estimate q more realiably. Note that the value of the spatial salience parameter A is unusually large to compensate for the very small values of ||V||||C||-0.54. In particular, the size of this parameter cannot be used to compare the importance of the spatial component between models. Rabinowitz, Macdonald, and Listhaug (1993) suggest, however, that we might discriminate between these models in a cross-national analysis, since discounting – but not directional voting – might be reduced in a highly disciplined polity such as that of Britain in which one party is expected to form a majority government. Because of concomitant factors, we are not sure that this method would be feasible.

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Table 7.4. Parameter estimates for the unified model of voter choice and its submodels (two issue dimensions): Norwegian Election Study 1993 (n = 1,446) Model

A

B

q

b

Log-likelihood

d

Estimates from voter-specific placements of candidates: Proximity–RM

.8 (.003)

— —

.46 (.07)

— —

— —

-2,391.4

— —

2.60 (.06)

— —

— —

— —

-1,823.9

Proximity– party ID

.06 (.003)

2.51 (.07)

1 —

— —

—

-1,607.4

RM– party ID

.06 (.003)

2.47 (.07)

0 —

— —

— —

-1,621.8

Proximity–RM– party ID

.06 (.004)

2.49 (.07)

.68 (.11)

1 —

— —

-1,603.1

Proximity–Matt.– party ID

.28 (.04)

2.48 (.07)

.13 (.03)

0

—

-1,590.1

Grofman discounting

.08 (.01)

2.49 (.07)

— —

— —

.80 (.11)

-1,596.9

Unified model

.26 (.05)

2.48 (.07)

.13 (.04)

.09 (.14)

— —

-1,589.8

Party ID

Notes: The two issues are left–right and opinion on the EEC. Parameter values in italics are set by design.

however, if voter behavior is interpreted as discounting, that discounting is substantial in 1989 and may have been moderately strong in 1993 if estimation is based on the two most prominent issues. The Grofman discounting model specifies, however, that the discounting occurs relative to the status quo point, which need not be the neutral point. Lacking any direct estimate of the status quo point from the Norwegian data, we attempted to estimate the location of the status quo from the data using maximal likelihood. For 1989 and 1993, respectively, these estimates are 0.2 and 1.0 units to the left of the center point of the 1–10 left–right scale. Only the latter

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Part I Models of Voter Behavior

displacement is significant (barely) at the 0.05 level. In neither year were the estimates of the other parameters affected. Although a Labor government was in place at the time of each of these elections, in each case it was a minority government that had come to power after a nonsocialist coalition government had failed8 (Aardal, 1990, 19949). Given this uncertainty, it makes sense that the displacements of the status quo from the neutral point are relatively small but to the left. The smaller displacement in 1989 may reflect the focus of the campaign that year in which both the Labor and Conservative parties sparred with the Progress party on social issues rather than between themselves on traditional economic issues captured by the left–right scale. In the Norwegian election of 1993 a major new issue arose: Should Norway join the European Economic Community (EEC)? This issue produced cleavages among the parties (see Aardal, 1994; Valen, 1994; Strøm and Svåsand, 1997; Macdonald, Rabinowitz, and Listhaug, 1998b) quite different from those traditionally generated by the left–right dimension associated most strongly with economic outlook (see Figure 7.1). Perhaps the strongest stand (against entry into EEC) was taken by the Center party, whose very name signals a middleof-the-road position (on the economic dimension). At the same time, Labor adopted a position (for entry into EEC) that placed it near the two traditionally conservative parties (Conservative and Progress). A conditional logit mixed proximity–RM model with two issues: left–right and opinion on EEC yields an estimate of 0.68 for b (see Table 7.4), which is intermediate between the rather extreme estimates of 1.19 and -0.28, respectively, for one-dimensional models with left–right alone and EEC alone. We comment further on these estimates below. According to the pure deterministic RM model, the two most extreme parties in Norway – the Socialist and Progress – should receive all the votes. In fact, their combined total is only 9.6 percent of the vote in 1985, 24.5 percent in 1989, and 14.7 percent in 1993. Referring to the 1985 election, Rabinowitz, Macdonald, and Listhaug (1991) classify both of these parties as being “beyond the region of 8

9

Under Norwegian procedure, the failure of a government does not precipitate a new election; rather, elections are held on a four-year schedule. The title of the Aardal (1994) paper incorrectly refers to the Storting election of 1993 as the election of 1994.

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Figure 7.1 Scatter plot of Norwegian parties in 1993 with respect to left–right and EEC scales.

acceptability.” Although they consider the “region of acceptability” (see Chapter 3) to be part of the definition of their directional model, the location of the region is arbitrary. Although the parties they exclude are more extreme than those they include, the specified selection is not endogenous to the model. A mixed proximity–RM model, however, if interpreted as incorporation of a proximity constraint into a directional model, builds into the latter model a penalty for extremism (or excessive intensity) and it does so in a way that is voter specific, rather than imposing it via a single ad hoc region of acceptability that applies equally to all voters. A secondary question is still of interest: Why is the RM component of the mixed proximity–RM model for the 1989 Norwegian election marginally greater than that in 1993? Let us first focus on the election of 1989. In that election the two extreme parties – Socialist and Progress – strikingly increased their share of the vote (compared to that in 1985), as the RM model predicts, cutting into the support of the more moderate Conservative and Labor parties. Although Norwegian politics remained relatively stable by inter-

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national comparison, panel studies analyzed by Aardal (1990) suggest unusual volatility among voters during the period 1985–89. As the 1989 election approached, the economic outlook became bleaker than it had been for many years, with unemployment rising to its highest level during the postwar period while immigration from Third World countries increased dramatically. The Progress party – the only party to oppose the official immigration policy – clearly benefited from the uncertainty in the minds of many voters, whereas the Socialists benefited from the economic unrest as well as a realignment of the Liberal party with the nonsocialist coalition. By contrast, in 1993 the traditional extreme parties (Socialist and Progress) lost their hold on the periphery due to the new issue of EEC entry. The Center party, with its strong opposition to EEC entry, reaped the benefits among opponents to entry. Estimates of b for 1993 reflect the proximity model on the left–right dimension in part because the vote share of the traditional extremist parties fell while that of the Center party increased. By contrast, on the EEC dimension, estimates for b reflect a directional model because of the increased strength of the Center party with its newly “extremist” position on the EEC issue. Merrill (1995) – in a study of Scandinavian elections – incorporates strategic voting (but not party ID) into a probabilistic model, under the assumption that voters, ceteris paribus, are more likely to vote for larger parties.10 He obtains support for a utility function intermediate between proximity and directional motivations (see Appendix 7.3). We prefer the conditional logit model because it is based on voterspecific information about historical voter choice as opposed to the aggregate information used in the strategic model and because strategic considerations are only one reason why a voter might choose to support a previously strong party. Using conditional logit, Iversen (1994) also fits the proximity and RM models for data sets from five multiparty European democracies: Belgium, Britain, Denmark, West Germany, and the Netherlands. He avoids the methodological questions concerning voter placements of 10

Such an assumption is based on the rationale that votes for smaller parties might in some cases be considered wasted, or of less value. Schofield (1993) and Schofield et al. (1998a) place the emphasis on the motivations of party leaders for postelection coalition building.

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candidates by using mean placement of candidates by a presumably more knowledgeable, elite group (party officials), which he obtains from the European Political Party Middle Elite study of delegates to the European party conference in 1979.11 Using binary independent variables for intensity and proximity (but not controlling for party ID), Iversen obtains slightly larger coefficients for intensity for Belgium and Britain and much larger coefficients for intensity than proximity for the other three electorates. He also performs a binary logit analysis for each individual party, obtaining a higher intensity coefficient for 13 of the 18 parties studied. He concludes that both proximity and directional effects are important for voting behavior. Because of his use of binary independent variables, however, his results are hard to compare with those of others. 7.4 Fitting the Conditional Logit Model to French Data To analyze the data in the 1988 French Presidential Election Survey (Pierce, 1995b), we focus on the five major candidates: Andre Lajoinie (Communist), François Mitterrand (Socialist), Raymond Barre (UDF), Jacques Chirac (RPR), and Jean-Marie Le Pen (National Front). Respondents were asked to place themselves and the five candidates on four seven-point scales12 representing left–right, church/ schools, public sector (government intervention in the economy), and immigration. Limiting respondents to those who reported selfplacements and all five candidate placements on the left–right dimension reduced the data set from 1,013 to 677 cases. For this four-dimensional model, Table 7.5 reports maximum likelihood estimates for the same range of model specifications used in the Norwegian study. Perhaps the most striking observation is the strong similarity between the French and Norwegian results. For the mixed proximity–RM model (with party ID control), the estimate of 0.59 for b suggests a model intermediate between the two pure extremes.13 11

12

13

Whatever the objective merits of elite placements, they need not be the locations upon which individual voters make judgments, and hence may introduce measurement error. Each scale was rescaled to range from -3 to +3, with -3 representing the leftmost position and +3 the rightmost position. Use of mean rather than idiosyncratic placements of candidates yields an almost identical estimate of 0.63 for the mixing parameter b, as indicated in Table 7.5.

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Part I Models of Voter Behavior

Table 7.5. Parameter estimates for the unified model of voter choice and its submodels (four issue dimensions): French Presidential Election Study 1988 (n = 677) Model

A

B

q

b

Log-likelihood

d

Estimates from voter-specific placements of candidates: Proximity–RM

.34 (.02)

— —

.92 (.06)

— —

— —

-733.0

— —

2.77 (.11)

— —

— —

— —

-619.6

Proximity– party ID

.19 (.02)

2.04 (.11)

1 —

— —

— —

-535.6

RM– party ID

.18 (.02)

2.32 (.11)

0 —

— —

— —

-540.9

Proximity–RM– party ID

.21 (.02)

2.09 (.12)

.59 (.12)

1 —

— —

-530.2

Proximity–Matt.– party ID

.69 (.10)

2.00 (.11)

.10 (.05)

0 —

— —

-519.2

Grofman discounting

.33 (.07)

2.06 (.12)

— —

— —

.66 (.12)

-527.4

Unified model

.72 (.13)

2.00 (.12)

.10 (.05)

-.06 (.16)

— —

-519.1

Party ID

Estimates from mean placements of candidates: Proximity–RM– party ID

.22 (.03)

2.33 (.12)

.63 (.18)

1 —

— —

-563.5

Unified model

.29 (.12)

2.31 (.12)

.44 (.28)

.55 (.66)

— —

-563.4

Notes: The four issues are left–right, church/schools, public sector, and immigration. Parameter values in italics are set by design.

Similarly, the estimate for the Grofman discounting model suggests moderate discounting (d = 0.66). This value is similar to those for Norway, with the exception of the five-issue analysis of the 1993 Norwegian election. Although the unified model estimates for q appear variable between Norway and France and between elections

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105

in Norway, we do not place great weight on multidimensional estimates for q, because, as we noted earlier, we suspect that a model with issue-specific salience parameters may be needed. 7.5 Discussion and Conclusions Overall, the conditional logit model of voter choice supports a mixed model in both Norway and France. Estimates of the mixing parameter suggest utilities intermediate between pure proximity and directional models – if not for every election, at least over a series of elections. These conclusions are sustained whether or not a party ID variable is included in the model and whether mean or vote-specific party placements are used. In fact, the latter dichotomy makes almost no difference at all, as can be seen by inspection of the estimates of b for Norway in 1989 and France in 1988 (see Tables 7.2 and 7.5). This contrasts with the significant impact of mean versus voterspecific candidate placement when thermometer scores are used in a voter utility model (see Chapters 4 and 5). Thus, this sometimesdisputed methodological question of mean placement versus voterspecific placement has little impact on the empirical assessment of voter choice models.14 In Chapter 10, we shall see yet another confirmation of the use of a party choice model and the mixed model it predicts. These results suggest that over a period of elections, both proximity and directional aspects are significant in explaining voter choice. When new issues arise in an election, a directional utility may better model enhanced support at the extremes. Given long-term stability, however, an election such as that of 1989 may be followed by a return to support for the center or near-center and is best explained by the proximity model. The analysis in this chapter provides further evidence that both the pure proximity and the pure directional models fall short of accounting for the voter preferences. In most cases, a probabilistic model with a party ID, strategic, and/or other component is necessary to obtain an adequate fit to the data. A mixed model, involving a trade-off 14

Likewise, the adjustment by subtracting means used by Westholm (see Appendix 5.3) has little impact, because voter choice models are largely independent of interpersonal comparisons.

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Part I Models of Voter Behavior

between the centrifugal effects of the RM model and the centripetal tendencies of the proximity model, may be one step in explaining the success of “moderately extreme” parties. We will return to this point in our analysis of party strategy in Chapter 10.

PART II

Models of Party or Candidate Behavior and Strategy

CHAPTER 8

Equilibrium Strategies for Two-Candidate Directional Spatial Models Things fall apart; the center cannot hold. William Butler Yeats, “The Second Coming”

8.1 Stable Strategies In Part I of this book, we focused on the effects of issues on the behavior of voters. Much of our motivation for doing so rests on the implications of this behavior on candidate strategy. What do our conclusions about models of voter choice say about whether candidates and parties would benefit from choosing identical ideological positions as opposed to divergent ones? We have seen that the RM model is more likely to predict divergent strategies than the proximity model; mixed models and the Grofman discounting model make intermediate predictions. A more fundamental question is: Are the strategies taken jointly by two or more candidates or parties stable, or does one party always have an incentive to defect? Otherwise put, does a set of stable strategies exist? Does the existence of stable strategies depend on the location of the neutral or status quo point, on the number of parties, on the dimension of the spatial model, or on the relative proportion of issue and non-issue voting? In this chapter we investigate stable strategies for each of two candidates in a single two-candidate contest, i.e., choices of policy positions from which neither candidate alone would have an incentive to deviate. Such strategies constitute a Nash equilibrium. In Chapter 9, we extend the idea of stability from a single election to a sequence of elections between two parties, developing a dynamic equilibrium that is implied by the discounting model, and, by extension, a mixed RM–proximity model. This equilibrium predicts strategies that are neither convergent nor wildly divergent. Chapter 10 investigates equi109

110

Part II Models of Party/Candidate Behavior

librium strategies for multicandidate competition. Such equilibria may exist and may be divergent in the presence of partisan voting. As we will see, comparison of equilibrium predictions for various models with the empirically observed strategies of the parties provides strong support for a mixed RM–proximity model. First, in the present chapter, we give conditions for the existence of both convergent and nonconvergent Nash equilibria in onedimensional versions of the Grofman discounting model and the RM model with proximity constraint. These conditions suggest that stable strategies may be intermediate between the convergent strategies (at the median) of the proximity model and the divergent strategies of the RM model. Second, we investigate Nash equilibrium strategies for the Matthews directional model. We provide a geometric characterization of the set of potential neutral points for which there is a directional equilibrium. 8.2 Nash Equilibrium under the Grofman Discounting Model and Constrained Directional Models In the one-dimensional case, for two candidates and the proximity model, the median voter theorem implies that stability occurs only when both candidates place themselves at the location of the median voter. Any position other than the median location can be beaten by a candidate at the median, whereas if both assume that position, no unilateral deviation would advantage either party. By contrast, under the RM (scalar product) directional model, the median position is not a stable point unless it happens to coincide with the neutral point. In general, ever more extreme positions on the median side of the neutral point are vote maximizing, unless a circle of acceptability is imposed, in which case a point on the circle is the equilibrium point. Using the shadow methodology introduced in Chapter 3, we will show that Nash equilibria can be specified for a wide class of models, including directional and discounting models. Except under strong symmetry assumptions, the Grofman discounting model and certain directional models involving a mixture of proximity and directional components predict moderate divergence, a more realistic expectation

Equilibrium Strategies for Two-Candidate Models

111

Figure 8.1 Nash equilibrium under the Grofman discounting model. (Source: Merrill and Grofman, 1998a: Figure 3.)

than the strict convergence of the pure proximity model or the strong divergence of the pure RM model. For higher dimensions, the shadow methodology is extended via the concept of the yolk. First we extend the median voter theorem about candidate equilibria in a one-dimensional Downsian setting to the discounting model (see Merrill and Grofman, 1998a). Under the Grofman discounting model for one dimension, a Nash equilibrium is obtained if the candidates locate at points such that the electorate expects them, after discounting, to implement policy at the position of the median voter, because in turn the voters will behave as if both candidates are located at the median, M.1 If S is taken as the origin, then both candidates must be positioned so that M is the discounted position of each. M may be interpreted as the (common) shadow of both candidates. Accordingly, the Nash equilibrium strategies may be divergent, but typically by only a modest amount if discounting is not excessive (see the example depicted in Figure 8.1, where the respective discount factors are 0.75 and 0.5).2 If the discounting factor is the same for both candidates, then the equilibrium strategies are convergent, i.e., the same for both candidates.Thus, the optimal strategies for the Grofman discounting model appear more like that of the proximity model than 1

The Nash equilibrium occurs when candidates locate at points Ci, i = 1,2, such that M = di Ci + (1 - di )S

2

where M is the median voter, di is the discounting factor for candidate Ci, i = 1, 2, and S is the status quo (see Figure 8.1). The Nash equilibrium strategies are given by Ci = S + ( M - S ) di and thus differ by the quantity 1 1 - ˆ Ë d1 d2 ¯

( M - S )Ê

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Part II Models of Party/Candidate Behavior

the extreme predictions of the RM theory. We summarize in the following proposition. Proposition 8.1. Under the Grofman discounting model in one dimension, a Nash equilibrium exists and is given by Ci = S + (M - S)/di. If the discounting factors are between 0 and 1, the equiM -S . librium strategies cannot diverge by more than the quantity d1d2 We have seen that voter utilities under the Grofman discounting model with common discount factor d and S = N are equivalent to those for the directional model with proximity constraint, provided b = d. It follows that the equilibrium analysis above for the Grofman discounting model applies equally to these constrained directional models, yielding the following proposition. Proposition 8.2. Under the RM model with proximity constraint and mixing parameter b, a convergent Nash equilibrium exists in one dimension in which each candidate is displaced from the neutral point by the quantity (M - N)/b. Only if the voter median and the neutral point are identical will the Nash equilibrium be the same under a mixed RM–proximity model as it is under a pure proximity model. Iversen (1994) redefines the ideal point, V, for each voter as the point V/b. It is in this sense that he states that the median voter result is unchanged under a mixed RM–proximity model; in fact, the equilibrium – although convergent – need not occur at the median voter, as usually defined.3 Existence of a Nash equilibrium does not in general extend to proximity models in two or more dimensions, because median lines (planes) need not intersect at a single point. The concept of the yolk – the smallest disc that intersects all medians (McKelvey, 1986) – however, retains some vestigial properties of an equilibrium. For example, a candidate located at any point outside the yolk can be beaten (or tied) by some candidate in the yolk. The shadow and yolk constructions can be combined to yield 3

Iversen’s further claim that the median voter theorem holds for the pure RM model with region of acceptability also depends on his redefinition; in conventional terms, the equilibrium for any asymmetrical electorate occurs on the boundary of the region of acceptability.

