When winning is the only thing: pure strategy Nash equilibria in a three-candidate spatial voting model

It is well known that there are no pure strategy Nash equilibria (PSNE) in the standard three-candidate spatial voting model when candidates maximize their share of the vote. When all that matters to the candidates is winning the election, however, we show that PSNE do exist. We provide a complete characterization of such equilibria and then extend our results to elections with an arbitrary number of candidates.     

Costly voting when both information and preferences differ: is turnout too high or too low?

We study a model of costly voting over two alternatives, where agents’ preferences are determined by both (i) a private preference in favour of one alternative e.g. candidates’ policies, and (ii) heterogeneous information in the form of noisy signals about a commonly valued state of the world e.g. candidate competence. We show that depending on the level of the personal bias (weight on private preference), voting is either according to private preferences or according to signals. When voting takes place according to private preferences, there is an unique equilibrium with inefficiently high turnout. In contrast, when voting takes place according to signals, turnout is locally too low. Multiple Pareto-ranked voting equilibria may exist and in particular, compulsory voting may Pareto dominate voluntary voting. Moreover, an increase in personal bias can cause turnout to rise or fall, and an increase in the accuracy of information may cause a switch to voting on the basis of signals and thus lower turnout, even though it increases welfare. This is a substantially revised version of Department of Economics University of Warwick Working Paper 670, “Information Aggregation, Costly Voting and Common Values”, January 2003. We would like to thank B. Dutta, M. Morelli, C. Perrroni, V. Bhaskar and seminar participants at Warwick, Nottingham and the ESRC Workshop in Game Theory for their comments. We would also like to thank the editor and an anonymous referee for their comments.     

Policy convergence in a two-candidate probabilistic voting model

We propose a generalization of the probabilistic voting model in two-candidate elections. We allow the candidates have general von Neumann–Morgenstern utility functions defined over the voting outcomes. We show that the candidates will choose identical policy positions only if the electoral competition game is constant-sum, such as when both candidates are probability-of-win maximizers or vote share maximizers, or for a small set of functions that for each voter define the probability of voting for each candidate, given candidate policy positions. At the same time, a pure-strategy local Nash equilibrium (in which the candidates do not necessarily choose identical positions) exists for a large set of such functions. Hence, if the candidate payoffs are unrestricted, the “mean voter theorem” for probabilistic voting models is shown to hold only for a small set of probability of vote functions.     

The spatial model with non-policy factors: a theory of policy-motivated candidates

The theoretical literature on two candidate elections is dominated by symmetric contests and vote-maximizing candidates. These models fail to capture two important features of real elections. First, most elections pit a stronger candidate against a weaker one. Second, candidates care not only about holding office, but also about policy outcomes. Ignoring any one of these features means we will fail to capture an important dynamic—strong candidates must balance their desire to change policy with their need to win the election. We provide conditions for the existence of an equilibrium in the spatial model with non-policy factors, when candidates are policy motivated. We provide a characterization of ‘regular’ equilibria and show that there exists at most one regular equilibrium. We provide conditions that guarantee that all equilibria are regular. We derive comparative statics for the model and show that increasing a candidate’s non-policy advantage causes that candidate to move towards his ideal point.     

Parties with Policy Preferences and Uncertainty over Voter Behavior

The equilibrium redistributive policy proposals of two parties with policy preferences are studied. Each party’s ideal policy coincides with that of citizens having a particular income level, and the party’s utility function reflects its attitude to the trade-off between the choice of preferred policy and the likelihood of victory. When parties face uncertainty about citizens’ abstention from voting, divergent equilibrium proposals are derived which are more moderate than their contrasting ideal policies. Political equilibria under different prior beliefs on abstention are then compared. It is shown that a lower likelihood of abstention in a particular income group induces both parties to make proposals catering to that group, in equilibrium.An erratum to this article can be found at     

Aggregate uncertainty in the citizen candidate model yields extremist parties

We extend the citizen candidate model of electoral competition with sincere voting to allow for k ≥ 2 states of aggregate uncertainty. We discuss and characterize the equilibrium set in this framework. We provide conditions for the existence of two-party equilibria when k = 2 and show that the policies of the two parties in any such equilibrium are not only divergent but that the parties are extremist: when the political mood is left-wing, the left-wing party wins decisively with a platform that is to the left of the left-wing median voter, while when the political mood is right-wing, the right-wing party wins decisively with a platform that is to the right of the right-wing median voter. We then provide conditions under which such equilibria remain robust for an arbitrary value of k.     