Equilibrium Strategies for Two-Candidate Models

113

Figure 8.2 Yolk and pseudo-yolk.

similar information about discounting and directional models. Given a status quo point, which we take to be the origin, and a common discounting factor, we define the pseudo-yolk as the set of possible candidate locations whose discounted positions lie in the yolk (see Figure 8.2). Hence a candidate lies in the pseudo-yolk if and only if voters believe she will implement policy in the yolk. In other words, the yolk plays the role of shadow of the pseudo-yolk. The following proposition is then immediate. Proposition 8.3. For any number of dimensions, under the Grofman discounting model or RM model with proximity constraint, any candidate not in the pseudo-yolk can be beaten (or tied) by some candidate in the pseudo-yolk. For a one-dimensional RM model with proximity constraint, a Nash equilibrium is obtained if the candidates locate at a common point whose shadow position falls on the median voter M, because the voters will behave as if both candidates are located at M. Thus, the Nash equilibrium strategy for each candidate is given by the location M/b, which represents a dilation (by the factor 1/b). We have shown that, in one dimension, the RM model with proximity constraint has a convergent equilibrium offset from the median.This result can be compared and contrasted with the result about median convergence for the pure proximity model and extreme divergence for the pure RM model, respectively. In higher dimensions, insofar as com-

114

Part II Models of Party/Candidate Behavior

petition under proximity tends to be constrained to the yolk (or points reasonably close to it), competition in the directional and discounting models studied here will tend to occur in the pseudo-yolk. 8.3 Nash Equilibria under the Matthews Directional Model Under majority rule, an equilibrium (core) location is also known as a Condorcet winner, i.e., a position that defeats (or at least is not defeated by) any other alternative in a paired contest. Such a position ensures stability among rational players: If both players assume this position, neither has an incentive to deviate unilaterally. Likewise, there is no advantage in remaining away from such as equilibrium position, and generally there is a disadvantage in doing so. For two-candidate competition, we first focus on a directional analog to the Condorcet winner we call the Condorcet directional vector. We offer geometric conditions in terms of the number of medians through a point that are both necessary and sufficient for the existence of directional equilibria. Unlike equilibria in proximity voting, the existence or nonexistence of which is unconditional, equilibria in directional voting are dependent upon the exact location of the neutral point or status quo point from which the preferred direction of choice is calculated. We also show that all Condorcet directional vectors must pass through the yolk, which is generally a small and centrally located disc (McKelvey, l986; Feld, Grofman, and Miller, 1988; Tovey, l992). Historical notes are given in Appendix 8.1. The remainder of this chapter is more technical than the rest of the book. Readers who prefer to move directly to the dynamic analysis in Chapter 9 or the multicandiate analysis in Chapter 10 may do so without loss of continuity. It is well known that, in two (or more) dimensions, equilibrium under the standard proximity model occurs only if all median lines pass through a single point (Davis, DeGroot, and Hinich, 1972). Such a complete median will occur only under very restrictive symmetry conditions on the set of voter ideal points.4 4

For example, Plott (1967) showed that an equilibrium (complete median) M occurs in a two-dimensional electorate consisting of a (odd) finite number of voters if and only if one voter is located at M and all other voters come in pairs, with each pair on opposite sides of a line through M.

Equilibrium Strategies for Two-Candidate Models

115

In the Matthews directional model, our definition permits voters and candidates to assume any position in space but defines utility in terms of the normalized vectors of unit length from the neutral point N in the direction of V or C. Thus, political actors are assumed to behave as if they lay on the unit sphere centered at N.5 Accordingly, questions of equilibrium can be addressed by assuming that all voters and candidates have been replaced by unit vectors emanating from N. We assume that no voter is located at N. Unit vectors with endpoints on the unit sphere about N may be interpreted as directions in which a candidate may propose to move the status quo or in which a voter would like it moved. For convenience, while we consider a fixed neutral point N, we take N to be the origin. In one dimension, the question of existence of equilibria for the Matthews model is trivial: From any neutral point, the direction toward the median voter is optimal for both candidates (if the neutral point is the median, all strategies are in equilibrium). On the other hand, for three or more dimensions, instability is typical.6 Consider, for example, an electorate of three voters. From any neutral point outside the plane of this electorate, any direction can be beaten at least two to one by some other direction. We focus on the twodimensional case because we anticipate that in many settings political competition can be regarded as two-dimensional (see Hinich and Munger, 1994; Poole and Rosenthal, 1997).7 8.3.1 Characterization of Condorcet Directional Vectors in Two Dimensions Definition. A vector (candidate) C* relative to the neutral point N is a Condorcet directional vector if for any other vector C 5

6

7

In the RM model with circle of acceptability, any candidate A lying in the interior (or the exterior) of the circle but not at the neutral point can be dominated (or tied) by a second candidate B in the same (or opposite) direction but on the circle. Thus, all candidates – assuming rationality – can be expected to move to the circle of acceptability. Accepting this to be the case, voters should behave as they would under the Matthews model, so that in this sense, the equilibrium analysis we present applies also to the RM model with circle of acceptability. We are indebted to N. Schofield (personal communication) for calling to our attention previous work of his, upon which this observation is based. See Section 10.4 concerning the existence of equilibria for the Matthews model in multicandidate contests.

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Part II Models of Party/Candidate Behavior

Figure 8.3 Indifference line for candidates, C* and C, and characteristic vector, A, of the support set of C*.

relative to N, the proportion of the voters favoring C* is greater than or equal to the proportion favoring C. Under a directional model, the support set for a fixed candidate C* with opponent C is the half-plane whose boundary passes through the neutral point and represents the indifference line between the candidates. The support function for C* assigns to each such half-plane (i.e., to each potential opposing candidate) the number of voter points in the corresponding support set (see Figure 8.3). In this setting, a line is called a median if the number of voter points in each (closed) halfplane determined by the line is at least n/2, where n is the total number of voter points. A point N is a complete median if and only if all lines through N are medians. The following theorem is adapted from Matthews (1979). Proposition 8.4. A vector C* (emanating from N) is a Condorcet directional vector if and only if the proportion of voters in any half-plane whose normal vector lies within 90 degrees of C* is a majority.

Equilibrium Strategies for Two-Candidate Models

117

Proof. See Appendix 8.1. Existence of a Condorcet directional vector (i.e., a dominant position under a directional model) is more likely than existence of a complete median (i.e., a dominant position for a proximity model). Unlike the condition for the existence of a complete median, this condition is not knife-edge in nature, because the majority of voters specified in Proposition 8.4 may be a strict majority even if no voter points lie on the boundary of the half-plane. In two dimensions, Proposition 8.4 has the following interpretation. Any half-plane of support uniquely specifies an angle, g, representing the direction normal to the boundary of the half-plane. For each g, -p < g £ p, we define the value of the support function, g(g ), as the number of voters in the associated half-plane. The necessary and sufficient condition of Proposition 8.4 for C* to be a Condorcet directional vector becomes g(g ) ≥ g(g + p )

for

g 0 -p 2 £ g £ g 0 +p 2

where g0 is the directional angle of the vector C*. Geometrical conditions in terms of the number of median lines for there to be a Condorcet directional vector for a two-dimensional configuration of n voter points, P1, . . . , Pn, and the direction in which that vector will point are presented in Propositions 8.5 and 8.6 and their corollaries below. The necessary and sufficient conditions depend on the location of the neutral point – which we now allow to vary – and the parity of the number of voter points. The proofs are in Merrill, Grofman, and Feld (1999). Note that the points P1, . . . , Pn form a star (see Figure 8.4 and Appendix 8.1 for further details). Definition. The Condorcet vacuum is the set of possible neutral points N for which there exists no Condorcet directional vector. Proposition 8.5. Given any configuration of n voters, where n is odd, and a neutral point N other than one of the voter points, the following are equivalent. (i) N lies in the Condorcet vacuum. (ii) N lies in more than one open star angle. (iii) N lies inside the interior of the star.

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Part II Models of Party/Candidate Behavior

Figure 8.4 Star angles for small electorates. (Source: Merrill, Grofman, and Feld, 1999: Figure 1.)

If these conditions do not hold, a Condorcet directional vector exists and lies in the direction of the vertex of the star angle containing N (or whose vertical angle contains N), i.e., the Condorcet directional vector must point toward some particular voter. As we will see below, the Condorcet directional vector must pass through the yolk. Schofield and Tovey (1992) have shown that for sufficiently large electorates sampled from an underlying symmetric distribution, the measure of the Condorcet vacuum shrinks to zero. Even if the distribution is not symmetric about each neutral point, however, simulation results (see Merrill et al., 1999) suggest that, in two dimensions, Condorcet directional vectors will exist for N located at most points inside the Pareto set8 (see Appendix 8.1 for some of these simulation results). Moreover, for even moderately large odd n, the star angles will be narrow (see Figure 8.5, which presents a star for 201 voters generated uniformly on the unit disc). Disregarding the very thin lines radiating from the central portion of the star as being too narrow to be observable in practice and hence largely irrelevant, the region of the space in which candidates might be expected to locate is small and quite centrally located. 8

The Pareto set is the smallest convex set containing all voter points. For every point outside this set, some point inside is preferred by all voters.

Equilibrium Strategies for Two-Candidate Models

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Figure 8.5 Star for 201 voters uniformly distributed on disc. (Source: Merrill, Grofman, and Feld, 1999: Figure 2.)

The illustrative configuration of five voter points depicted in Figure 8.4 shows the star angles and labels the (odd) number of median lines passing through a point in each open region. N has no Condorcet directional vector if and only if N lies inside the open star. For n even, the condition for existence of a Condorcet directional vector is quite different from that for odd n. Let n = 2k. If no three voter points are collinear, through each voter point there passes one (or more) median, called an opposing median, which partitions the voter points into two exactly equal sets. Suppose that N does not lie

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on any opposing median and assume, without loss of generality, that N is the origin. Proposition 8.6. Given any configuration of n voters, where n is even, then a sufficient condition that a point N, not on an opposing median, not have a Condorcet directional vector is that N lie in some triangle formed by the intersection of three opposing medians, say m1, m2, and m3. If all voter points lie on a circle centered on N, this condition is also a necessary condition for N not to be a Condorcet directional vector. Figure 8.6 shows a plot of opposing medians for 200 voters generated uniformly on the unit disc. The pattern is remarkably similar to that of the star for a comparable odd number of voters. In two dimensions, the yolk is the smallest disc that intersects all median lines. In two dimensions, the directional yolk is the smallest disc that intersects all lines containing Condorcet directional vectors. Merrill et al. (1999) show that the directional yolk is the same as the yolk, so that all Condorcet directional vectors pass through the yolk. For two dimensions, the region specified by Proposition 8.6 within which no Condorcet directional vectors exist is the same as the electoral heart defined by Schofield (1993, 1996) as the union of the core and the cycle set.9 Schofield observes that around any point within the triangle formed by three intersecting medians there will exist a cycle. As we have seen in the directional context, from such a point no direction is undominated. Schofield (1996) suggests the heart as a solution set for a class of voting models including the directional ones considered here. For two dimensions, from any point outside the heart, the Condorcet directional vector exists and points toward the yolk. Instability is restricted to points internal to the heart. 8.3.2 The Condorcet Vacuum for American and Norwegian Data To assess empirically the patterns suggested in Figure 8.5 and 8.6, we look at a sample of 201 respondents from the 1992 American National 9

We thank N. Schofield for pointing out the mathematical identity of these two concepts.

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Figure 8.6 Star for 200 voters uniformly distributed on disc. (Source: Merrill, Grofman, and Feld, 1999: Figure 4.)

Election Study, making use of self-placements on a liberal–conservative scale and on a government services scale for the X and Y coordinates (see Figure 8.7).10 The basic pattern – together with the small central region representing instability – is similar between the simulated and empirical data. If, however, the electorate is far from symmetric – directionally tripolar for example – the Condorcet vacuum is much larger and in this case roughly triangular in shape, as seen in Figure 8.8. 10

To smooth the graininess of the seven-point scales, respondent positions were subjected to random scatter, e.g., individuals giving a response of 2 were scattered uniformly between 1.5 and 2.5. Scales were centered at 0 by subtracting 4.

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Figure 8.7 Star for 201 voters from the 1992 American NES.

An intermediate pattern emerges for a sample of 201 respondents in the 1989 Norwegian Election Study (see Chapter 7) using selfplacements on the left–right and alcohol policy scales for the X and Y coordinates (see Figure 8.9).11 These data were chosen because the scatter plot shows that the distribution is partially tripolar. In general, we infer that the central Condorcet vacuum (as opposed to the spokes) diminishes rapidly with electorate size as long as a significant number of voter points are interior to the Pareto set. This pattern can be partially disrupted, however, by marked tripolarity of 11

The ten-point scale was centered on zero by subtracting 5.5; positions were subjected to random scatter as described above.

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Figure 8.8 Star for 201 voters following a tripolar distribution.

the electorate. An alternative approach to equilibrium analysis is presented in Appendix 8.2. 8.4 Strategies when Different Models Govern Each Candidate The preferred strategies of candidates depend on the evaluative model or models with which voters view them. Heretofore, we have assumed that evaluation of both candidates is governed by the same model. Empirical evidence (see Chapters 4 and 5) concerning the theoretical models developed in Chapters 2 and 3 suggests that even

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Figure 8.9 Star for 201 voters from the 1989 Norwegian Election Study.

with only two candidates, voters may use different models in assessing each one, so that strategies need not be symmetric. First consider a two-candidate race with a single issue dimension. We denote by N the neutral or status quo point and by M the location of the median voter. Under a pure proximity model for both candidates, M is the optimal strategy for both candidates and constitutes a Nash equilibrium. Under a pure RM (scalar product) model for both candidates with no circle of acceptability, there is no equilibrium or stability: Any position can be beaten by a candidate on the median side of the original position (if M = N, all positions result in a tie). If the pure scalar product model is modified by a circle of acceptability,

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then the boundary of the circle on the median side of N is optimal for both players. Under the Matthews model, however, movement toward the median side of N is optimal for both so that any pair of locations on that side of N constitutes an equilibrium. For a one-dimensional RM model with proximity constraint (and mixing parameter b ), there is a Nash equilibrium (see Section 8.2), but it is neither at the median nor at the neutral point, but rather is shifted from the neutral point by a distance that is proportional to the difference between the median and the neutral point and inversely proportional to the size of the mixing parameter.12 For b strictly between 0 and 1, the equilibrium is further from N than M is; for example, if b = 1/2, it is twice as far from N as is M and in the same direction (cf. Figure 8.1). Thus, as the directional component of the model increases, the equilibrium is drawn away from the median but is not destroyed, becoming more extreme as b decreases to zero. Exactly the same equilibrium obtains for a Grofman discounting model with status quo S = N and common discounting factor equal to b. Overall, the proximity model for two candidates in one dimension implies convergence to the median; a directional model implies divergence, whereas a mixed model implies convergence to a position more extreme than the median. We now permit response to the two candidates to follow different models. For simplicity we will refer to the candidates as I and C, which may be interpreted as incumbent and challenger, respectively, but need not be. Strategic analysis in this setting does not lend itself to neat solutions. The observations that follow are speculative in nature. More definitive statements await further work. Suppose that voters evaluate I according to the proximity model but C according to the RM model. The utility functions for the respective candidates are no longer comparable, so that we cannot locate the indifference point between the candidates, nor determine which strategy is winning. What we can assess are the relative effects of movements of one candidate or the other. Suppose, for example, that both candidates start at the median position and that – without loss of generality – M is to the right of N (see 12

The location of the equilibrium is given by E = N + (M - N)/b.

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Figure 8.10 Strategy for challenger.

Figure 8.10). If C moves to the right, C (who depends on the RM utility) gains in utility for all voters to the right of N and loses for voters to the left. Other things being equal, this is an advantage to C because the median and hence most voters are to the right of N. But more significantly, C polarizes the issue-oriented electorate by moving further to the right, causing issue-oriented voter choice to become an increasing component of voter decision making. If C is behind her opponent in non-issue-based voter support, providing such an incentive for voters to make issue-based choices may help overcome this shortfall in support. Accordingly, challengers, who are often behind at least initially, tend to polarize the electorate by taking more extreme stands. If, instead, C is ahead in non-issue-based choices, polarizing the electorate may not help on balance; in other words, it may be best not to rock the boat. If, on the other hand, C moves to the left of the median, utility gains and losses reverse, with the contest becoming most bland as C crosses the neutral point (representing indifference). By moving in either direction from the median but without overtaking C, candidate I (dependent on the proximity utility) will lose utility from more voters than those from whom she will gain utility, so that movement may be unwise. If, however, I can get to the right of C, then C’s scalar-product-based support will be reduced (since M is to the right of N). Successfully doing this will depend on the credibility of such a move in light of I’s past positions, although simply moving to a position near that of C may be sufficient to neutralize C’s scalar-product advantage. For example, Bill Clinton – the incumbent in the 1996 U.S. presidential race – took a number of positions that appeared to be at least slightly to be right of center, with the effect of neutralizing potential Republican advantages. In summary, if I follows the proximity model and C the scalar product, I will typically choose a strategy near the median while C may benefit from moving away from the median, especially if C finds

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herself initially behind in the race. Similarly, if C depends on a mixed proximity–scalar-product model, such movement away from the center, but of a more limited nature, may be called for. 8.5 Conclusions We have begun to address the question of conditions under which stable or equilibrium strategies exist for two candidates or parties. For a single dimension, the Grofman discounting model implies that an equilibrium exists, but in general at a point more displaced from the status quo than is the median voter location. As we have seen, the same must hold for a mixed RM–proximity model. Although such an equilibrium may not persist in higher dimensions, a region (termed the pseudo-yolk) does exist that dominates points outside its boundaries. For the Matthews directional model in two dimensions, we can characterize those potential neutral points for which no direction constitutes an equilibrium as a star-shaped region with a relatively small central core. For neutral points outside this region, equilibrium directions appear to point into the region, attracting political activity into the center, within which instability can occur. Our results also show that when it makes sense to distinguish incumbents and challengers because voters perceive the two in different ways, then the response by parties and candidates may lead to different optimal positions for incumbents and challengers.

CHAPTER 9

Long-Term Dynamics of Voter Choice and Party Strategy Our political solar system, in short, has been characterized not by two equally competing suns, but by a sun and a moon. Lubell, The Future of American Politics (1952: 200)

9.1 Why Is There Limited Polarization and Alternation of Parties? Contrary to the expectations of the median voter theorem in one dimension, empirical evidence implies that the policies of the two major parties in American politics do not in fact converge (Brady and Lynn, 1973; Bullock and Brady, 1983; Poole and Rosenthal, 1984b; Alesina and Rosenthal, 1995: section 2.6). Why is this so? Alesina and Rosenthal (1995: 17) distinguish between the Downsian model in which “parties choose policy in order to win elections” and a partisan model in which “the parties want to be elected in order to choose policies.” This idea suggests a trade-off between a desire to win and a desire to implement policy that motivated Wittman (1977, 1983), Calvert (1985), and Brams and Merrill (1991). Their models also take account of uncertainty about the distribution of voter preferences. Each of these analyses assumes a Euclidean loss function for each party as the spatial location of implemented policy recedes from that of desired policy, and each draws a conclusion of partial convergence, i.e., optimal party strategies lie intermediate between those implied by pure vote seeking and pure policy seeking. Wittman emphasizes the breakdown of convergence under these assumptions, while Calvert points out that the departure from convergence is only gradual as the traditional assumptions are continuously relaxed. 128

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Alesina and Rosenthal (1995) emphasize the divergence aspect of partial convergence, particularly if parties’ claims of moderation during a campaign cannot be believed. This may occur because parties – once elected – have little incentive to move toward the center, but rather may revert to their preferred, more extreme ideological positions. In repeated elections, however, parties may stand to gain vote share in the long run by establishing a centrist reputation, although individual candidates may heavily discount the future. A variety of equilibria – convergent or divergent – may occur, depending on the amount of discounting of future election consequences. These authors also give several institutional reasons for party polarization, such as the influence of primaries and activists and the threat of entry of third parties (Palfrey, 1984), but develop most fully the balancing effect of divided government, whereby voters achieve moderation by supporting parties with different partisan objectives in different branches of government. Alesina and Rosenthal (1995) argue that in American politics, voting outcomes in both on-year and off-year elections are determined largely by the desire of moderate voters to obtain divided government and thereby avoid the extremes of governmental dominance by either party. They state that “Since policy outcomes depend on the results of both elections, voter behavior in one contest is not independent of the expected result in the other” (italics in original). Their model is driven by uncertainty in the minds of voters about who will win the presidential election. In deciding how to vote in an on-year election without this information, a moderate voter, they argue, does not know which legislative party to favor, whereas in off-years, the holder of the presidency is well-known. Those voters who misjudged the previous presidential election and voted for the party for congress that eventually won the presidency are likely to switch parties in the off-year congressional race. A corollary to this argument is that midterm switching should be greater, the greater the uncertainty in the previous presidential election. Thus, for example, if the outcome of a presidential election is a near certainty, we would expect little misjudging and no need to switch legislative support two years later. We can approximate the degree of certainty in a presidential race by the margin of victory of the eventual winner. Contrary to expectations of the model, we find midterm

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switching1 highest, not after closely contested elections, but rather after landslide elections. For the period 1948–90, the correlation between the midterm switch and the margin of victory is 0.58, a positive value, not negative as might be expected from the Alesina and Rosenthal theory. Of course, it is still impressive, as Alesina and Rosenthal point out, that all midterm switches since 1918 have been negative, including those following close presidential elections. But the correlation given above suggests that strategizing by moderate voters is only part of the story in explaining voting behavior in American presidential elections – much less in sequential contests in general in two-party electorates. Indeed, in addition to any desire to attain divided government, we would expect that strong swings to the right or the left would be followed by corrections in the political marketplace. Although considerable work has been addressed to explaining the lack of convergence in party strategies, the search for an internal mechanism – without resort to institutional structures – that drives alternation in office has been largely ignored. In the next section, we develop a model of two-party competition that provides such a mechanism for alternation in office and whose spatial prediction may be characterized as either partial convergence or limited polarization – depending on one’s perspective. This model is based on the Grofman discounting model in which voters evaluate candidates by proximity but only after discounting their claims or their capacity to move the status quo to their preferred policy position. Here we assume further that the voters are correct in their assessment; that is, the discounted movement represents how far the respective parties can move. Our dynamic model predicts that implemented policy positions will be divergent, but less so than would be the case if party candidates implemented policy at the ideal points of the parties. The primary inferences from the model are, however, that party control will alternate over a series of elections and that different parameter assumptions imply different patterns of alternation. Although the model is dynamic, it does not require uncertainty about voters’ preferences. What it does assume is that parties are 1

Midterm switching is defined in Alesina and Rosenthal (1995: 84) as the change in the vote share of the president’s party between an on-year and an off-year election.