Voting games,indifference, and consistent sequential choice rules

We present four choice functions which characterize the stationary points of sequential search rules derived from a preference relation over outcomes. These functions are contrasted with others in the literature in terms of narrowness of choice as well as their ability to satisfy certain normative and consistency conditions, and it is shown how two of these sets arise as the set of equilibrium outcomes of a voting game under different tie-breaking assumptions.We would like to thank Nicholas Miller for introducing us and two anonymous referees for valuable comments and suggestions.     

Modeling large electorates with Fourier series, with applications to Nash equilibria in proximity and directional models of spatial competition

 In this paper we introduce harmonic analysis (Fourier series) as a tool for characterizing the existence of Nash equilibria in two-dimensional spatial majority rule voting games with large electorates. We apply our methods both to traditional proximity models and to directional models. In the latter voters exhibit preferences over directions rather than over alternatives, per se. A directional equilibrium can be characterized as a Condorcet direction, in analogy to the Condorcet (majority) winner in the usual voting models, i.e., a direction which is preferred by a majority to (or at least is not beaten by) any other direction. We provide a parallel treatment of the total median condition for equilibrium under proximity voting and equilibrium conditions for directional voting that shows that the former result is in terms of a strict equality (a knife-edge result very unlikely to hold) while the latter is in terms of an inequality which is relatively easy to satisfy. For the Matthews [3] directional model and a variant of the Rabinowitz and Macdonald [7] directional model, we present a sufficiency condition for the existence of a Condorcet directional vector in terms of the odd-numbered components of the Fourier series representing the density distribution of the voter points. We interpret our theoretical results by looking at real-world voter distributions and direction fields among voter points derived from U.S. and Norwegian survey data. Received: 7 July 1995 / Accepted: 14 May 1996     

Electoral competition in 2-dimensional ideology space with unidimensional commitment

We study a model of electoral competition between two candidates with two orthogonal issues, where candidates are office motivated and committed to a particular position in one of the dimensions, while having the freedom to select (credibly) any position on the other dimension. We analyse two settings: one where both candidates are committed to the same dimension, and the other where each candidate is committed to a different dimension. We focus on characterisation and existence of pure strategy Nash equilibria when the core is empty. We show that if the distribution of voters’ ideal policies is continuously differentiable and has a bounded support, then an equilibrium exists if the candidates are differentiated enough. Our results for the case where the candidates have a common committed issue have implications for the literature on valence.     

Information efficiency and majority decisions

The analysis is concerned with the characterization of equilibria of a two-stage voting game involving private information acquisition. Rational ignorance and information efficiency are identified. It turns out that information efficiency is not always desirable. By restricting the payoff domain, we are able to characterize completely the set of equilibria. In this case, information acquisition by few voters benefits a majority, or even the whole community. Economic Literature Classification Number D70, D71, D80.The author would like to thank Peter Bernholz, MartinHellwig, Tilman Börgers, Harald Nedwed and a referee for helpful comments and suggestions.     

Approval voting and positional voting methods: Inference,relationship, examples

Approval voting is the voting method recently adopted by the Society for Social Choice and Welfare. Positional voting methods include the famous plurality, antiplurality, and Borda methods. We extend the inference framework of Tsetlin and Regenwetter (2003) from majority rule to approval voting and all positional voting methods. We also establish a link between approval voting and positional voting methods whenever Falmagne et al.s (1996) size-independent model of approval voting holds: In all such cases, approval voting mimics some positional voting method. We illustrate our inference framework by analyzing approval voting and ranking data, with and without the assumption of the size-independent model. For many of the existing data, including the Society for Social Choice and Welfare election analyzed by Brams and Fishburn (2001) and Saari (2001), low turnout implies that inferences drawn from such elections carry low (statistical) confidence. Whenever solid inferences are possible, we find that, under certain statistical assumptions, approval voting tends to agree with positional voting methods, and with Borda, in particular.Michel Regenwetter thanks the National Science Foundation for funding this research through NSF grant SBR 97-30076. Both authors thank the Fuqua School of Business for financially supporting their collaboration. Most of this research was done while Regenwetter was a faculty member at Fuqua. We thank Prof. Steven Brams for his valuable comments as a discussant of a previous version of this paper, given at the 2002 Public Choice meeting, and Prof. Donald Saari for his helpful comments in conversations and on another draft. We also thank the editor in charge and a referee for their valuable comments. Tsetlin acknowledges the support of the Centre for Decision Making and Risk Analysis at INSEAD.     