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Figure 9.1 Base dynamic model: Median voter at 0. Note: Left and Right’s discounted positions are at L¢ and R¢. Because L¢ is closer to the voter median at 0, Left wins and L¢ becomes the next status quo.

unable to implement the full extent of their objectives, but rather only able to move the status quo in the direction of those objectives. Such constraint is to be expected and may be caused by a variety of factors, including bureaucratic inertia, divided government, and the pressure of special-interest groups. 9.2 Base Dynamic Model under Discounting We will assume throughout a one-dimensional model with two parties, which we call Left and Right. Left’s ideal point is to the left of Right’s, so without loss of generality, we may assume the two ideal points are L = -1 and R = 1, respectively. To focus initially on our main argument, we begin with a base model in which it is assumed that the positions declared by the parties are in fact their ideal (preferred) positions but that they end up implementing policy at the discounted locations recognized by the voters. As part of this base model, we also assume that the median voter is located at 0, midway between the party ideal points, and that the two discounting factors for the parties are the same. Later we relax these assumptions, investigating among other things how party strategy affects our results. In each of a sequence of elections, either Left or Right is elected by majority vote. Let S0 be the initial status quo (see Figure 9.1). During the campaign for the (k + 1)st election, the status quo is at Sk, the policy position achieved by the incumbent party (the party chosen in the kth election).According to the Grofman discounting model (see Chapter 3), each party’s capacity to move policy from the previous

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status quo, Sk-1, to its ideal location is discounted by a factor d, common to both parties. Thus, the location of the status quo Sk is the position implemented by the party who won the previous election. In each election, voters choose that party whose discounted spatial position is closer. The factor d by which the potential movement is multiplied is initially assumed common to both parties. For example, if the potential movement of the status quo is discounted by 20 percent, the value of d (which reflects the residual movement) would be 0.8. Suppose that the initial status quo point S0 lies between -1 and +1. Let L¢ and R¢, respectively, denote the points at which the parties are perceived to implement policy. We note that, in addition to representing the perceived locations of implementable policy for the respective parties, L¢ and R¢ may represent the locations to which the parties actually move in the campaign as realistic responses to what they and the voters believe actually can be accomplished. In any case, we will speak of Left being drawn to position L¢ and similarly Right being drawn to R¢. If S0 is positive, Left’s perceived policy is drawn further (from -1) than Right’s is (from 1) so that party Left is closer to the median voter and wins, implementing policy at what becomes the new status quo, S1; if S0 is negative, Right is drawn in further and wins.2 It follows that Sk = Sk -1 + d(I k - Sk -1 )

(9.1)

where Ik is the ideal location of the winning party in the kth election, i.e., Ik = 1 if Sk-1 is negative and -1 if Sk-1 is positive. Although initially one party may win several times in a row, once victory switches to the other party, strict alternation occurs from then on. To understand this, note that if, say, Left wins the kth election but Right wins the (k + 1)st election, we must have Sk-1 > 0 and Sk < 0. But then, by applying eqn. (9.1) twice, we have 2

If d satisfies 0.5 £ d < 1.0, discounted policy cannot cross the origin, so that strict alternation between parties occurs from the beginning, with Right always implementing a positive policy and Left a negative one. If d lies between 0 and 0.5, a party, say Left, may win by moving across the origin, i.e., by implementing a positive policy closer to the ideal position of the other party. In this case, Left may win more than once in succession, but eventually the discounting will bring Left to a negative policy, after which alternation will occur.We thank R. Potthoff for helpful comments that improved these arguments.

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2

Sk +1 = (1 - d) Sk -1 + d 2 > 0 so that Left must win the (k + 2)nd election. A dual argument holds if Right wins first, then Left. If we start (after alternation begins) with S0 positive and write k = 2n for k even, then, in particular S2n = S2n -1 + d(1 - S2n -1 ) and S2n -1 = S2n - 2 - d(1 + S2n - 2 ) It follows (after some algebra) that 2

S2n = (1 - d) S2n - 2 + d 2

(9.2)

and, similarly, 2

S2n +1 = (1 - d) S2n -1 - d 2

(9.3)

Because strict alternation occurs, eqs. (9.2) and (9.3) specify the sequence of status quo points (policy positions) achieved by Right and Left, respectively. 9.3 Convergence to Separate Points of Stability for Each Party under the Base Model Equation (9.2) defines a first-order, nonhomogeneous difference equation. The general solution of such an equation (see, e.g., Henrici, 1964) is given by S2n = c(1 - d)

2n

+

d 2-d

(9.4)

where the constant, c, is determined by the initial status quo S0.3 In any event, since d < 1, the limiting value Sc of S2n is given by 3

The general solution of the corresponding homogeneous difference equation is of the form S2n = c(1 - d)2n, with the general solution of the nonhomogeneous equation obtained by adding to this solution a particular solution of the nonhomogeneous equation. Since the nonhomogeneous part of eqn. (9.2) is a constant, the particular solution of eqn. (9.2) is normally also a constant, say a. Substituting a in eqn. (9.2) yields (after some algebra) a = d/(2 - d).

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SR =

d 2-d

(9.5)

and is independent of the starting value.4 We summarize in the following proposition. Proposition 9.1. Suppose that a one-dimensional model is scaled so that the ideal positions of two parties, Left and Right, are located at -1 and +1, respectively, and that the median voter is at 0. If the initial status quo lies strictly between -1 and +1, and if d (0 < d < 1) denotes the (common) discounting factor, then (except for a possible initial sequence of wins by one party): (i)

the sequence of status quo points alternates between parties, d (ii) the positions of the Right party converge to the point SR = , 2-d -d . (iii) the positions of the Left party converge to the point SL = 2-d Unless the discount factor d is small, the form of eq. (9.4) suggests that the convergence is quite fast, as is illustrated in Figure 9.2, which shows the sequence of positions that would be implemented by party Right in the alternating elections that party Right would win. Note that this sequence settles down to the same location regardless of the initial status quo. The persistent oscillatory behavior of the status quo points (and hence of the policy positions achieved) is apparent even before convergence (to separate points of stability) is reached (see Figure 9.3). For a reasonable range of possible status quo points, the model predicts alternation of parties in power and alternation of policy positions achieved between distinct locations on either side of the voter median, but never approaching that median. On the other hand, if no 4

The constant c = S0 - SC. Similar calculations using eqn. (9.3) show that 2

2n

S 2 n +1 = ( 1 - d ) S 2 n - 1 - d 2 = c ( 1 - d ) -

d 2-d

-d , with c = S1 - SL. Note that if the even 2-d terms S2n exceed SR in absolute value, then the corresponding odd terms, S2n+1, are smaller in absolute value than SL, and vice versa. and that the odd terms converge to SL =

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Figure 9.2 Base dynamic model (median voter at 0; d = 0.5): Location of party Right in elections won by Right, for three starting values of the status quo.

discounting occurs (i.e., d = 1), then (in this base model) the location of the status quo is irrelevant to both voting behavior and party strategy and the parties split the vote equally. If the assumption that the median voter is at the origin were relaxed, then the nearer party would always win. Recall that voter choice under the Grofman discounting model and the mixed proximity–RM model is the same if the status quo and neutral point are the same and if the mixing parameter b = d (see Chapter 3). Thus, the conclusions of this chapter also apply to the latter model as long as the neutral point tracks the status quo. 9.4 Party Strategy under Discounting The assumptions made in our base model ignore the effects of party strategy. We have assumed that each party’s declared policy location is its ideal point, -1 or +1, and that it plans to and will implement

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Figure 9.3 Base dynamic model (median voter at 0; d = 0.5): Movement of the location of the status quo with victories alternating between parties.

policy at the discounted positions indicated above. We now consider a strategic model in which parties have both policy-seeking and voteseeking motivations. We assume that each party wishes to maximize a utility function that incorporates both the value of the policy that is implemented and the value of being the party that implements it. We may write such a function (for party Right) as 2

U ( x2 ) = -( x2 - I ) + bP ( x1 , x2 ) where x1 represents the location of party Left, x2 that of party Right, I is the ideal position of party Right, b is a coefficient representing the value of winning, and P(x1, x2) denotes the probability that party Right wins (see, e.g.,Alesina and Rosenthal, 1995). Calvert (1985) shows that parties typically optimize this trade-off by choosing a location intermediate between their ideal location and the median voter. If b = 0,

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Figure 9.4 Strategic dynamic model: Median voter at 0. Note: Because of discounting, Left cannot achieve its optimal point on its utility curve (at Loptimal), so must be content with L¢. Right could move the status quo as far as R¢, but it moves only to Roptimal, where it can attain higher utility.As in the base model, L¢ is closer to the median than Roptimal and is implemented by Left, who wins.

the optimum is I; if b is arbitrarily large, the optimum approaches the location of the median voter. Consider the situation in Figure 9.4, where the initial status quo S0 is to the right of center. Under the strategic model, parties may move, but within limits. We assume that a party is capable of moving policy up to but not beyond the discounted location. Thus, locations L¢ and R¢ now represent barriers beyond which the respective parties are incapable of moving the status quo (see Figure 9.4). One or both of these barriers may be less extreme than the optimal locations (Loptimal and Roptimal) under the strategic model. Because, in our example, L¢ is closer to the median, it is more likely to be binding than is R¢.5 If so, Left’s best strategy is L¢, the nearest feasible point to its unconstrained optimum, Loptimal. 5

Note that the locations of the parties refer to their discounted locations, i.e., the locations of policy that they would implement if elected. Thus their declared policies in the campaign would typically be more extreme, namely, those policies that, after discounting, would be those that they intend to implement. As the scenario above unfurls, we will see that the declared policy for Left is just its ideal policy at -1; the declared policy for Right is somewhat less extreme than its ideal policy of +1. This suggests a plausible situation in which the challenging party announces its ideal position while the incumbent party announces a position near the status quo, which it itself has created.

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For Right, on the other hand, the optimum6 is more likely to be in its feasible set, so that Right’s optimum position, Roptimal, is less extreme than R¢. But this value will still be further from the median than L¢,7 resulting in an expected win for Left, who will implement policy at S1 (= L¢), just as in our base model. This suggests that the same pattern of alternation will emerge when parties are strategic, but with less certainty. 9.5 Modifications of the Model for Asymmetric Parties and Disparate Discount Factors Returning to our base model, we now relax the assumption that the median voter is at the origin, that is, the midpoint between the parties’ ideal points. Denote by M the median voter’s location and suppose that M is between 0 and 1. If the status quo, S, is sufficiently low but positive, the midpoint between L¢ and R¢ will fall between 0 and M. Thus, party Right wins at least twice in a row and will continue to win until S (= R¢) moves sufficiently to the right that L¢ is once again closer to M. In fact, alternation of parties occurs precisely if S>

M 1-d

(9.6)

As we can see from Figure 9.5, if s falls below this critical value, then party Right wins until subsequent status quo points move above M/(1 - d). Thus the party closer to the median voter may win several times in succession, and then lose once to its opponent before winning several times again. If, however, SR > M/(1 - d), the status quo values never reach the critical value and strict alternation is the rule. This latter condition is equivalent to M<

d(1 - d) 2-d

(9.7)

We summarize in the next proposition. 6

7

The optimum strategy for Right under the trade-off model depends on Left’s strategy, so Right’s will be shifted slightly from its equilibrium value if Left is prevented from moving to its optimum because of discounting. If the two utility curves are symmetric with respect to 0, the point (Roptimal) at which Right’s utility curve peaks will be further from 0 than L¢, which is to the right of the point (Loptimal) where Left’s utility curve peaks (see Figure 9.4).

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Figure 9.5 Unbalanced dynamic model: Median voter offset from 0. Note: Because the status quo is between the median voter and the point specified by M/(1 - d), R¢ is closer to the median voter than L¢ and so party Right wins.

Proposition 9.2. Assume the conditions of Proposition 9.1, except that the median M need not be equidistant from the party ideal points. Except for a possible initial period, alternation of parties occurs as long as condition 9.7 holds. If it does not, strict alternation fails to occur, to the advantage of the party whose ideal point is nearer the median. This threshold value for the failure of alternation varies from near 0 when d is near 0 or 1 to about 0.17. Hence, ceteris paribus, we would expect alternation of parties as long as the median voter is not too far from the party midpoint, but some repeat wins for a party substantially closer to that median. Figure 9.6 portrays the sequence of policies implemented (by both parties) in an example with M = 0.25 and d = 0.05. Note that the status quo points (after stability is reached) are -1/7, 3/7, and 5/7, so that party Right has two wins for each single win by party Left. Second, we return to the assumption that M = 0, but relax the assumption that the discount factors are the same for both parties. Denote by dL and dR, respectively, the discount factors for the two parties, and assume that dL > dR. If Right wins and the status quo S is positive, it can be shown that Right can win at least twice in a row if S is sufficiently small.8 8

The midpoint between L¢ and R¢ is SÊ 1 Ë

dL + d R ˆ dL - d R 2 ¯ 2

which in turn is negative if and only if

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Figure 9.6 Unbalanced dynamic model: Movement of the status quo for voter median at 0.25 (d = 0.5). Note: Each win by party Left is followed by two wins by party Right.

Substantively, small values of d (i.e., heavy discounting of a party’s policy position) work to the advantage of the party, in the sense that the party can expect to win more than half the elections. Insofar, however, as the values of the status quo represent policy actually implemented, such a party pays a price because implemented policy is drawn further from the party’s ideal location. S<

dL - d R 2 - (dL + d R )

Since Right wins if the midpoint is negative, Right can win at least twice in a row if S is sufficiently small. If Right wins exactly twice in a row before Left wins, the equilibrium value of S closest to the origin is given by S=

d R (1 - dL )(1 - d R ) - dL (1 - d R ) + d R 2 1 - (1 - dL )(1 - d R )

Thus, for example, if dL = 0.8 and dR = 0.5, the status quo cycles through 3/19, 11/19, and -13/19. We thank R. Potthoff for this example.

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Similar arguments show that if both disparity of the discount factors and asymmetry of the median voter occur, Right can win twice in a row if M > (dR - dL)/2. The two factors can counteract each other if, for example, the median voter is closer to Right – tending to make repeat wins by Right more likely – but Left is more heavily discounted (dL < dR) – tending to make repeat wins by Left more likely. 9.6 Discussion American politics in the mid 1990s, on first sight, seems as puzzling as it is familiar. In 1992 American voters sought a change from twelve years of Republican rule and chose a Democrat, Bill Clinton, as president.Yet only two years later, they elected the first Republican House of Representatives in 40 years, which included 71 freshman Republicans – many of them uncompromisingly conservative with a fast-paced agenda called the Contract with America. Within another two years, the Republican leader of the House, Newt Gingrich, had become one of the least popular politicians in America, the Contract was only partially implemented, and Clinton rode easily to reelection. Are voters really as fickle as these reversals might suggest? Or can their behavior be explained as a rational reaction to the shifts in the policies advocated by Clinton and the Republican congressional leaders and the voters’ changing perceptions of the direction in which the country was moving? How might both Clinton and the Republicans decide what positions to take? How might each seek to modify their positions in response to new information about what voters really wanted? The argument in this paper suggests that in repeated elections between two parties, discounting alone can imply a sequence of implemented policies that do not converge to the voter median, but typically switch back and forth on either side of this median. No real shifts in voter policy preferences are needed to maintain this pattern. It need not depend on variations in vote-seeking and policy-seeking strategies of the parties. The location of each party’s implemented policy converges over time to its own point of stability on its side of the voter median. The proximity of these points to the median voter cannot be obtained by simply discounting the distance from the party’s ideal location to the voter median by the voters’ discount factor; in fact, the

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points of stability are generally closer to the median, as indicated by eqn. (9.5). Alternation of single victories by each party is expected if the median is at the midpoint between the desired policy locations of the parties and the discounting factors for the two parties are equal. Onesided alternation is expected if the median is substantially to one side or if the discounting factors are unequal, with more than one victory in a row favoring the party closer to the median or the party more heavily discounted. The partial dominance of one party that results from one-sided alternation is encapsulated in Lubell’s (1952) metaphor of the two American parties as a sun and a moon. From a normative point of view, an electorate may enjoy several advantages from moderately separate policy positions at stability. Such a situation offers a choice between distinct policy positions. Over a period of time the parties can implement a greater variety of policies than if both converged to the median and implemented identical policies. The disruption attendant on wild swings between extreme policies, however, is avoided. In the face of significant factors other than discounting – such as swings in the economy, changing salience of issues, and the personalities and perceived competencies of candidates – we cannot expect to predict the outcome of elections by the oscillation model alone. We can observe, however, that the perceived locations in presidential elections of each of the two major American party candidates on the liberal–conservative scale as measured by the NES surveys have remained relatively constant over a number of elections (see Table 9.1). If these perceived locations are interpreted as the ideal positions of the candidates and the parties they represent, the Democratic and Republican locations can be identified with -1 and +1, respectively, in our model. The standardized median voter location (self-placement) is close to 0 on this scale except for 1988 (M = 0.20) and 1996 (M = 0.13). The location of implemented policy – which reflects the efforts of both the president and congress – is harder to specify. But with the benefit of hindsight, we may venture some guesses as to what our model may say about the outcomes of recent elections. In 1980 the weak economy and the (slightly) liberal status quo both favored the challenger, Reagan, who won. In 1984 a robust economy

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Table 9.1. Medians, means, and standard deviations for self- and candidate placement: American NES 1980–96 Medians

Means

Standard Deviations

Election

Self

Rep.

Dem.

Self

Rep.

Dem.

Self

Rep.

Dem.

1980 1984 1988 1992 1996

.53 .30 .45 .18 .36

1.94 1.90 1.81 1.82 1.84

-.91 -1.26 -1.59 -1.34 -1.55

.45 .27 .39 .13 .34

1.84 1.71 1.64 1.66 1.72

-.90 -1.09 -1.36 -1.28 -1.40

1.47 1.40 1.45 1.50 1.47

.90 1.04 1.00 .97 .86

1.19 1.22 1.21 1.04 1.06

offset the effect of a decidedly conservative status quo, leading to a second Reagan victory. In 1988 Bush – representing the incumbent Republican administration – was closer to the median voter than Dukakis, facilitating a repeat win by the Republicans. Four years later, the median voter was equidistant between the candidates and the Republican status quo (along with a weak economy) favored Clinton, who won in 1992. The expected swing back to the Republicans came quickly this time, in the Congressional elections of 1994, to be reversed again in 1996 with a strong economy offsetting the effect of a slightly conservative median voter. Over the series of elections the pendulum of implemented policy swings back and forth as voters respond to the changing status quo. The implications of this model, along with institutional inertia and the checks and balances of the federal system, ensure that – relative to the ideal positions of the parties – the amplitude of these swings is greatly damped.

CHAPTER 10

Strategy and Equilibria in Multicandidate Elections We know what happens to people who stay in the middle of the road; they get run down. Aneurin Bevan, Observer (December 9, 1953)

10.1 Multicandidate Equilibria In one dimension, two-candidate equilibrium strategies are convergent for the proximity model. In this chapter, we investigate the existence and nature of equilibria for three or more candidates. Under a proximity model – even for one dimension – no equilibria exist for three candidates (the two extreme candidates always have a unilateral incentive to move inward, but eventually the squeezed center candidate can benefit by leapfrogging; this cycle repeats with no end). In fact, for plurality-maximizing candidates, in one dimension, no equilibrium occurs for any fixed odd number of candidates three or higher.1 For four candidates, assuming a uniform distribution of voters, an equilibrium does occur, with two candidates located at the 25th percentile of the distribution and two at the 75th percentile.2 In any event, such an equilibrium is far from convergent at the median and is quite implausible empirically. It is an artifact of deterministic assumptions. 1

2

A plurality-maximizing candidate maximizes her vote share relative to the highest vote share obtained by any of the other candidates. For pure vote-share maximizing candidates, an equilibrium is possible for five or more candidates, but requires that the candidates be spread out over the voter distribution. See Eaton and Lipsey (1975) for the existence and nonexistence of multiparty equilibria; see also Brams and Straffin (1982) and Feddersen, Sened, and Wright (1990) concerning the effects of entry of new candidates. In general, for an even number of candidates, equilibria occur with a pair at each of the quantiles 1/K, 3/K, . . . , (K - 1)/K of the voter distribution (Cox, 1987, 1990). This pairing of candidates occurs because the most extreme candidate on either end always has an incentive to move in and join the next most extreme candidate. Equilibria among candidates are not to be confused with equilibria among voters, as studied by Myerson and Weber (1993).

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145

In Chapter 6 we introduced a probabilistic model of voter choice, fitting it first to data from a two-party electorate and then to data from a multicandidate electorate in Chapter 7. As we will see, a probabilistic model – even one based on proximity – can lead to a convergent multicandidate equilibrium, which extends to a multidimensional setting. If, however, a party ID component is added, i.e., vote choice is in part determined by partisan identification of the voters, a convergent equilibrium is obtained in two-party competition; but for three or more parties, the equilibrium is divergent, i.e., the parties settle on distinct optimal strategies. We can also show that these optimal strategies spread further apart as directional aspects are added to the model. We will fit this probabilistic model to data from the Norwegian National Election Studies of 1989 and 1993 to compare the actual spatial locations of parties with the predictions derived from equilibrium theory reported here and also with that predicted by the unified model we previously used in Chapter 7. Our results are based on the work of Adams (1997b, forthcoming a, b) and Adams and Merrill (forthcoming, 1999). Under a two-candidate, two-dimensional (deterministic) Matthews directional model, we saw in Chapter 8 that undominated directions exist for a neutral point outside a star-shaped region labeled the Condorcet vacuum. Such an undominated direction represents an equilibrium direction: If both candidates move in that direction, neither has an incentive to deviate. When there are more than two candidates, we will show below that equilibria are often possible in two dimensions – even without any probabilistic assumption or assumption that vote choice is in part determined by partisan identification of the voters – and contrast this situation with the divergent or nonexistent equilibria in a (deterministic) proximity model of multicandidate competition. 10.2 A Multidimensional Convergent Equilibrium By adding a probabilistic component representing non-issue voting to a one-dimensional, multicandidate proximity model, de Palma, Ginsberg, Labbe, and Thisse (1989) show that if the probabilistic component is sufficiently large, a convergent equilibrium occurs when the voter distribution is uniform and candidates seek to maximize their

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share of the vote. They also show that if the probabilistic component is sufficiently small, a divergent equilibrium occurs, and they specify its degree of dispersion as a function of the size of the probabilistic component. For the two-candidate case, Enelow and Hinich (1989) give general conditions that imply that the expected vote share is concave. They apply this result to show that for a quadratic proximity model with a normally distributed probabilistic component, equilibrium at the mean voter location occurs if the variance of the probabilistic component is sufficiently large. Extending this fundamental work to multicandidate elections, Lin, Enelow, and Dorussen (1999) obtain a sufficient condition for a convergent equilibrium based on concavity of the expected vote share, which depends on the size of the probabilistic component. The latter depends, in turn, on the scale, the voter distribution, and the number of candidates, but the details are left unresolved. Building on this work, Adams (forthcoming b, 1999) obtains some remarkable results that relate multicandidate equilibria to interpretable and testable measures of uncertainty. Using a conditional logit model (of the form described in Chapter 6), he proves that for sufficiently high amplitude of the probabilistic variate, the candidate who maximizes social utility (i.e., achieves the highest average utility over all voters) is asymptotically (in the number of voters) certain to win the largest share of the vote. There is no restriction on dimension. In particular, if the proximity utility is defined by a city-block metric,3 then the point whose coordinates are medians in the respective dimensions constitutes an equilibrium. If the quadratic Euclidean metric (Chapter 2) is used, the multidimensional mean is the equilibrium. Figure 10.1 depicts an example of the Adams setting for three candidates and one dimension. In this example, the voter distribution follows a continuous triangular distribution on the interval from -3 to +3 that peaks at 0 and reflects roughly the frequency distribution of respondents in NES surveys. Candidates A, B, and C are located at -1, 0, and +1, respectively (see arrows in Figure 10.1). Using the city3

The city-block metric is defined by the utility function n

U ( V, C) = -Â vi - ci . i =1

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Figure 10.1 Distribution of support in the Adams probabilistic model.

block metric, candidate B is squeezed and defeated by A and C in a deterministic model, but wins (on an expected-value basis) in a probabilistic model with a sufficiently high standard deviation. The three plots in the figure represent the distribution of support under a probabilistic model. It may seem ironic that these results hold for high, not low, values of probabilistic noise, i.e., for situations in which issue voting constitutes only a small (but nonzero) portion of voting behavior. As Adams points out, behavior that is primarily issue based causes centrist candidates to be squeezed in the multicandidate setting. In the presence, however, of only a small amount of issue voting, a centrist candidate can gain support (on an expected-value basis) from voters to whom she is not closest and win on the basis of overall or average nearness.4 Of course, the vote share of the various candidates will differ by little under these assumptions; variation in the expected non-issue support of candidates might dominate the competition. Adams’s theorem assumes that the mean of the nonpolicy utility 4

In a related paper, Adams (forthcoming b) assesses policy representation and Condorcet efficiency using a Monte Carlo simulation. Condorcet efficiency is the percent of elections having a Condorcet candidate for which this candidate is chosen (see Chamberlin and Cohen, 1978; Merrill, 1984, 1988). He finds that both are higher for a mixture of policy and nonpolicy voting than for either pure issue-oriented voting or pure randomness. The strong performance under limited policy voting – with regard to both Condorcet and policy representation criteria – appears to bode well for application of these results to real electorates.

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component is the same for all candidates,5 i.e., that all candidates are equally “popular,” but this is an assumption unlikely to hold in practice. Without this assumption, the conclusion of the theorem no longer follows, because the outcome of the election tends to be dominated by systematic nonpolicy preferences. For example, the status of a candidate with a substantial nonpolicy lead is unlikely to be affected by a small policy component. The winner may still be the Condorcet candidate, but the chances that she best represents the voters on issues (in the sense of social utility) is at best random. In general, to affect the outcome, policy voting must be sufficient to offset systematic nonpolicy disparities between candidates. Hence the conclusion of Adams’s theorem – which assumes very limited policy voting – will not necessarily hold in the presence of a sufficient variation in nonpolicy preferences. However, Adams (personal communication) has done simulations that suggest that the conclusion of the theorem may remain valid for a moderate (as opposed to a slight) level of policy voting. The latter is not so easily upset by nonpolicy voting. 10.3 Divergent Equilibria with Partisan Voting and the Effect of a Directional Component Building on the work of Coughlin and Nitzan (1981a, b), Erikson and Romero (1990)6 and others (de Palma et al., 1989; de Palma, Hong, and Thisse, 1990; Nixon, Olomoki, Schofield, and Sened, forthcoming; Lomborg, 1997; Lin et al., 1999), Adams (forthcoming a, 1997b) investigate equilibria for multicandidate contests using partisanship as an explicit non-issue variable.7 Adams (forthcoming a) simulates the 1992 U. S. three-cornered presidential election between Clinton, Bush, and Perot, finding a nonconvergent equilibrium that is robust to considerable variation in assumptions and model parameters. Adams 5 6

7

This assumption is relaxed in Adams (forthcoming c) and Hug (1995). Erikson and Romero (1990) derive equilibrium results for two-candidate contests by incorporating a measurable, non-issue variable (e.g., partisanship) as well as a probabilistic component into a proximity model. Using a quadratic utility function, they obtain a local equilibrium at the coordinate mean, where each component mean is weighted by both issue salience and the elasticity or marginality of the voter. Thus, for example, uncommitted voters have more weight and more clout in drawing the equilibrium point to their position. The Adams model is the probabilistic, partisan, proximity model described in Chapters 6 and 7. In fact his use of this model motivated our use of it.

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(1997b) proves that if voters’ partisanship is correlated with their policy preferences, a divergent equilibrium will occur for votemaximizing candidates as long as issue-salience to voters is sufficiently low. Substantively, Adams finds that a candidate – by moving away from her party’s traditional issue position – loses more of her own party’s partisans than she gains among the partisans of other parties. This effect is particularly marked with the presence of an independent candidate located between the two major parties, as either major-party contender is more prone to lose voters to the independent than to the other major-party candidate. As Adams observes, in a multicandidate contest, different voters are marginal with respect to different subsets of candidates. Hence, unlike a two-candidate contest in which both candidates can focus on the same set of marginal or uncommitted voters, different candidates focus on different groups, making a nonconvergent equilibrium more likely. Adams and Merrill (forthcoming, 1999) evaluate these results empirically by first estimating parameters of a (one-dimensional) probabilistic, partisan conditional logit model for the data from the Norwegian Election Studies, as is done here in Chapter 7. Using a cross-tabulation of self-placement on the left–right scale with recalled vote from the previous election to reflect the relation between partisan bias and spatial location, they find that equilibrium positions exist for the seven major parties and that these locations are not convergent but distinct from one another.8 Similar results can be expected not only for party identification but also for any measured nonissue variable that influences the vote and is correlated with issue position. The distinctness of the party positions at equilibrium contrasts with the clumped (and less-realistic) equilibrium locations reported by Nixon et al. (1995: Table 5) and Schofield, Sened, and Nixon (1998b: 8

Vote-share maximization – assumed as the party objective in the Adams–Merrill study – may not always be appropriate, particularly under proportional representation. Schofield, Sened, and Nixon (1998b) argue, for example, that in the 1992 Israeli election, the two largest parties (Labor and Likud) were vote-maximizers, whereas the smaller parties may not have been. The closeness of the 1989 Norwegian election – in which the left bloc (Labor and the Socialists) wound up three votes short of a majority – suggests, however, that vote-maximization may be a reasonable assumption for that election.

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Table 6), who use a logit model for six parties in the 1992 elections for the Israeli Knesset.9 These latter studies employ party-specific intercepts but do not account for a partisan or comparable bias correlated with spatial position. If the spatial component of the probabilistic model is pure proximity, the party positions at equilibrium found by Adams and Merrill for the 1989 Norwegian Election Study are distinct from one another but fairly tightly clustered near the center of the 1–10 left–right scale, much more so than are the actual (mean) placements of the parties by the respondents. These equilibrium positions range from Labor on the left at 4.80 to the Conservative party on the right at 6.65. By contrast, the actual placements range from the Socialist party at 2.42 to the Progress party at 9.02. If, instead, the spatial component is the pure RM model, all party equilibrium locations are at the boundaries of the region of acceptability,10 which these authors take to be the limits of the scale, namely 1 and 10. All liberal parties locate at 1 whereas all conservative parties locate at 10, again an unrealistic scenario. Using, however, a mixed proximity–RM model with mixing parameter, b = 0.52, estimated from the data by maximum likelihood, Adams and Merrill obtain equilibrium positions ranging from the Socialist party at 2.55 to the Progress party at 9.00 – positions that are almost identical to the actual party placements estimated by mean placements over respondents and in exactly the same order from left to right. The distributions of equilibrium positions for the proximity, RM, and mixed models are depicted in Figure 10.2, in relation to actual party placement. Each plot can be compared with the reference line (the 45 degree line in Figure 10.2), which indicates where party locations at equilibrium would be if they followed actual party placements exactly. Clearly, the equilibrium locations under the mixed proximity–RM model (directional model with proximity constraint) most closely follow the actual placements, providing strong support 9

10

The Adams–Merrill result also contrasts with the convergent configuration of party optima (at the mean voter location) obtained by Schofield et al. (1998b) using a multinomial probit analysis and sociodemographic control variables for the Dutch elections of 1977 and 1981. Schofield et al. argue, however, that such convergence may not be optimal, for it may reduce the power to implement desired policies or to bargain for those policies with potential members of a coalition government. Without a region of acceptability, the parties all have an incentive to be arbitrarily extreme, so there is no equilibrium.

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Figure 10.2 Model predictions of party locations versus actual (mean) party placements by model type: Norway 1989. Note: For the mixed proximity-RM model, the points are closer to the 45 degree reference line, which represents equality between actual party location (mean placement) and predicted party location. (Source: Adapted from Adams and Merrill, forthcoming: Figure 6.)

for a model comprising both proximity and directional aspects.11 The mixed model correctly predicts where the parties actually locate, whereas the pure proximity and pure RM models do not. 10.4 Regions of Candidate Support for Directional Models for More than Two Candidates We now turn to what can be said about the spatial regions of support for candidates under directional models. How the locations of candidates affect both the number of voters who support them and whether those voters tend to be centrists or extremists has a major influence on candidate strategy and potential equilibria. 11

This analysis applied to the unified model leads to a similar equilibrium configuration.

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Part II Models of Party/Candidate Behavior

A preliminary investigation of regions of support and candidate strategy for the Matthews model indicates that – for two dimensions – the directional locations of candidates must be relatively spaced out for an equilibrium to occur. In particular, the (empirically implausible) pairing of candidate locations – characteristic of equilibria under the proximity model for an even number of candidates – does not occur in a two-dimensional Matthews model because the contested space is a circle rather than a line. This is true because in the Matthews model there is no such thing as an extreme position along the circle. For similar reasons, the Matthews model is free of the instability characteristic of the proximity model for an odd number of candidates. Coughlin (1992), under probabilistic assumptions for a generalized, two-candidate logit model (binary Luce model), proves that the Matthews directional assumptions always imply existence of an equilibrium, which – in Coughlin’s model – may consist of staying at the status quo point. He also gives a condition for the latter equilibrium. We will focus, instead, on the RM model. It was noted in Chapter 2 that in an RM directional model in a race between two candidates, each receives the votes of voters in a half-plane bounded by the perpendicular to the line segment joining them and through the neutral point.12 If the voters are distributed symmetrically about the neutral point, then each candidate receives support from one half the electorate. If the distribution is not symmetric, the candidate with the greater number of voters in the associated half-plane receives the greater support. The regions of support and the strategies to which they lead for contests with three or more candidates are more complex. We assume that candidates seek to maximize their support relative to that of other candidates. In a one-dimensional model, only the two most extreme candidates receive any votes; all those in the interior of the candidate interval are squeezed out. By the logic of the scalar product, the rightmost candidate receives all votes to the right of N while the leftmost candidate gets all those to the left. Next consider a two-dimensional model with, say, four candidates labeled A, B, C, and D. If none of these candidates lies in the interior of the triangle formed by the other three (see Figure 10.3a), voters 12

In this section, we do not assume a “circle of acceptability.”

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a

b Figure 10.3 Regions of support for the RM and proximity models: No candidate in the interior of the convex hull of the others. a. RM Model b. Proximity Model

who prefer A to B lie in a half-plane bounded by the perpendicular to the line segment AB and passing through the neutral point, N. Likewise, voters who prefer A to D lie on one side of the perpendicular to AD through N. Thus, the region of support for A is the sector of the plane consisting of the intersection of these half-planes.13 Ceteris paribus, the proportion of the space supporting A is 13

The half-plane determined by the perpendicular to the line segment AC (connecting A to the nonadjacent vertex C) includes the intersection of the other half-planes and is hence redundant.

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proportional to the size of the angle formed by this region at N. In fact the two bounding perpendiculars and the line segments AB and AD form a quadrilateral with two right angles. It follows that the angle of this quadrilateral at N is the supplement to the angle at A, so that the size of the region of support of A is inversely related to the size of the angle at A. In particular, that candidate lying at the most acute angle will receive the largest region of support and, for an electorate symmetric about N, the greatest numerical support. Other things being equal, this advantage will tend to be associated with more extreme candidates (e.g., candidate B in Figure 10.3a). It is possible that a candidate, such as C in Figure 10.3a, may lie outside her own region of support. This will occur when the perpendicular to a line segment connecting two candidates (CB in Figure 10.3a) intersects not the segment itself but its extension. In this case a voter located at C and, hence, agreeing perfectly with C would nevertheless vote for B over C. In fact, were C also a voter, the model would predict that she would vote against herself. This feature of the pure RM model does not make much sense. By contrast, for a proximity model, the bounding perpendiculars bisect the line segments joining candidates and need not meet in a point (see Figure 10.3b). For the proximity model, the relation between extremity and extent of support is much less clear-cut than it is for the RM model. Now suppose that one candidate, say D, lies in the interior of the triangle formed by the other three (see Figure 10.4a). Under the RM model, D has no region of support and hence receives no votes whatever; candidates A, B, and C partition the electorate among themselves. Figure 10.4b indicates the division of the electorate according to the proximity model. Candidate D is guaranteed some region of support, although this region may be small, especially if D lies near the center of a small triangle formed by the other candidates. To describe in general the regions of support when the number n of dimensions is two or more, we need several definitions. Given a finite set of points C1, . . . , CK in Rn, the convex hull H of C1, . . . , CK is the polyhedral region (solid figure) determined by C1, . . . , CK.14 An 14

To be precise, a set S in n-dimensional space Rn is convex if, for any pair of points A and B in S, the line segment from A to B is also contained in S. Given a finite set

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155

a

b Figure 10.4 Regions of support for the RM and proximity models: One candidate in the interior of the convex hull of the others. a. RM Model b. Proximity Model

exposed point of the convex hull is a corner point or vertex.15 The regions of support are given by the following result, the proof of which is in Merrill (1993).

15

of points C1, . . . , CK in Rn, the convex hull H of C1, . . . , CK is the intersection of all convex sets containing C1, . . . , CK. A point E of a convex set S is an exposed point of S if there exists a hyperplane that bounds S and intersects it at the single point E. If a convex set is the intersection of half-spaces, the exposed points are usually called vertices.

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Proposition 10.1. In the RM directional model, if there are K candidates positioned at C1, . . . , CK, then only those candidates positioned at the exposed points of the convex hull of C1, . . . , CK receive any support from voters. Each exposed point is the location of a candidate, Ci, who receives support from a conical region with apex at the neutral point, N. Corollary 10.1. Under the RM model, a candidate should vote for herself if and only if that candidate is located at an exposed point of the convex hull of all the candidates. Note that the size of the cones specified in the theorem will vary inversely with the size of the polyhedral angle formed by the cone at Ci. Accordingly, a candidate whose position is extreme relative to that of the other candidates and is thus positioned on a sharp point of the convex hull will tend to be favored. Finally, note that it is the positions of the candidates relative to each other and not their positions relative to those of the voters that determines the regions of support. For example, a rigid shift of the candidate set has no effect on the regions of support under the RM model since all bounding hyperplanes must pass through the neutral point. 10.5 Discussion and Conclusions The RM model (without circle of acceptability) implies wild divergence; in particular, candidates at the periphery can expect the greatest support. The implausibility of this implication was, of course, the original reason that Rabinowitz and Macdonald introduced the region of acceptability.16 Enelow and Hinich (1989: 109), however, describe the stability that we can expect when a nonpolicy component is added to the proximity model: The kind of disharmony of views that destabilizes elections when these views concern policies can stabilize elections when these views concern the non16

As noted earlier, we have argued – as does Iversen (1994) – that this objective is better attained through the RM model with proximity constraint.

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157

policy attributes of the candidates. Sufficient subjectivism among voters is conducive to electoral stability.

We have seen that – for a one-dimensional proximity model – introduction of a probabilistic component can extend the twocandidate median voter theorem to apply to a multicandidate setting. Indeed we have seen that the convergent equilibrium obtained holds also in higher dimensions.17 In one dimension, a convergent equilibrium is maintained for two-candidate competition when a partisan component is added (i.e., vote choice is assumed in part determined by partisan identification of the voters), but the equilibrium becomes divergent in multiparty competition. Thus, in this latter equilibrium, strategies are not identical but spread out over the issue space. This spreading of party locations increases when a directional component is included in the mix. For multicandidate elections in one dimension, neither the proximity nor the directional model predicts convergence at the median, yet the directional model overpredicts the observed degree of party dispersion and the proximity model underpredicts it. In contrast, à la “Goldilocks and the Three Bears,” the degree of mixture estimated by the probabilistic model of voter choice in Chapter 7 for Norwegian data predicts a degree of spread of party locations that is “just right,” i.e., closely resembles actual party placements for the Norwegian election of 1989. This data analysis provides the strongest empirical support available for a mixed model. 17

For the Matthews model, however, even without a probabilistic or partisan component, the existence of three or more candidates may lend stability, but this stability does not extend beyond two dimensions.

CHAPTER 11

Strategy under Alternative Multicandidate Voting Procedures [A] voter finds party ideologies useful because . . . ideologies help him focus attention on the differences between parties. Anthony Downs, An Economic Theory of Democracy (1957: 98)

11.1 Alternative Voting Procedures Numerous voting procedures have been suggested for use in multicandidate elections, including the Borda count, the single-transferable vote (STV), plurality with runoff, and approval voting.1 That the choice of system can affect the outcome and the strategies of both voters and candidates and can advantage certain types of candidates – for example, centrists or extremists – has been well documented (Rae, 1971; Straffin, 1980; Riker, 1982; Bogdanor and Butler, 1983; Brams and Fishburn, 1983; Merrill, 1988; Cox, 1997). We discuss briefly – for directional and proximity assumptions – the regions of support and the strategies they imply for these alternative voting systems. This chapter is based on Merrill (1993). Under approval voting, each voter casts a single vote each for as many candidates as he wishes. The candidate with the largest vote total wins. We assume that each voter chooses that strategy that will maximize his expected utility (see Merrill, 1988: 60). If the candidates are denoted by C1, . . . , CK, the voter votes for Ci if his utility for Ci exceeds his average utility for all the candidates. For a twodimensional RM model, each candidate receives votes from all voters 1

The STV system – for single-member districts – is used in the Australian House of Representatives, where it is called the alternative vote. Approval voting is used in a number of professional organizations, including the Institute of Electrical and Electronics Engineers, the Mathematical Association of America, the American Statistical Association, and the Institute for Operations Research and Management Science.

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159

in a half-plane bounded by a line that passes through the neutral point 0.2 Thus, for a symmetric voter distribution, each candidate receives support from exactly half the voters. Under the Borda count, each voter ranks the candidates, giving higher scores to preferred candidates. The candidate with the highest average score wins. We may assume, without loss of generality, that the scores are K - 1, K - 3, . . . , - (K - 1). Thus, a voter’s score for a candidate equals the number of candidates who are less preferred minus the number more preferred. Again assuming a symmetric electorate, in each pairwise comparison, each of the two candidates is preferred by an equal number of voters according to the RM model. It follows that the total preferences and, in turn, the total scores for all candidates are the same, so that the Borda scores for all candidates are identical. Under the STV system for a single-winner race, each voter provides a complete preference ordering. If no candidate obtains a majority of first-place votes, the candidate with the fewest first-place votes is eliminated, and the second-place votes for her supporters are transferred to augment the first-place totals of the remaining candidates. This process is repeated until one candidate attains a majority and becomes the winner. According to the arguments in Section 10.4, interior candidates are eliminated first, followed by exposed-point candidates at the least-sharp corners of the unfolding convex hull. This process is likely to lead to the eventual selection of a relatively extreme candidate (see simulations below). 2

Utility of the voter V for a candidate C is given by U(V, C) = V · C. Hence V should vote for Ci if K

V ◊Ci > (1 K )Â V ◊C j j =1

that is, if K È ˘ V◊ÍCi - (1 K ) C j ˙ > 0 Î ˚ j =1

Hence Ci receives votes from all voters in a half-plane bounded by the line defined by the equation K È ˘ V◊ÍCi - (1 K ) C j ˙ = 0 Î ˚ j =1

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Part II Models of Party/Candidate Behavior

11.2 Are Centrists or Extremists Favored? Whether centrist or extremist candidates are advantaged depends upon the voting system and the model. Assume, for the moment, that the voter distribution is symmetric with respect to the neutral point.3 Under the plurality system, assumption of a pure RM model implies that the candidate located at the most pointed vertex (the sharpest acute angle in a two-dimensional model) of the polygonal region spanning the candidate set receives the greatest support. Such a candidate is likely, although not necessarily, to be the most extreme, that is, farthest from the neutral point. The Condorcet candidate is indeterminate in an RM directional model because all pairwise contests are ties. Under a proximity model, however, either centrists or extremists may win, depending on the degree to which centrists may be squeezed by surrounding candidates. Now consider a mixed proximity–RM model. For any pair of candidates, the plane of indifference lies intermediate between its positions for the pure proximity and RM models, so the candidate nearer the center commands a majority of the voters. Accordingly, for a multicandidate election assuming a mixed model with a nonzero proximity component, the most centrist candidate is favored in all pairwise contests and hence is – by definition – the Condorcet candidate. Under approval voting, all candidates receive equal votes under the RM model. For a proximity model, the median is an equilibrium in a one-dimensional model (Cox, 1987) and for higher dimensions, candidate positions toward the center of concentration of candidates tend to be favored (see Merrill, 1988: 43, n. 5). For the Borda count, under a proximity model, the median is again an equilibrium for one dimension (Cox, 1987). Generally centrist candidates tend to be preferred by majorities over many other candidates, and are more likely to receive the highest Borda score. Under the RM model, as we have seen, all candidates are equally favored. Intermediate results can be expected for mixed models (see simulation results below). 3

We assume throughout that for each region, the probability that a voter lies in the region is greater than zero.

Strategy under Alternative Voting Procedures

161

Suppose, now, that the voter distribution is asymmetric. For example, consider a two-dimensional model in which the voter distribution is symmetric with respect to the point (m, 0) for some m > 0. Given two candidates under the RM model, if C1 lies to the right of C2, the line of indifference is perpendicular to (C1 - C2) and passes through the origin. The region of support for C1 includes the center of symmetry (m, 0) and hence C1 is supported by more than half of the electorate. Thus, the Condorcet candidate is the rightmost candidate in a multicandidate race under the RM model. For a proximity or mixed model with sufficiently large mixing parameter b, the candidate nearest (m, 0) will be favored. For approval voting and the RM model, those candidates will be ¯ be the mean favored who lie in the direction of the asymmetry. Let C ¯, the line vector for all candidate positions. If C1 lies to the right of C of indifference between voting for and not voting for C1 is perpen¯) and passes through the origin. Thus, C1 is supported dicular to (C1 - C by more than half of the electorate because the region of support of C1 includes the center of symmetry, (m, 0). The most favorable posi¯) is nearest to the horition is the one for which the vector (C1 - C zontal and in the direction of the asymmetry. 11.3 Simulation Results To gain greater insight into the regions of support for alternative voting systems, computer simulations were conducted to estimate the degree of support for candidates as a function of distance from the neutral point.4 Table 11.1 reports the tendency of a voting system to choose centrist candidates as a function of the model of voter decision making. Under either the RM (b = 0) or a mixed proximity–RM 4

Each simulation specified a two-dimensional spatial model, in which both voters and candidates follow an uncorrelated bivariate normal distribution with marginal standard deviations equal to 1. Each simulated run consisted of 1,000 elections (each of 200 candidate sets is associated with 5 different electorates). Each election involved 5 candidates and an electorate size of 101. Separate runs were conducted for each of three levels of the mixing parameter b and for two levels of asymmetry (m = 0 representing no asymmetry and m = 0.5 representing moderate asymmetry). Under approval voting, each voter was assumed to vote for those candidates whose utilities exceeded the mean utility for that voter. This strategy maximizes expected utility under the assumption of ignorance about the votes of the remaining electorate (see Weber, 1977; Merrill, 1981).

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Part II Models of Party/Candidate Behavior

Table 11.1. Mean rank of winning candidate by voting system, ranked by nearness to neutral point Asymmetric Electorate (m = 0.5)

Symmetric Electorate System

b=0

b = 0.5

b = 1.0

b=0

b = 0.5

b = 1.0

Plurality Approval voting Borda STV

4.10 3.18 3.44 3.92

2.65 1.43 1.35 2.15

1.65 1.38 1.23 1.39

3.93 3.13 3.51 3.81

3.16 1.81 2.04 2.70

2.00 1.51 1.49 1.74

Notes: The RM directional model is represented by b = 0; the pure proximity model, by b = 1.0. Rank 1 is nearest to neutral point; rank 5 is farthest. Source: Merrill (1993: Table 1).

(b = 0.5) model, approval voting (AV) and the Borda count are more likely to choose a centrist candidate than plurality and STV; for a pure proximity model (b = 1.0), this differential is considerably reduced. For a symmetric electorate both approval voting and Borda produce tied results for the RM model, so for a finite sample, the mean rank of five candidates is near 3. For an asymmetric electorate under RM, approval voting and Borda tend to choose the most extreme candidate in the direction of the asymmetry although that may not be the most extreme candidate overall. In contrast, plurality – under the RM model – chooses the candidate at the sharpest point of the convex hull of the candidates, which is likely to be the most extreme. STV first eliminates the candidates in the interior of the convex hull, then those at the least sharp vertices, choosing in the end a relatively extreme candidate. 11.4 Conclusions The simplest extension to multicandidate elections of majority-rule voting is the plurality system. Under the RM directional model, only candidates positioned at exposed points of the convex hull of the candidate set receive any votes under plurality or – at each stage –

Strategy under Alternative Voting Procedures

163

under STV systems. Winning candidates under these systems tend to be located in relatively extreme positions. By contrast, under the approval voting and Borda systems, the RM model implies equal support for all candidates, at least for a symmetric electorate. These implications appear unrealistic, but they are alleviated by adding a proximity component to the model. Under an RM model with proximity constraint, centrist candidates are favored under approval voting and Borda, but not necessarily under the plurality and STV systems. Regardless of voting system, as expected, the RM model is far more likely to advantage extremist candidates than voting based on either the mixed or pure proximity model. None of these effects changes greatly upon the introduction of asymmetry in the distribution of voter ideal points into the model.

POSTSCRIPT

Taking Stock of What’s Been Done and What Still Needs to Be Done In a democracy it is frequently necessary to enter the polling booth, holding one’s nose. Bernard Levin, Quote, Unquote (1989: 88)

The aim of this book has been to offer a unified theory of voter choice. At the outset, we asked two major questions: (1) How do voters translate information about the issue positions advocated by the candidates into voting decisions based on the voters’ preferences for outcomes? (2) What are the implications of such voting behavior for the strategy of candidates and parties? For example, how do candidates and parties adopt their issue positions in the light of what they come to know/believe about voter decision rules and voter issue preferences? Under what conditions will there be candidate/party equilibria with respect to issue positions, i.e., sets of platforms that, once adopted, will remain more or less the same because no candidate (or party) believes that she can improve her vote share by shifting her proposed issue positions as long as the other candidates/parties do not change theirs? These questions have been at the heart of a vast body of research stemming from the seminal work of Anthony Downs (1957). No work of scholarship stands alone; we have built on the work of many others. As we see it, the heart of our own contribution to earlier work seeking to answer these two questions1 is fourfold. First, we have provided an empirically practical way of integrating the Downsian perspective on issue proximity with later work – emphasizing directionality of choice – by offering a unified model of 1

Other critical questions stemming from Downs (1957) include, “When will people choose to vote?” and “When (and to what extent) will people choose to become politically informed?” These are questions that we do not deal with in this volume.

164

Postscript

165

voter choice. Our model, incorporating proximity, direction, and intensity, allows us to directly examine the relative weight of each of these factors on voter behavior and to test for the statistical significance of each.2 In this framework, earlier models such as those of Downs (1957), Matthews (1979), and Rabinowitz and Macdonald (1989) become special cases.3 Moreover, we have shown that the extent of voter discounting of candidates’ proposed issue positions in terms of change from the status quo (Grofman, 1985) is mathematically analogous to the degree of directionality and intensity in our unified model of voter utility. Second, like the moderate voter who, according to the traditional spatial theory, prefers a moderate course, we defend – both theoretically and empirically – a moderate theory of voting – one that is intermediate between the extremes of the pure directional models on the one hand and the pure proximity model on the other. Such a stance may lack the directness of an uncomplicated theory but, we believe, is more consonant with reality. Following up on a hypothesis offered by Enelow, Endersby, and Munger (1993), we have found support for a difference in voter decision rules for different types of candidates. In American politics, both agreement on issue location and direction of change appear important in predicting voter attitudes toward all candidates, including incumbents, whereas the third factor of our model, intensity, appears significant only for challengers. Third, considerably extending ideas in Grofman (1985), we have shown how the location of the status quo can play a vital role in affecting voter decisions among candidates and parties. As the status quo shifts, so too may voter choices, even though voter issue preferences remain unchanged. We have used this discounting idea to model the dynamics of party alternation in office. Finally, we have developed an extended version of our initial unified model of voter choice and applied it to model strategic inter2

3

Incorporating several pure models into a single unified model permits us to see relationships between pure models, to see each of them as poles on continua representing mixed models of different degrees of mixture, and to simultaneously test all the models and their combinations with the others acting as controls. Relatedly, a mixture of proximity and directional–intensity components may be interpreted as a directional model with proximity constraint – thus operationalizing Rabinowitz and Macdonald’s concept of candidate acceptability.

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Postscript

action between parties. The two key modifications are as follows: (1) Following the ideas of Hinich (1977) and Coughlin (1992), we permit voters to choose probabilistically, and (2) following ideas in Erikson and Romero (1990) and Adams (1997a, b) we incorporate into the voter choice function an element of partisan identification that acts to bias the voter in favor of the party for which he has previous loyalty. Though the perspective in this book has been heavily influenced by the classic work of Downs (1957) and follow-up work in spatial social choice, the incorporation of the concept of partisan affiliation as creating a form of partisan bias links our work to what is informally known as the Michigan school and the equally seminal The American Voter (Campbell et al., 1960).4 When we model voter preferences probabilistically and include a biasing component for partisan identification, stable strategic choices in policy space, i.e., equilibria, do exist – even in a multicandidate setting.5 Equilibria predicted under pure proximity assumptions are tightly clustered, whereas those expected under pure Rabinowitz–Macdonald assumptions are highly dispersed.6 Yet neither extreme dispersion nor extreme concentration is empirically observed. An intermediate probabilistic model, i.e., a directional model with proximity constraint and partisan bias, however, can realistically predict party behavior, as our empirical analysis of the Norwegian electorate indicates. This work is being extended to other multiparty polities.7 Future Work Though we believe that we have made some important advances in modeling how voters decide how to vote and how candidate and parties structure the choices open to the voters, and in testing competing models, we make no claims that we have incorporated all of the critical variables. Several chapters of the present book contain 4

5 6

7

See also Fiorina (1981); Miller and Shanks (1996). Cf. Grofman (1987); Feld and Grofman (1991). These results are very different from those of most other multiparty models. In the first case, equilibrium locations are concentrated toward the median; in the latter case, they press against an arbitrarily defined circle of acceptability. Similar results have been obtained for the 1988 French presidential election (see Adams and Merrill, 1998).

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167

models with party identification as a concomitant variable. In joint work with James Adams we are pursuing a fuller incorporation of social-psychological and sociological variables into the unified model of voter and party choice. A follow-up volume is planned. Also, we have seen that it may be difficult to distinguish voting behavior under the mixed proximity–RM model from voting behavior under the Grofman discounting model. Teasing out this distinction will have to be left as an unanswered puzzle.8 Our work on two-party competition should be seen as complementary in perspective to three books that also offer a synthesis and extension of Downsian ideas: Enelow and Hinich (1984), Hinich and Munger (1994), and Alesina and Rosenthal (1995). Each of these works stresses some factors that we have largely or totally neglected. For example, both Enelow and Hinich and Hinich and Munger consider how candidate characteristics can influence voter choice; Alesina and Rosenthal consider the impact of economic conditions on voter choice and look at how voters might balance off outcomes in one election contest against outcomes in another in deciding what set of candidates to vote for; and Hinich and Munger consider in a very sophisticated way the link between the multiplicity of issues and an underlying ideological space of lower dimensionality.9 Two other important issues are the degree of ambiguity/clarity of candidate positions and differences in issue salience among issues and among different sets of voters. The former is addressed in Hinich and 8

9

Yet this distinction may not matter for the analysis of party response, because either model predicts the same voter behavior – and it is this behavior and the optimal party response it implies that we have shown to occur empirically, as in our analysis of the Norwegian electorate. Traditionally, each dimension of a spatial model has been associated with an issue. Voters have been assumed to compare their positions on an issue with those of the candidates. However, Hinich and Munger (1994) argue that primarily voters make decisions not on the basis of a multitude of issues but on the basis of ideologies, i.e., abstract worldviews or sets of internally consistent principles associated with candidates or parties. Such an ideology is at once a cue to forecasting policy positions on a number of issues and also the basis of a credible and consistent commitment for the candidate or party who espouses it. Hence, they argue, issue space is governed by a space of ideology. Unlike issue positions, ideological stance (and the number of ideological dimensions) are not directly observable but may be ascertained through statistical methods such as factor analysis from observed issue positions. Consideration of the difference between issue dimensions and policy dimensions and the question of the dimensionality of the spaces we are likely to encounter in real political competiton is outside the scope of this book.

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Munger and also in another recent book in the Downsian tradition, Alvarez (1997). The latter is addressed in a thought-provoking paper by Hammond and Humes (1993). For example, although the utility functions as we have defined them reflect relative salience among issues, explicit salience parameters need to be incorporated directly into the models (see Adams and Merrill, 1998). We regard all of these complicating factors as important in understanding voter choice, and we plan in future research to build some or all of them into the unified model of voter choice proposed in this volume. We might also note that our work on two-party competition makes two other important simplifications. First, we neglect the role of party primaries (or other mechanisms) in shaping candidate choice (see, e.g., Aranson and Ordeshook, 1972; Coleman, 1972; Owen and Grofman, 1995). Second, we neglect the importance of multiple constituencies in which elections are simultaneously taking place (see, e.g., Shvetsova, 1997; Grofman, Koetzle, McDonald, and Brunell, forthcoming). Just as we do not regard our work on two-party competition as definitive, we regard our approach to multicandidate and multiparty competition and our attempts to deal with the implications of alternative voting systems as promising, but very far from the last word. Our work on these topics should be seen as complementary to recent work by scholars such as Kenneth Shepsle (1991), Norman Schofield (1993, 1996), and Gary Cox (1997). Shepsle provides a review of recent modeling and equilibrium analysis on multiparty competition10; Schofield introduces the concept of the heart as a focal point for optimum party location and emphasizes the importance of postelection coalition formation; Cox considers strategic factors that may influence voting choices as a function of type of electoral system in use. Finally we might note that, like most researchers, we have focused primarily on voter–candidate interaction in a static setting. In Chapter 9, however, we introduced a model that is dynamic over elections. We showed that discounting in a one-dimensional, two-party proximity model is sufficient to imply that the two parties have stable, but distinct, strategies. The existence and distinctness of these strategies is 10

See also Enelow and Hinich (1989).

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169

rather robust with respect to a variety of assumptions. Party victory can be expected to switch back and forth between the two partisan groups, but with the party whose ideal point is nearer the median voter getting the lion’s share of the spoils. But the dynamics of a single campaign also need to be better addressed, with appropriate panel data to track the effect of persuasion from candidates, parties, and opinion leaders on voter decisions (see, e.g., Skaperdas and Grofman, 1995; Regenwetter, Falmagne, and Grofman, 1996; Alvarez, 1997). Persuasion must be understood to include not only a candidate’s own positioning but also the often effective labeling of the position of one’s opponent, and other forms of negative advertising. We regard persuasion as a major frontier for research on voter choice. Still, in summing up, we prefer to stress how much progress we and other scholars in the Downsian tradition have made, both theoretically and empirically, rather than ending on what might seem to be a note of “so many ideas still to be explored, so little known.” Downs’s fundamental insight was that democratic politics is about the interactive links between party platforms, government policy performance, voter preferences, and voter choices. It is our hope that this book will further contribute to uncovering the nature of those links.

Appendices

Appendix 3.1 Mixed Proximity–RM Models Following Iversen’s (1994) argument that voter utility in a directional model should be idiosyncratically constrained, we have defined the RM model with proximity constraint as a mixed directional and proximity model where b is a mixing parameter. In fact, our formulation, the mixed model of Rabinowitz and Macdonald (1989), and Iversen’s (1994) formulation are mathematically equivalent.1 The formulation we have presented has several advantages over the others. First, the 1

Both the original mixed model of Rabinowitz and Macdonald (who credit the original formula to Howard Rosenthal): 2

U ( V, C) = 2 r1 V ◊C - r2 [ V + C

2

]

(A.1)

and the Iversen (1994) formulation U ( V, C) = sV ◊C - (1 - s ) V - C

2

(A.2)

are equivalent to the formula we have given for the RM model with proximity constraint U ( V, C) = 2(1 - b )V ◊C - b V - C

2

(A.3)

To see that eqn. (A.3) is equivalent to the Rabinowitz–Macdonald form, note that eqn. (A.3) may be expanded as 2

2 V ◊ C - 2 b V ◊ C - b[ V - 2 V ◊ C + C 2

= 2 V ◊ C - 2 b V ◊ C - b[ V + C 2

= 2 V ◊ C - b[ V + C

2

]

2

2

]

] + 2 b V◊ C

Dividing both sides by b and writing r = 1/b, we have

170

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171

parameter b is easily interpreted as a mixing parameter anchored at 0 and 1 for the pure RM and pure proximity models, respectively. Second, b changes linearly as the indifference plane between the candidates moves from its location for a pure RM model to that for a pure proximity model.2 Third, the two pure models correspond to straightforward null hypotheses that are convenient to test statistically; namely, b = 0 for the pure RM model and b = 1 for the pure proximity model. By contrast, Rabinowitz and Macdonald’s model ratio (equivalent to our 1/b) would have the value of +• for a pure RM model, which makes the corresponding null hypothesis awkward to test. A compromise between Rabinowitz and Macdonald’s original idea of an arbitrary circle of acceptability and Iversen’s idiosyncratic constraint is the following simpler model, which we call the RM model with centered constraint. It is defined by subtracting from the scalar product utility a multiple of the squared distance from the neutral point to the candidate location. Accordingly, the constraint on the voters’ utilities for a candidate is universal, not dependent on the voter’s location. The model is specified by the utility function U (V, C) = 2(1 - b )V ◊C - b C

2

where b is estimated from the data. The subtracted term imposes a diminution in utility as candidates recede from the neutral point, the 2

2 rV ◊C - [ V + C

2

]

(A.4)

Now dividing both sides of eq. (A.1) by r2, and defining the model ratio as r = r1/r2, we likewise obtain eq. (A.4). Thus b is simply the reciprocal of Rabinowitz and Macdonald’s model ratio. To see that eqn. (A.3) is equivalent to the Iversen form, set b = 2(1 - s)/(2 - s) and note that this makes 1 - b = s/(2 - s). Thus, (A.3) becomes 2s 2( 1 - s ) V◊ C V-C 2-s 2-s 2

2

=

2 [ sV◊ C - (1 - s) V - C 2-s

2

]

which is equivalent to the Iversen expression. Thus, for example, a value of b equal to 0.5 corresponds to a model ratio of 2.0 in Rabinowitz and Macdonald’s notation, which they would interpret as a rejection of the pure proximity model. In fact, it is equally a rejection of the pure RM model. In addition, their scale tends to exaggerate any dominance of the RM component.

172

Appendices

effect of which is slight for small deviations but increasingly pronounced as the distance becomes larger.3 Indifference occurs for the RM model with centered constraint if V lies on the perpendicular bisector of the segment connecting A9 and B9, where A9 =

b A 1-b

B9 =

b B, 1-b

and

i.e., if the discounting factor is replaced by

b . 1-b Thus, for b < 1/2, voters may behave as if the candidates are more clustered about the origin and are hence more likely to support relatively extreme candidates than under a pure proximity model, where b = 1/2. Appendix 4.1 Methodology: Data Analysis To facilitate interpretation of the 1–7 scales of the American NES, scales were reversed when necessary so that “1” represents the most liberal and “7” the most conservative position on each issue. The four-point abortion scale in 1992 was expanded to a seven-point range. All scales were shifted by subtracting 4 in order to be centered 3

For a fixed voter V, utility under this model is a quadratic function of C with maximum at È1- b ˘ C=Í V Î b ˙˚ i.e., in the same direction as the voter. Maximum utility occurs at C = V if b = 1/2 (as the proximity model would predict). It occurs at more extreme points as b decreases, tending to infinity as b approaches zero.

Appendices

173

at zero. Thus each scale varies from -3 on the left to +3 on the right. Following Markus and Converse (1979), Rabinowitz and Macdonald (1989), and Merrill (l995), scalar products and quadratic distances were normalized – for each voter – by dividing by the number of issues with usable data. This maximizes the number of usable cases, by including cases with incomplete data, while at the same time weighting all cases equally. Respondents were restricted to those who reported voting for one of the two major candidates and could place themselves and the Democratic and Republican nominees on the seven-point liberal–conservative scale (and who did not place the Democrat to the right of the Republican). Respondents placing the Democrat to the right of the Republican were omitted because of apparent misunderstanding of the meaning of the scale. Partial analysis with all respondents included suggests that this omission has little effect on the results (see also Dow, 1998a). Although control variables (in the form of voter demographics) have been used (see, e.g., Rabinowitz and Macdonald, 1989; Macdonald, Rabinowitz, and Listhaug, 1995; Dow, 1998a) in assessing spatial models through regression or other methods, we believe their use is not always compelling. We have chosen not to use control variables in reporting the results of our voter utility models, although we have done a dual analysis with controls in some test cases. Introduction of simple control variables – such as race, region, or party – appears not to substantially alter the conclusions concerning voter utility. We will use a control variable representing party identification in our study of voter choice in Chapters 6 and 7, because we expect choice to be more heavily dependent on party ID than is utility. Appendix 4.2 Methodology: Linear versus Quadratic Utility Functions Because – in American elections – we are most interested in the choice that voters make between two candidates, we might consider the thermometer-score difference between a pair of candidates. Such a difference, however, plots as a straight line, not only according to the RM model, but also according to the proximity model when a

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quadratic Euclidean metric is used.4 Hence the predicted difference of utilities is equivalent, and we cannot expect to distinguish between the two models. If, instead, we use a linear Euclidean metric for proximity, the predicted difference between utilities plots, not as a straight line, but rather as a backward Z-shape, as in Figure A.4.1a. Is there empirical evidence to prefer the quadratic or linear form? In Figure A.4.1b, we plot the thermometer-score difference for three pairs of candidates from the 1984 American NES. The evidence is ambiguous, but seems more suggestive of the backward Z-shape of a linear metric, at least for the two nominees, Reagan and Mondale. If the two moderates, Ford and Hart, are located at approximately +1 and -1, respectively, then under a linear metric, we would expect the plot to level off above +1 and below -1, which is not the case. Likewise, for the two Democrats, Kennedy and Hart, we would expect the plot to be level above -1, again contrary to fact. If, however, the true model is more nearly a mixture of RM and proximity components (see Chapter 5), the intermediate nature of the plots between constant sloping lines and backward Zs could be explained in terms of a linear metric, but not a quadratic one. Despite this tentative evidence for a linear as opposed to quadratic metric, we use a quadratic metric in our analysis because of the interpretations that it permits for a mixing parameter in the unified model that we introduced in Chapter 3 and test in Chapter 5. Empirical effects of the difference between the linear and quadratic utility will be noted as we go along. Appendix 4.3 Methodology: Mean versus Voter-Specific Placement of Candidates Rabinowitz and Macdonald (1989); Macdonald, Listhaug, and Rabinowitz (l991); and Rabinowitz, Macdonald, and Listhaug (l991, 1993) use mean ideological placements of the candidates, rather than 4

For the proximity model, the difference in utility between candidates C1 and C2 is U ( V, C1 ) - U ( V, C 2 ) = -( V - C1 ) ◊ ( V - C1 ) + ( V - C 2 ) ◊ ( V - C 2 ) = 2 V ◊ (C1 - C 2 ) + C 2

2

- C1

2

which differs from the difference between RM utility functions by the constant ||C2||2 - ||C1||2. The plot of both utility differences is a straight line.

a

b

Figure A.4.1 Utility differences by model. a. Utility differences for hypothetical candidates for three models. b. Thermometer score differences for selected candidates versus liberal– conservative position of the respondent (1984 American NES). 175

176

Appendices

voter-specific (idiosyncratic) placements. These authors find a substantially better fit for the pure RM model in fitting a regression model to explain thermometer scores for American, British, and Scandinavian data than is obtained using voter-specific placements. Others, using voter-specific placement of candidates, find this preference for the pure RM model over the pure proximity model greatly reduced or reversed (Pierce, 1993, 1995a, 1997; Merrill, 1994, 1995; Krämer, 1996; Gilljam, 1997; Krämer and Rattinger, 1997; Merrill and Grofman, 1997a; Westholm, 1997; Dow, 1998a). Voter-specific placement has been used in other contexts (see, e.g., Palfrey and Poole, 1987: 522; Enelow, Endersby, and Munger, l993: 132). The methodological differences that lead to these disparate results are discussed below, along with the biases introduced by both methods and attempts at their resolution. Macdonald, Rabinowitz, and Listhaug (1995) argue that only a single objective measure of party or candidate location must be used to link candidate strategy with popular support. But shifts in candidate position can be expected to be associated with corresponding shifts in the distribution of perceptions of a candidate’s position by voters. Thus the linkage of candidate strategy and voter support does not require the association of a single numerical value with candidate strategy. Furthermore, Enelow and Hinich (1994: 168) find that correlated voter–candidate data are “. . . completely consistent with a spatial model in which candidates have fixed, stable locations on a set of underlying predictive dimensions.” Macdonald and Rabinowitz (1997) argue that voters process political stimuli on an on-line basis, i.e., each stimulus may affect their opinion but is not stored in memory (Lodge, McGraw, and Stroh, 1989; Lodge, Steenbergen, and Brau, 1995). Hence voter-specific candidate placements are suspect. Although data from such placements are soft, we agree with Maddens (1996) that it is difficult to see how validity can be improved by using mean placements over all voters. We argue that use of voter-specific placement is preferable and in particular avoids a bias in favor of the RM model. It would seem obvious that a voter’s evaluations are more closely attuned to the voter’s own assessment of a candidate’s position than to the national mean placement, which is not known to the voter. As Dow (1998a) points out, spatial utility theory requires that a voter’s issue position

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177

be compared with his perception of a candidate’s issue position, not with someone else’s perception of that position. Not all voters will locate the candidate in the same position – especially since there will be subjective interpretations of the scale presented by the survey instrument and because different voters are exposed to different information flows. For example, suppose a labor party candidate’s mean placement is 3 on a 1–7 left–right scale, but a particular voter – who supports the labor party and agrees with the party on the issues – interprets the scale in such a way that he places both himself and the party at 1. Using mean placement leads to a misleadingly low proximity utility for this voter, whereas his RM utility will be appropriately high. We would argue that there is more likely to be consistency among voters in judging whether the voter is close or not close to a candidate than in interpreting absolute position on an arbitrary scale. The voter in our example correctly judges distance but is inconsistent with the conventional (average) estimate of candidate position. For such voters, voter-special placement is more accurate than mean placement. As Lewis and King (1998) point out, if candidate location really is variable, use of mean placement introduces measurement error. Use of mean placements biases the results in favor of a directional model, because random guessing by uninformed voters (see Powell, 1989) tends to draw mean placements toward the neutral point. If computation is based on the mean, thermometer scores of voters who place themselves between the mean placement of the candidate and their own placement of the candidate may appear to correlate well with directional predictions but poorly with proximity predictions because the mean misspecifies candidate position as viewed by the voter. For example, suppose an informed voter (correctly) places both himself and the candidate at location 3 on a scale from -3 to +3. Both models predict high utility for this candidate. The candidate’s mean position is likely to be less extreme, say 1. The directional model still assigns a higher utility (scalar product) to the voter at location 3 than at any other position, whereas the proximity model assigns a low utility relative to that for voters at or near 1. Thus, in this case, if mean perceived position for candidates is used in the analysis, the directional utility function appears, incorrectly, to correlate more closely

178

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with the respondent’s thermometer score than does the proximity utility function. Use of mean placement introduces a bias that is associated with the interpersonal comparisons in voter utility models – a bias that can be partially alleviated by the Westholm adjustment (see Appendix 5.3). Use of voter-specific placement introduces its own bias, however, because of projection or rationalization effects in which a voter adjusts the policy location of the candidate he prefers so as to move it closer to the voter’s own ideal point. We deal with this issue in Chapter 5. The strong substantive preference for the RM model obtained by Rabinowitz et al. is dependent on use of mean candidate placement; it disappears with voter-specific placement even after adjustment for projection. Platt, Poole, and Rosenthal (l992) attempt to locate alternatives accurately without having to choose between mean or individual placements by uninformed voters. These researchers, by analyzing members of the U.S. Congress rather than a mass public, rely on a presumably well-informed elite to place candidates. Substantively, they find the proximity model superior to the RM model. Several studies have compared proximity and RM models under both mean and voter-specific placements of candidates. They confirm the findings of Rabinowitz and his colleagues for mean placements and generally find the results for voter-specific placements much more favorable to the proximity model. For example, Krämer and Rattinger (1997) repeat the meanplacement calculations of Rabinowitz and Macdonald for the American NES during the period 1968–92. Whereas with mean placement, values are generally higher for the RM model (except for Nixon in 1968 and Carter in 1980), the reverse is true when voter-specific placement is used. Pierce (1993, 1995a, 1997) finds that the RM model fits better for French data for the 1988 election when mean placement is used but obtains mixed results when voter-specific placement is used. For Swedish data, Gilljam (1997) finds the RM model superior in more cases using mean placement, but the reverse for voter-specific placement. Using mean placement for the Norwegian data, he confirms the results of Macdonald, Listhaug, and Rabinowitz (1991) that the correlation is higher for each party under the RM model (the mean difference between the correlation coefficients for the two

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179

models is 0.093). Using voter-specific placement, this preference drops (the mean difference in coefficients is 0.026). In a methodological study, Lewis and King (1998) compare mean and voter-specific placement for the same Norwegian data analyzed by Macdonald, Listhaug, and Rabinowitz (1991), Macdonald, Rabinowitz, and Listhaug (1998a), and Westholm (1997) as well as by us in Chapter 7. They obtain the usual preference for the RM model using mean placement, but not for voter-specific placements. Lewis and King explain the contrasting results in terms of what parameters are estimable under the differing assumptions, in the presence of intercept terms representing unmeasured, candidate-specific effects. We address this question in Chapter 7 when we deal with voter-choice models for multicandidate electorates (see, especially, Appendix 7.1). Appendix 5.1 The Nature and Magnitude of Projection Effects As we have indicated, projection occurs when a voter places a favored candidate nearer to himself than might objectively be determined.The extent to which projection occurs can be assessed by decomposing the variance of candidate placement over voters. In a one-dimensional ¯ denotes model, if V denotes the respondent’s self-placement and C the mean candidate placement, the effect of projection might be ¯) for a favored candidate and expected to be proportional to (V - C the negative of this quantity for a disfavored candidate. Such an analysis was performed by Markus and Converse (1979), who employed the thermometer score less 50 to distinguish between favored and disfavored candidates. They found that projection, although statistically significant, accounted for from near zero to only 8 percent of the variation over five issues and two candidates in the 1976 NES data set. Merrill and Grofman (1997a) performed a similar analysis for the eight presidential nominees in the 1980–92 NES data, using placements on the liberal–conservative (L/C) scale for V and C and reported vote choice as the determinant of favored/disfavored status. Thus, for example, projection for the Republican is expected to be positive (i.e., in the conservative direction) for a voter who reports voting for the Republican and places herself more conservative than

180

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the Republican’s mean placement. Projection for the Democrat is expected to be negative (i.e., in the liberal direction) for a voter who reports voting for the Democrat and places himself more liberal than the Democrat’s mean placement.5 Although statistically significant, the projection factor found by Merrill and Grofman explains only 6–18 percent of the variance of candidate placement. These consistently small values – over several elections and issues – suggest that projection constitutes only a small part of variation in candidate placement over voters. Still, conceivably, projection effects might have a more sizable impact on measures of the relative contributions of the proximity and directional models, especially for issues that are less well-defined. In an attempt to neutralize the effects of projection, they adjusted each respondent’s placement of a candidate by subtracting the regression ¯), where s denotes some term obtained by regressing C on s(V - C measure of the voter’s support for the candidate. Merrill and Grofman take s = ±1 according to whether or not the voter voted for the candidate. In other words, the best estimate of the respondent’s projection was deleted from the respondent’s candidate placement. Technically, the projection-adjustment procedure is as follows. Using the entire sample of respondents, fit the model C = b0 + b1s(V - C¯) + e, where s = 1 if the voter voted for the candidate and s = -1 ¯), where bˆ1 is the estimate if not. We then replace C by C - bˆ1s(V - C ˆ for b1. The projection parameter, b1, is estimated separately for each issue. To assess the effect of the correction for projection, we have included in Table 5.3 in Chapter 5 comparisons of estimates for the intensity parameter q and the mixing parameter b for both voterspecific placements and mean placements, as well as for projectionadjusted rescoring of voter-specific placements, so that we can readily compare the three bases of estimation. Estimates for q and b for adjusted and unadjusted voter-specific placements are quite similar to each other but both are substantially different from those for mean placement. In the unified model, the incumbency pattern for the intensity 5

Vote choice rather than thermometer score is used to specify candidate support since otherwise the thermometer score would appear on both sides of the regression equation in the adjusted analysis, rendering the statistical results meaningless.

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181

parameter q is apparent for voter-specific placements whether adjusted for projection or not (see Table 5.3).6 It is obscured, however, when using mean placements. Appendix 5.2 Interpretation of Model Parameters When the unified model includes the intensity factor q, the value of b loses the simple interpretation it has in an RM model with proximity constraint. No longer is the location of the indifference plane between the candidates a linear function of b nor is it a linear function of q. The small values of b, however, do not imply that proximity plays a smaller role than direction or intensity in explaining utility. A rough assessment of the relative importance of the proximity and directional components can be made by performing a linear regression with thermometer score as dependent variable and with X1 = -||V - C||2 and X2 = 2

V◊ C V C

[V

q√

C]

as independent variables, where qˆ is the estimate for q obtained by nonlinear regression. Comparison of the standardized regression coefficients obtained gives an approximate measure of the relative roles of the two components. The results suggest a larger contribution by the directional component. Appendix 5.3 The Westholm Adjustment for Interpersonal Comparisons Discrimination between voter utility models for a fixed candidate relies entirely on interpersonal comparison of utilities when candidate location is based on mean placement, and partly does so for voterspecific placement. If candidate location is a constant over voters, both the proximity and RM utilities are entirely functions of voter location. One prediction of the RM theory is a greater spread in utility 6

The incumbency effect is significant at the 0.02 level for unadjusted voter-specific placements and at the 0.01 level for projection-adjusted placements.

182

Appendices

between candidates for extreme voters than for moderate voters. Although this expectation is supported empirically, it completely ignores what we and Westholm (1997) argue is the more important implication of the RM theory – namely, that a fixed voter prefers more extreme candidates to moderate ones as long as both are on his side of the neutral point. Use of voter-specific candidate placements permits the RM utility function, V · C, to be a function of both V and C, but in our voter utility model, the value of C reflects only differing placements of a single candidate rather than different candidates. Thus, a stronger test of alternative models would involve several candidates simultaneouly, with at least two on the same side of the neutral point. We do just that in Chapter 7 when we focus on models of voter choice as opposed to voter utility. Westholm (1997) introduces a method to test models of voter utility while avoiding interpersonal comparisons; his method is primarily effective for multicandidate elections. He normalizes each respondent’s model-predicted utility by subtracting the respondent’s mean utility over all parties (candidates). Thermometer scores are adjusted similarly. We briefly motivate Westholm’s technique and outline his results below. For a fixed candidate, following the Westholm adjustment, the utility functions for the proximity and RM models differ only by a constant, so the method is of limited usefulness for distinguishing between the models for a single candidate. For a fixed voter, however, the adjusted utility functions over candidates retain their distinct shapes, although the distinction between the RM and proximity models can still be confounded with nonspatial factors that affect the candidate-specific intercepts. Such confounding can be avoided by omitting the intercepts from the model; the bias thus introduced appears prohibitive for two-candidate contests but acceptable in elections with several candidates (see a full discussion of this point in Appendix 7.1). Thus, the Westholm method is best used in multicandidate elections. Failure to make Westholm’s adjustment in models of voter utility tends to bias the results against the proximity model (for either method of candidate placement) because mean utilities over candidates as predicted by the proximity model are strongly dependent on the location of the voter, dropping off sharply for extreme voters, who

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183

are far from most candidates. By contrast, mean utilities as predicted by the RM model are relatively constant over voter locations, with positive and negative utilities balancing each other on average. Yet, empirically, the mean thermometer scores for a set of candidates are fairly constant as a function of voter locations, with only a modest rise near the middle of the scale. This constancy is presumably a result of respondents’ adjustment to the survey instrument, in which each respondent tends to give a candidate whom he rates average a thermometer score near 50. It could, of course, be argued that this observation is evidence for the RM model. Westholm’s analysis, however – stratified by degree of eccentricity of the voter so that the interpersonal adjustment described above is diminished – shows a distinct preference for the proximity model for each level of eccentricity, dropping off smoothly as eccentricity increases. This is not what would be expected if the RM model were correct. Furthermore, in the American NES, most voters who place both candidates on the opposite side of the liberal– conservative scale from themselves nevertheless give favorable (i.e., greater than 50) thermometer scores for their preferred candidate. This is to be expected as a scale-adjusted response but not as a consequence of the RM model. Accordingly, Westholm favors a simpler explanation: voters – when confronted with a thermometer feeling scale from 0 to 100 and instructed to interpret ratings between 50 and 100 as warm (favorable) feelings but ratings between 0 and 50 as not feeling favorable – at least in part adjust their response to the scale and how they rate other candidates rather than to how they might perceive their utilities to compare with those of other voters. Westholm applies his interpersonal adjustment to the 1989 Norwegian Election Study, pooling together respondent evaluation and/or placement of seven major parties (see Table A.5.1).7 Using the adjustment, Westholm obtains a higher Pearson correlation between thermometer (sympathy) scores and the proximity model (0.62) than between thermometer scores and the RM model (0.55). Using standardized regression coefficients, he shows that interpersonal adjustment exerts a strikingly greater effect on the results than either 7

See Section 7.3 for a description of the data analyzed by Westholm.

184

Appendices

Table A.5.1. Standardized regression coefficients for the 1989 Norwegian Election Study for various modeling choices for proximity and RM utilities Method of party placement Voter specific

Mean

Linear

Quadratic

Linear

Quadratic

Interpersonal adjustment Proximity utility RM utility

.56 .07

.48 .12

.46 .09

.42 .13

No interpersonal adjustment Proximity utility RM utility

.32 .26

.27 .29

.23 .30

.22 .30

Source: Adapted from Westholm (1997: Table 3).

the method of party placement (voter-specific or mean) or the choice of metric (linear Euclidean or quadratic Euclidean). Discrimination between alternative spatial models depends on a number of methodological issues that may affect the results, including variation in the method of determining candidates’ issue positions, projection and rationalization, interpersonal bias, and the mathematical form of the proximity utility function. Of these factors – for the American NES data studied – allowing for voter-specific placement of candidates has the largest effect on the relative contribution of the competing models (although the possible effect of interpersonal bias is difficult to assess for two-party elections). By contrast, for the multiparty data from Norway and Sweden, Westholm shows that interpersonal adjustment has by far the greatest effect among these factors, although the predictive performance of competing models also differs significantly between methods of candidate placement. The Westholm adjustment for interpersonal bias depends on both placement and thermometer scores for candidates other than the focal candidate. In a two-candidate race – typical in the United States – such adjusted values for the two candidates are simply negatives of each

Appendices

185

other,8 with the adjusted RM and proximity functions differing only by a constant. Some NES studies (e.g., those of 1980 and 1988) include individuals other than the Democratic and Republican nominees, but the selection of those persons is often arbitrary and many such individuals are not actual candidates or, even if they are, necessary data may not be available (e.g., for Ross Perot in 1992). In any event, many of these persons do not represent independent parties, but rather the same Democratic and Republican parties represented by the nominees. In the 1988 American NES, placement data were available for Ronald Reagan and Jesse Jackson in addition to Bush and Dukakis. Estimates of b for Reagan and Jackson in the mixed proximity– RM model are almost identical to those for their respective party nominees. It would be useful to develop a theoretically based interpersonal adjustment not dependent on specific candidates or pseudocandidates external to the focal candidate. Such appears possible by subtracting from the model-predicted utility the expected value of that utility over potential candidates. If we assume that the expected value of candidate position is zero and that the standard deviation, si, for each issue dimension is finite, the expected value9 of the proximity-predicted utility is -||V||2 - Ssi2. Since the constant, - Ssi2, is immaterial in defining utility, we may adjust proximity-predicted utility by subtracting the quantity -||V||2, yielding an adjusted utility equal to -||V - C||2 - (-||V||2) = 2V · C - ||C||2. Multiplying the ||C||2 term by a parameter yields an adjusted mixed proximity–RM utility.10 Like the Westholm adjustment, this utility function can be distinguished from 8 9

For any two values x1 and x2, with mean x¯ = (x1 + x2), x1 - x¯ = -(x2 - x¯). The expected value of the proximity-predicted utility is E[- V - C

2

2

2

] = -E[Â v - 2Â v c + Â c ] = -Â E[v ] - Â E[c ] = - V - Âs i

i i

2

i

2

i

i

2

2

i

10

The model investigated by Lewis and King (1998) is equivalent to 2

2

U ( V, C) = -bv V - b c C + 2 V ◊ C They find for the Norwegian data analyzed by Westholm (and in Chapter 7 of this book) that the estimate for bv is near zero. This is similar to the adjusted utility we have just described.

186

Appendices

the RM utility function only by an intercandidate comparison and – because of potential confounding – is most useful for multicandidate elections. Appendix 7.1 Methodology: The Lewis and King Critique As we saw in Chapter 6, in attempting to discriminate between the proximity and RM models in a two-candidate contest, any intercept term will tend to be confounded with the effects of model type on voter choice, as both model type and intercept term shift the location of the indifference point between the two candidates. Thus, omission of the intercept term leads to biased estimates; inclusion of the term permits model differences to be confounded with other effects (see also Macdonald, Rabinowitz, and Listhaug, 1995).11 In a contest with K candidates (K > 2), there are potentially K - 1 intercept terms. It is still possible for omission of intercepts to create bias, but such bias is increasingly unlikely as the number of candidates increases. For example, if proximity is the correct model, bias in the direction of the RM model due to omission of intercepts could occur only if more extreme candidates systematically have higher intercept terms, i.e., if nonspatial (and nonpartisan) party attractiveness is correlated with spatial extremity. Were such a relation strong enough, the likelihood of candidate selection might conceivably increase with extremity rather than peak near the voter. Conversely, if the RM model is correct, bias toward the proximity model would occur only to the extent that nonspatial party attractiveness is correlated with spatial centrism. Accordingly, we performed Monte Carlo simulations on artificial data that were constructed to resemble those found in the Norwegian and French national studies. A conditional logit model of voter choice for a mixed proximity–RM model was used to obtain estimates of b. The bias in the mixing parameter occurs insofar as the partyspecific intercepts are correlated with the eccentricity of party location (distance of the party location from the neutral point). We refer to this correlation as the eccentricity correlation. A positive eccentricity correlation implies bias toward the directional model (values of b 11

If mean party placements are used, b becomes inestimable, i.e., the model is unidentified.

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187

near 0); negative correlation implies bias toward the proximity model (values of b near 1). There seems no way to test whether there are systematic nonspatial reasons why, say, extremist parties might be attractive or nonattractive; in any event, eccentricity correlation might occur by chance in a particular election. Still, because the partyspecific intercepts are likely to be the result of a number of unrelated factors, it seems unlikely that the eccentricity correlation would be high if there are a large number of parties. Simulation runs were performed on artificially generated data with a specified value of the mixing parameter (b = 0.5) and specified values for party intercepts and party locations (intercepts were generated as normal variates with mean 0; party locations are uniformly distributed and centered on the neutral point). Each run of the simulation generated 1,000 voters and seven parties and yielded an estimate bˆ ; the difference b - bˆ is an estimate of bias. Runs were done for various values of the eccentricity correlation and for various values of the standard deviation of the party intercepts (a measure of the size of the intercepts). Regression of the estimates of bias against various possible predictors suggests that it is approximately proportional to the product of the eccentricity correlation (R) and the standard deviation (S) of the party intercepts, with a proportionality coefficient of about 0.19 (see Figure A.7.1, in which R*S denotes the product of R and S). Thus, for a correlation of, say, 0.5, and a standard deviation of the intercepts of about 1.0,12 the bias would be on the order of (0.19)(0.5)(1.0) = 0.095. This added uncertainty is comparable in size to the nominal standard error (about 0.05 for the 1989 Norwegian data with N = 1,565 and 0.12 for the 1988 French data with N = 677). Thus, the size of these values suggests that the correct model is still a mixture of proximity and directional components. These results help explain why estimates of the mixing parameters for a family of models each based on a different set of control parameters (so that the intercepts are likely to be shuffled) give fairly 12

One of the factors of the independent variable in regression is the standard deviation (S) of the party intercepts. To get a sense of its order of magnitude, we estimated intercepts (for the 1989 Norwegian data) in reduced models in each of which a specific value of b was imposed (so that each reduced model is identified). The results suggest that S is of the order of magnitude of 1 or less for reasonable values of b.

188

Appendices

Figure A.7.1 Regression of estimated bias in the mixing parameter for simulated data. Note: R*S is the product of R and S. R is the correlation between partyspecific intercepts and party eccentricity; S is the standard deviation of the party intercepts. Bias is the bias in estimating the mixing parameter b.

similar values. For example, for the 1988 French presidential election, Adams and Merrill (1998) obtain estimates of the mixing parameter ranging from 0.62 to 0.76 for various combinations of control variables, including party ID and sociodemographic variables, using either conditional logit or modifications of it. Returning to the empirical study of Norwegian data, we note that eccentricity correlation does not appear likely to be strong (either positively or negatively) for the seven Norwegian parties, because we have no reason to expect that parties with nonpolicy and nonpartisan advantages are systematically centrist or extremist. For the five major candidates in the 1988 French election, there is some correlation between party strength and centrism – although the most centrist candidate (Raymond Barre) came in third, not first – opening a possible bias toward the proximity model.Thus, certainly for Norway and probably for France, we feel fairly safe proceeding with inferences from models without intercept terms. Our simulation results suggest that omitting intercepts will lead to no undue danger that model differences will be substantially confounded by other effects.

Appendices

189

Appendix 7.2 Methodology: English Translations of Questions from the Norwegian Election Studies Party choice in the 1985, 1989, and 1993 Norwegian studies was based on the question: “What party or which list (or pact) did you vote for?” Placement on the left–right scale was based on a question for which the 1993 version is typical: “Here we have a scale running from 1 on the left – representing those who stand on the extreme Left politically – to 10 on the right – representing those who stand on the extreme Right politically.Where would you place yourself on this scale? Where would you place the political parties on such a scale?” In the 1989 study, the immigration scale extended from (1) “make it easier for immigrants to come to Norway” to (10) “the number of immigrants should be restricted more strongly than at present.” Alcohol policy (reverse coded) specified that (1) denotes “the position that alcohol should be sold without restrictions and at greatly reduced prices” while (10) “means that the sale and production of alcohol should be more strongly regulated than it is today.” In the 1993 study, the EEC question (also reverse coded) specified that “The value 1 expresses the view that Norway absolutely should become a member of EEC; while the value 10 expresses the view that Norway absolutely should not become a member of EEC.” Appendix 7.3 A Strategic Probabilistic Model of Voter Choice In a study of Scandinavian electorates, Merrill (1994) – following Enelow and Hinich (1982, 1984: 84) – models the random component with normal distributions with mean zero and common standard deviation, s. Rather than control for party ID effect, Merrill attempts to account for historical party strength by multiplying utility by a factor, pa, where p represents the historical proportion of the vote received by a given party and a is a parameter reflecting the strength of strategic voting. The strategic model starts with utilities, U(Vj, Ci), defined by U(Vj, Ci) + dij + Xij, where dij is the (deterministic) mixed proximity–RM utility function (see Chapter 3) and Xij are identically distributed normal variates with mean 0 and common standard deviation, s.

190

Appendices

These utilities are multiplied by strategic factors, pia, where pi is the average proportion of the vote received by party Ci in the three previous elections and a is a parameter to be estimated (see Merrill, 1981, 1988: 61–62, 68–69). The probability pij that party Ci is preferred to all other parties by a voter Vj can be expressed as a function of the deterministic utility dij, the standard deviation of the stochastic component s, and the strategic parameter a.13 This function can be evaluated by numerical integration. The mixed model predicts long-term stability for the current distribution of party support – a historically accurate prediction for Norway despite aberrant elections such as that of 1989 – whereas the pure models do not. Maximum likelihood estimates for the parameters of Merrill’s strategic model using only the left–right dimension14 are given in Table A.7.1 for one Swedish15 and three Norwegian elections. Overall, the results suggest support for a model intermediate between the proximity and the RM models. The strategic-model estimates are more favorable to the RM model (in part because of the use of mean placements). To make the models comparable, we recomputed the estimates for the conditional logit model for only one or two dimensions without an intensity parameter and used mean placements. For 1989, the estimates for b based on the left–right issue alone are 0.11 for the strategic model and 0.25 for the conditional logit model. For the 1993 model using the two major issues, left–right and EEC, those estimates are 0.55 and 0.78. Although these estimates vary between the models, both models reflect contributions from proximity and direction.16 13

[

a

]

a

p ij = p pi Uij > max pk U (Vj , Ck ) k πi

a

a

Ê x - pk dkj ˆ Ê x - pi dij ˆ a = Ú (1 pi s ) f Á ˜ dx ˜ ’ FÁ a Ë pk a s ¯ Ë ¯ p s k πi i -• •

14

15

16

where f and F denote the density and distribution function, respectively, of the standard normal probability. The proof is given in Merrill (1994). Unlike the conditional logit analysis, which uses voter-specific placement of candidates, the strategic analysis used mean placement. In Sweden, five parties are analyzed, from left to right: Communist, Social Democrat, Liberal, Center, and Conservative. Voters and parties were placed on a unidimensional left–right scale (0–10). The number of usable respondents for Sweden in 1979 was 2,328. Election results from previous elections for both Norway and Sweden were obtained from Mackie and Rose (1991). Note that the contributions from the proximity model are substantial even though we are using mean party placements.

Appendices

191

Table A.7.1. Parameter estimates for Merrill’s strategic voting model for Norway and Sweden Election (Issue)

b

s

a

Norway 1985 (left–right)

.51

7.7

.55

Norway 1989 (left–right) Norway 1989 (left–right)

.11 .40

9.6 9.8

.36 0

Norway 1993 (left–right)

1.07 (.06) .00 (.12) .55 (.07) 1.12 (.05)

12.0 (.5) 27.0 (1.3) 21.7 (.8) 12.5 (.6)

.52 (.03) .92 (.04) .95 (.03) 0 —

9.5

.70

Norway 1993 (EEC) Norway 1993 (L/R & EEC) Norway 1993 (left–right) Sweden 1979 (left–right)

.33

Using the likelihood ratio statistic as a measure of fit for the strategic model, Merrill (1994) also shows that a blending of the two models more nearly reconstructs the distribution of vote share. None of the pure models is, by itself, sufficient to account for the distribution of votes by party. In a model incorporating strategic voting via historical vote share, however, a pure proximity or a pure RM model will – on a long-term basis – disrupt that distribution of vote share, whereas a mixed model does not.

Appendix 8.1 Notes on Equilibrium Analysis Historical Notes The set of Condorcet directional vectors at a point N is called the directional core in Cohen and Matthews (1980). Using results of Schofield (1977, 1978), Cohen and Matthews observe that the directional core is nonempty at a point N if Schofield’s (1978) null dual condition does not hold for N.

192

Appendices

There has been work on the conditions for equilibrium17 for various specific directional models. Coughlin and Nitzan (1981b) show, under probabilistic assumptions, that at every status quo point there is a directional equilibrium. Rabinowitz and Macdonald (1989) show that – given a circle of acceptability – a dominant position exists if the electorate is symmetrically distributed about a voter point other than the neutral point. Macdonald and Rabinowitz (1993) – under the assumption that voters are restricted to a square (or hypercube) – prove that a dominant strategy exists if there is a quadrant such that every halfspace containing that quadrant has more voters than the complementary half-space. The material in this chapter is based largely on Schofield (1977, 1978, 1983, 1985), Matthews (1979), and Merrill, Grofman, and Feld (1999). Proof of Proposition 8.4. Note that if C is any vector other than C*, the vector A = C* - C is perpendicular to the indifference plane between C and C* and (normalized to be of unit length) is the normal vector of the support set of C* (see Figure 8.3 for an example in two dimensions). By the Cauchy–Schwarz inequality, A · C* = ||C*||2 - C · C* ≥ 0. If the condition of the proposition holds, C* is a Condorcet directional vector. Conversely, if C* is a Condorcet directional vector, let A have length one and be such that A · C* ≥ 0. Set C = (C* - A)/||C* - A|| so that A is the normal vector of the support set of C* (when opposed by C). The condition of the proposition follows since C* is a Condorcet directional vector. The Star Angle. The star angle (see Grofman, Owen, Noviello, and Glazer, 1987; Shapley and Owen, 1989) with respect to a neutral point N with vertex at a voter point Pi is the angle of the form PjPiPk (or the union of such angles) such that all lines through Pi and within the angle are medians (see Figure 8.4a). If none of the Pi are interior to the convex hull of P1, . . . , Pn, then the line segments — of the form P¯ jPi and PkP i defining the star angles form a star with points at the Pi. The star angle for an interior point may, however, be composed of disjoint angles (see, e.g., the star angle at point, P1, in 17

Note that the equilibria obtained by Myerson and Weber (1993) involve configurations of voters in contrast to the configurations of candidates or parties investigated in this and succeeding chapters.

Appendices

193

Figure 8.4b, which is composed of the two angles P4P1P5 and XP1P6 — — where the segment, XP1, is an extension of the segment, P2P1 ). Simulation Analysis. For an odd number of voter points, simulation results show that for a uniform distribution on a disc, the probability that a Condorcet directional vector fails to exist varies from 8 percent (for n = 3) to 11 percent (for n near a dozen) and back to 9 percent (for n = 101 and n = 201). Standard errors in these simulations drop from about 0.7 to 0.3 percent over this range of n. For a normal distribution, probabilities are about 2 percentage points higher uniformly over n. For a tripolar distribution on the disc, they are about 4 percentage points higher, for virtually all n, than for a uniform distribution. Thus, for n up to at least 201, if the neutral points are drawn from the same distribution as that from which the voter points are drawn, simulation suggests that the probability that a neutral point lies in the portion of the space from which no Condorcet vectors exist is about 10 percent. Appendix 8.2 Use of Harmonic Decomposition to Determine Equilibria For large electorates in a two-dimensional model, equilibrium analysis can be expressed in terms of the directional modality of the electorate, i.e., the number of directions from a neutral or status quo point N in which there are concentrations of voters. To this end – without loss of information – we project each voter point, V = (x,y), onto the circle of radius one about N by replacing x by x/Îx2 + y2 and y by y/Îx2 + y2, respectively. Parameterizing this circle by q, -p < q £ p, we denote by f = f(q) the probability density of (projected) voters on the circle.18 Any voters located at N have probability zero. It is a remarkable fact that as long as the density function f is reasonably regular,19 it can be uniquely decomposed into the sum of 18

19

We assume throughout that this probability distribution is continuous and that the electorate is large enough that the assumption of continuity is not completely unreasonable. It is sufficient for f to be continuous and piecewise smooth. A continuous function is piecewise smooth if it has left and right derivatives at every point and – except possibly for a finite number of points – these are equal and define a continuous derivative.

194

Appendices

functions called harmonics of the form cos n(q - fn), where each n is an integer and each term is multiplied by appropriate coefficients or amplitudes. This sum is called a Fourier series.20 Even harmonics, those terms of the form cn cos n(q - fn) for even n, each have n/2 pairs of diametrically opposed humps (modes), i.e., concentrations of voters are symmetrically paired on either side of the neutral point. The corresponding odd harmonics are not symmetric. Merrill and Grofman (1997b) show that N is a complete median if and only if the Fourier series of the associated density function f on the unit circle about N has only even terms. Intuitively, N is a complete median if the electorate is “directionally symmetric.” 20

The Fourier series of f is given by f (q ) =

• 1 1 + c1 cos(q - f 1 ) + c 2 cos 2(q - f 2 ) + . . . = + Â c n cos n(q - f n ) 2p 2p n = 1

where cn is the amplitude and fn is the phase shift.

Glossary of Symbols

Euclidean distance: For any vector, X = (x1, . . . , xn), ||X|| denotes Euclidean length or distance, i.e., X =

n

Âx

2 1

. Thus, ||V - C|| is the Euclidean

i =1

distance between V and C. Scalar product (or dot product): For any vectors V = (v1, . . . , vn) and C = (c1, . . . , cn) in n-dimensional space, their scalar or dot product is n

given by V◊C = Â vi ci = v1c1 + . . . + vncn . i =1

195

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Index

Aardal, Bernt, 93n2, 100, 102 Aarts, Kees, 60, 76 Adams, James, 82, 85, 93n1, 145–50, 166–68, 188 alcohol policy, as an issue in Norway, 95, 97, 122, 189 Alesina, Alberto, 7n4, 128–30, 136, 167 Alliance party (Britain), 57 alternation in office, 130, 165 Alvarez, Michael, 82–83, 94n3, 168–69 ambiguity of candidate positions, 167 American National Election Studies, 3, 24, 40, 53–72, 76–79, 84, 120, 142, 146, 172–74, 178–79, 183–85 American politics, 3, 7, 9, 39, 67, 89, 128–29, 141, 165, 173 approval voting, 16, 158–63 Aranson, Peter, 41n3, 168 Arrow, Kenneth, 14n11 Axelrod, Robert, 14n11 Barre, Raymond, 92, 103, 188 Belgium, elections in, 53, 64, 77, 102 Bevan, Aneurin, 144 biased estimate of the mixing parameter, 186 Bogdanor, Vernon, 158 Borda count, 16, 158–62 Brady, David, 41, 128 Brams, Steven, 128, 144n1, 158 Britain, elections in, 53, 57, 60, 77, 93, 98, 102 Brunell, Thomas, 168

207

Bullock, Charles, 41, 128 Bush, George, 55, 59n6, 143, 148, 185 Butler, D., 158 Calvert, Randall, 128, 136 Campbell, Angus, 1, 6, 81 Carroll, Douglas, 23–24 Carter, Jimmy, 62, 76, 178 Cauchy-Schwarz inequality, 192 Center party (Norway), 95, 100, 102 Center party (Sweden), 190 centrists, 57, 151, 158, 160 challengers, relation to choice of model of, 13, 41, 62–64, 68–71, 74, 77–80, 84, 125–27, 165 Chamberlin, J.R., 147n4 Chirac, Jacques, 92, 103 Christian Democratic party (Belgium), 64 Christian party (Norway), 95 circle of acceptability, 38–40, 100–01, 110, 112, 115, 124, 150, 152, 156, 166, 171, 192 city-block metric, 146 Clinton, William, 9, 70–71, 126, 141, 143, 148 coalition government, 100, 150 Cohen, Linda, 191 Cohen, M.D., 147n4 Coleman, James, 41, 168 Communist party (France), 103 Communist party (Sweden), 190 complete median, 114, 116–17, 194

208

Index

computer simulation, 95, 118, 147, 160–61, 186–88, 193 conditional logit model, 13, 83–85, 91, 94, 100–02, 105, 146, 149, 186–90 Condorcet candidate, 147–48, 160–61 directional vector, 15, 114–20, 192–93 vacuum, 117–18, 121–22, 145 winner, 15, 114 Congress (United States), 9, 178 Conservative party (Britain), 57 Conservative party (Norway), 95, 100–01, 150 Conservative party (Sweden), 190 control variables, 62, 64, 150, 173, 188 Converse, Philip, 1, 7, 13 convex hull, 154, 156, 159, 162, 192 Coombs, Clyde, 23 core, 15 correlation, 59, 62, 65, 68, 74, 130, 178, 183, 187–88 Coughlin, Peter, 25n6, 82, 148, 152, 166, 192 Cox, Gary, 144n2, 158, 160, 168 Damgaard, E., 39 damped directional model, 42, 86 Davis, Otto, 114 de Palma, A., 145, 148 De Soete, Geert, 23, 24n4 Democratic party (United States), 3, 7, 9, 64, 71, 76, 84, 142, 173, 185 Denmark, elections in, 77, 102 deterministic models, 13, 81–85, 91–94, 100, 144–47, 189–90 directional core, 191 equilibrium, 110, 114, 192 models, 6–12, 24, 38–46, 50–51, 66–67, 71, 74, 78, 84–86, 89, 101–02, 110–12, 115–17, 125–127, 145, 150, 157, 165–66, 170, 177, 186, see also Matthews directional model, Rabinowitz-Macdonald directional model discounting, 3–15 discounting model, see Grofman discounting model divided government, 129–31

Dorussen, Han, 146 dot product, see scalar product Dow, Jay, 76, 79, 94n3, 173, 176 Downs, Anthony, 1, 4–5, 14n11, 19, 22, 158, 164–66, 169 Downsian model, 5, 8, 12, 20, 28, 38, 41, 44, 48, 50, 128 Dukakis, Michael, 55, 59n6, 143, 185 dynamic model, 130 Eaton, Curtis, 144n1 eccentricity correlation, 186 of party location, 186 of voter positions, 183 economic conditions, 1, 167 EEC, see European Economic Community Endersby, James, 67, 68, 84, 165, 176 Enelow, James, 2n, 67–68, 82–84, 146, 156, 165–68, 176, 189 equilibria, 14, 16, 129 for multicandidate elections, 144–52, 164, 166, 192 for two-candidate elections, 109–127 Erikson, Robert, 82, 148, 166 Euclidean distance, 21, 43, 195 length, 21, 26, 195 European Economic Community, 95, 98, 100, 102, 189–90 European Political Party Middle Elite study, 77, 103 exposed point, 155–56 extreme value distribution, 83 extremists, 151, 158, 160 Falmagne, Jean-Claude, 169 Feddersen, Timothy, 144n1 Feld, Scott, 14n10, 23n, 114, 117, 166n4, 192 Fiorina, Morris, 7n4, 166n4 Fishbein, M., 24 Fishburn, Peter, 158 Flanders, elections in, 64 Ford, Gerald, 54–57, 62, 174 Fourier series, 194 France, elections in, 4, 13, 16, 53, 77, 91–92, 104–05, 188

Index Franklin, Mark, 65 French Presidential Election Survey, 65, 91–92, 103 Germany, elections in, 53, 64–65, 77, 102 Gilljam, Mikael, 57, 65, 176, 178 Gingrich, Newton, 141 Ginsberg, V., 145 Glazer, Amihai, 41, 192 Granberg, Donald, 57 Griffin, Robert, 41 Grofman, Bernard, 6, 8, 11, 14n10, 41n3, 60n10, 65, 70n2, 114, 117, 165–69, 176, 179–80, 192, 194 Grofman discounting model, 9, 12, 22–25, 33, 44–51, 54, 84, 167 alternation in office under, 130–31, 135 definition of, 22–23 empirical tests for, 98–99, 104 equilibrium strategies for, 109–13, 125, 127 Hammond, Thomas, 168 harmonic decomposition, 193 Hart, Gary, 54–57, 59, 174 heart, electoral, 14, 120, 164, 168 Hinich, Melvin, 2n2, 81–82, 114–15, 146, 156, 166–68, 176, 189 Hong, G., 148 Hug, S., 148n5 Humes, Brian, 168 ideal point, 1–2, 4–7, 14, 22, 39, 44, 54, 112, 131, 135, 139, 169, 178 ideological space, 16, 167 idiosyncratic placement of candidates, see voter-specific placement immigration policy as an issue in France, 92, 103 as an issue in Norway, 95, 98, 102, 189 incumbents, relation to choice of model of, 13, 37, 41, 62–84, 125–27, 165 indifference line, 33, 35, 42, 44, 48, 50–51, 74, 116, 161 plane, 160 point, 33, 85–86, 125, 186 informed voters, 79, 177 intensity, as a factor in directional

209 models, 3, 23–51, 64–80, 89, 98, 101–03, 165, 180–81, 190 intensity parameter, 42 interpersonal comparison of utility, 54, 57, 60, 84, 86, 97, 105, 178, 181–82 Israel, elections in, 149n8, 150 Italy, elections in, 24 Iversen, Torben, 7, 12, 38–39, 43, 44n7, 77, 102, 112, 156n16, 170–71 Jackson, Jesse, 54–55, 57, 59n6, 185 Jones, Calvin, 7, 13, 82 Kennedy, Edward, 54–55, 57, 59, 174 King, Gary, 36n, 79, 94, 177, 179, 185n10, 186 Knesset (Israeli), 150 Koetzle, William, 168 Krämer, Jurgen, 62–65, 74–76, 176, 178 Labbe, M., 145 Labor party (Britain), 57 Labor party (Israel), 149n8 Labor party (Norway), 95, 97, 100–01, 149n8, 150 Lajoinie, Andre, 103 Laver, Michael, 14n11 Le Pen, Jean-Marie, 92, 103 leapfrogging, 144 Ledyard, John, 82 left/right scale, 59, 95, 97–100, 102–03, 122, 149–50, 190 Levin, Bernard, 164 Lewis, Jeffery, 36n16, 79, 94, 177, 179, 185n10, 186 Liberal party (Norway), 95, 97, 102 Liberal party (Sweden), 190 liberal/conservative scale, 19, 53–54, 70, 121, 142, 173, 183 Likud party (Israel), 149n8 Lin, Tse-min, 146, 148 linear proximity model, 62, 65 Lipsey, Richard, 144n1 Listhaug, Ola, 8, 49, 57, 60, 79, 92, 98n7, 100, 173–74, 176, 178–79, 186 Lodge, Milton, 176 logistic regression, 64 logit, 86, 103, 150, 152, 190, see also conditional logit model

210

Index

log-likelihood, 87–89, 95, 97–98 Lomborg, Bjorn, 148 Lubell, Samuel, 15n13, 128, 142 Macdonald, Stuart Elaine, 6, 8, 10–12, 23n3, 24, 29–30, 38–39, 43– 44, 49, 57–60, 68, 76, 79, 92, 98n7, 100, 156, 170–79, 186, 192 Mackie, T., 190n15 Maddens, Bart, 64–65, 176 Markus, Gregory, 7, 13 Martin, Andrew, 94n3 Matthews, Steven, 6, 11–12, 25, 110, 116, 165, 191–92 Matthews directional model, 12, 25–38, 41–42, 46, 50, 62, 66–67, 70–78 definition of, 25–26 equilibrium strategies for, 110, 114–23, 125–27, 145, 152 probabilistic version, 85–89, 98 maximum likelihood estimates, 70, 85–87, 95, 103, 190 McAllister, Ian, 93n1 McDonald, Michael, 168 McKelvey, Richard, 14n10, 112, 114 mean placement, 68, 77–78, 103, 105, 176–81, 190 median voter, 5, 14, 23, 41, 54, 110–15, 124, 127–43, 157, 160, 169 median voter theorem, 5, 110, 157 Merrill, Samuel, 8, 47, 56n5, 60n10, 65, 70n2, 79, 82, 102, 111, 117–18, 120, 128, 145, 147n4, 149–50, 155, 158, 160, 161n4, 166n, 168, 173, 176, 179–80, 188–92, 194 Michigan model, 1, 4, 8, 81, 166 mid-term switches, 129–30 Miller, Warren, 1, 166n4 Miller, Nicholas, 14n10, 114 Mitterrand, François, 103 mixed proximity and directional model, see mixed proximity/RM model mixed proximity-RM model, 12, 15, 39–40, 43–48, 50–51, 54–55, 76–77, 89, 96, 110–13, 150, 160, 163, 181 definition of, 39, 43– 44 equivalent formulations, 170–71 mixing parameter, for the mixed proximity-RM model, 51, 70–77, 80,

86, 94, 97–98, 103, 105, 112, 125, 135, 150, 161, 170, 174, 180, 186–88 equivalence with discounting factor, 47–51 model ratio, for the mixed proximityRM model, 170–71 models of coalition formation, 14 moderately extreme parties, 4, 106 Mondale, Walter, 54, 56–59, 62, 76, 85, 174 Morris, Irwin, 79n8 multicandidate election, 13, 90–91, 144, 146, 157–58, 162, 182 multidimensional space, 19 multinomial logit, 83 multinomial probit, 94n3 multiparty election, see multicandidate election Munger, Michael, 67–68, 81, 84, 115, 165, 167–68, 176 Myerson, Roger, 144n2, 192n17 Nagler, Jonathan, 82–83, 94n3 Nash equilibrium, 109 under the Grofman discounting model, 110–14 under the Matthews directional model, 124–25 National Front party (France), 103 NES, see American National Election Studies Netherlands, elections in, 60, 77, 102, 150n9 neutral point, 2, 11, 13, 23–57, 78, 80, 84–89, 98–100, 110–18, 125–26, 135, 145, 152–53, 156, 159–61, 171, 177, 182, 186–87, 192–94 Nitzan, S., 25n, 82, 148, 192 Nixon, David, 148, 149n8 Nixon, Richard, 178 non-linear regression, 181 non-policy factors, 147, 188 Norway, elections in, 4, 13, 16, 53, 59–60, 65, 91–105, 120–22, 145, 149–50, 157, 166–67, 178–79, 183–90 Norwegian National Election Studies, 59, 91, 93n2, 122, 145, 149–50, 183, 189 Noviello, Nicholas, 192 null dual condition, 191

Index Olomoki, D., 148 on-line processing of political stimuli, 176 Ordeshook, Peter, 41n3, 82, 168 Owen, Guillermo, 41n3, 168, 192 Page, Benjamin, 7, 13, 82 Palfrey, Thomas, 129, 176 Pareto set, 118, 122 party affiliation, 93–94 party identification, 1, 4, 6, 8, 16, 23, 60–62, 74–84, 91–96, 102–05, 145, 149, 167, 173, 188 party-specific intercepts, 80, 94, 150, 186 Perot, Ross, 148, 185 persuasion, 16, 169 Pierce, Roy, 7, 65, 92, 103, 176, 178 Platt, G., 7, 78, 178 Plott, Charles, 25n6, 114n4 plurality system of voting 16, 160, 162 plurality with runoff, 16, 158 plurality-maximizing candidate, 144 policy-seeking candidate, 128, 136, 141 Pollard, Walker, 82 Poole, Keith, 7, 40–41, 78, 115, 128, 176, 178 Potthoff, Richard, 41n3, 132n3, 140n8 Powell, Lynda, 177 probabilistic models, 13, 16, 53, 81–94, 102, 105, 145–152, 157, 166, 190, 192 unified model, 94 probability of vote choice, 82 Progress party (Norway), 95, 100–02, 150 projection, 8, 13, 51, 66–80, 178–184 adjustment, 74, 86 proximity model, 2–15, 55 as a component of the unified model, 38–51, 170 as a pure model, 19–37 definition of, 19–21 empirical tests for, 52–108, 174, 178, 182, 186 for alternative voting procedures, 160–62 in equilibrium analysis, 109–17, 144 pseudo-yolk, 113, 114 quadratic proximity model, see quadratic utility function

211 quadratic utility function, 21, 39, 65, 83, 86, 146–48, 174 Quinn, Kevin, 94n3 Rabinowitz, George, 6, 8, 10–12, 23n3, 24, 29–30, 38–39, 43–44, 49, 57–60, 68, 76, 79, 92, 98n7, 100, 156, 170–79, 186, 192 Rabinowitz-Macdonald directional model, 11–12, 165–66, see also mixed proximity/RM model as a component of the unified model, 38–51 as a pure model, 23–37 definition of, 29–31 empirical tests for, 52–108 in relation to candidate strategy, 156 Rae, Douglas, 158 random component of a model, 83, 94, 189 rationalization, 8, 13, 66, 178, 184 Rattinger, Hans, 62–65, 74, 76, 176, 178 Reagan, Ronald, 9, 54–57, 59n6, 62, 76, 85, 142–43, 174, 185 Regenwetter, Michel, 169 region of acceptability, see circle of acceptability region of support of candidates, 33, 152–56, 161 regression, 59, 68, 70, 74, 79, 173, 176, 180–83, 187 representation policy leadership model, 43 Republican party (United States), 3, 7, 9, 71, 74, 76, 84, 126, 141– 43, 173, 179–80, 185 Reston, James, 52 Reynolds, H.T., 23, 24 Riker, William, 15n13, 158 Rivers, Douglas, 85 RM model, see Rabinowitz-Macdonald directional model RM model with proximity constraint, see mixed proximity-RM model RM model with centered constraint, 171 Romero, David, 82, 148, 166 Rose, Richard, 93n, 190n

212

Index

Rosenthal, Howard, 7n4, 8, 40– 41, 78, 115, 128–30, 136, 167, 170, 178 RPR party (France), 103 salience coefficient, 83 scalar product, 24, 26, 31, 39, 42–43, 45, 47–48, 110, 124, 152, 171, 177, 195 Schattschneider, E.E., 91 Schofield, Norman, 14n10, 25n6, 94n3, 102n10, 115n6, 118, 120, 148–49, 150n9, 168, 191–92 Sened, Itai, 144n1, 148–49 shadow position, as a strategy, 47–51, 110–13 used to show equivalence of models, 47–51 Shanks, Merrill, 166n4 Shapley, Lloyd, 92 Shepsle, Kenneth, 168 Shvetsova, Olga, 168 single-transferable vote, 16, 158–63 Sloss, J., 25n6 social choice, 14 Social Democrat party (Sweden), 190 social utility, 146, 148 Socialist party (France), 103 Socialist party (Norway), 95, 100–03, 149n8, 150 spatial model, 4, 5, 11, 14, see also specific models star angle, 117, 118, 192 status quo, 2–12, 15, 22–25, 44–51, 54, 84, 98–99, 109–15, 124–27, 130–43, 152, 165, 192–93 stochastic model, see probabalistic model Stokes, Donald, 1, 7n4 Straffin, Philip, 144n1, 158 strategic model, 102, 136–37, 189–91 voting, 102, 189, 191 strategies convergent, 3, 5, 10, 14–16, 109–13, 129, 144–50 divergent, 3, 10, 15, 111–13, 144–50 moderately divergent, 110–14, 128–30 of candidates, 4–5, 9, 14, 36, 68, 80, 109, 151–52, 176 of parties, 3, 10–16, 32, 128, 130 stable, 109–10

Strøm, Kaare, 100 STV, see single-transferable vote submodels of the unified model, 86 support set for a candidate, 116 Svåsand, Lars, 100 Sweden, elections in, 23, 53, 57, 65, 91, 178, 184, 190 Swedish Election Study, 91 thermometer scores, 53–62, 65, 70, 78–79, 105, 176–77, 183–84 third parties, entry of, 129 Thisse, J.-F., 145 Tovey, Craig, 14n10, 114, 118 Train, Kenneth, 83 tripolar distribution of voters, 121–22, 193 UDF party (France), 103 unidentified model, 186n11 unified model, 3, 10–16, 36, 38–46, 51, 145, 164, 165–68 definition of, using voter choice, 82–83 definition of, using voter utility, 45–46 deterministic, 92 empirical tests for, using voter choice, 82–98, 104, 145 empirical tests for, using voter utility, 67–78, 174, 180–81 probabilistic, 94 uninformed voters, 78, 177, 178 United States, elections in, 3, 4, 8, 13, 15, 53, 60–67, 70–80, 84, 126, 184 utility, 1, 4, 5, 10–14 function, 19–37, 47, 67, 86, 170–74, 181–82 Valen, Henry, 100 van der Eijk, Cees, 65 Vlaams Blok party (Belgium), 64 voter choice model, 13, 89, 90, 97 voter sophistication, 77 voter utility model, 13, 97 voter-specific placement of candidates, 59–60, 65, 68, 76, 86–103, 105, 176–81, 184, 190 vote-seeking candidates, 128, 136, 141 vote-share maximizing candidates, 144, 149

Index wandering vector model, 23 Wayman, Frank, 8 Weber, Robert, 144n2, 161n4, 192n17 Westholm adjustment, 87, 97n5, 178, 181–85 Westholm, Anders, 8, 10, 59, 79n9, 105n14, 176, 179, 182–85

213 Whitford, Andrew, 94 Wittman, David, 128 Wright, S.G., 144n1 Yeats, William Butler, 67, 109 yolk, 14, 111–14, 118, 120, 127 directional, 120