Rejecter-proposer legislative bargaining with heterogeneous time and risk preferences

This article considers a legislative bargaining model in which the rejecter in the previous round becomes the proposer in the current round. We allow the time and risk preferences to differ across players and the voting quota to be a supermajority or submajority. We show that there exists a stationary subgame perfect equilibrium and that each player’s equilibrium payoff conditional on being a proposer is unique, and we explicitly derive the equilibria and equilibrium payoff. We compare a proposer’s equilibrium payoff when the time interval between two consecutive rounds tends to zero with respect to the protocols of the selection of proposers and the voting quota: we show that a proposer’s equilibrium payoff can be greater in this article’s rejecter-proposer model than in the Baron–Ferejohn random-proposer model; even though the voting quota increases, a proposer’s equilibrium payoff can increase.     

Relationship admitting families of candidates

A central theme in social choice is to determine when must there be a relationship among a group's sincere election rankings of several different subsets of candidates. This issue is completely resolved here for positional voting methods. Namely, necessary and sufficient conditions are derived for a family of subsets of candidates to determine when there is a choice of positional voting methods so that there are relationships among the election rankings. The same issue is resolved for a related family of social choice mappings. Then, in part, these necessary and sufficient conditions are used i) to analyze sequential voting procedures, ii) to show how to create new classes of axiomatic representations for social choice mappings that uniquely characterize the Borda Count, and iii) to determine the limits of indeterminacy for positional voting election outcomes.This research was supported in part by NSF grant IRI-8803505 and a Fellowship from the Guggenheim Memorial Foundation.     

Alienation,indifference and the choice of ideological position

In this paper we describe a simple model of individual voting behavior and present its implications for the candidate positioning problem under both vote and plurality maximization. Under our assumptions, some voters at the extremes of the ideological spectrum typically will not vote because they are alienated by the equilibrium location of candidates. There will also be some voters in the middle of the ideological spectrum who will not vote because they are indifferent between the equilibrium locations of the candidates. Both the abstention from alienation and from indifference arise explicitly from utility maximization. Once we allow for alienation and indifference, the two alternative candidate objective functions (vote maximization and plurality maximization) yield different outcomes. In particular, we show that under vote maximization the Median Voter (or Minimum Differentiation) outcome will not arise. On the other hand, under plurality maximization, the Median Voter outcome may or may not hold, depending on the distribution of voter preferences.We should like to thank Jerry Fusselman, Jon Hamilton, Mel Hinich, Charlie Holt and participants at seminars at the University of Virginia, Northwestern University and the 1989 Meetings of the Public Choice Society and the Economic Science Association for their helpful comments.     

A Signaling Model of Repeated Elections

I develop a two period model of elections in which voters’ first period actions affect candidates’ estimates of voter preferences and thus affect second period electoral and policy outcomes. I find an equilibrium in which centrist voters abstain in the first election, despite facing zero costs of voting and having a strict preference between the alternatives before them. The reason centrists abstain is to signal their preferences to future candidates and thereby promote future policy moderation.For helpful discussions and comments I thank David Austen-Smith, Tim Feddersen, Roger Myerson, Tom Palfrey, Ronny Razin, two anonymous reviewers, and seminar participants at Kellogg, NYU, and the 2000 World Congress of the Game Theory Society.     

Candidate Stability and Voting Correspondences

We extend the analysis of Dutta et al. (in Econometrica, 69:1013–1038, 2001) on strategic candidacy to multivalued environments. For each agenda and each profile of voters’ preferences over running candidates, a voting correspondence selects set of running candidates. A voting correspondence is candidate stable if no candidate ever has an incentive to withdraw her candidacy when all other potential candidates run for office. In the multivalued framework, candidates’ incentives to withdraw depend on candidates’ preferences over sets. If candidates cannot vote and they compare sets of candidates according to their expected utility conditional on some prior probability assessment, then a voting correspondence satisfies candidate stability and unanimity if and only if it is dictatorial. If the probability assessments are restricted to be uniform, candidates’ preferences over sets are consistent with leximin preferences, or candidates can vote, then possibility results are obtained.This paper is a revised version of the second chapter of my Ph.D. Dissertation submitted to the Universitat Autònoma de Barcelona. I am indebted to my supervisor Salvador Barberà for his advice, encouragement, and patience. I thank the hospitality of the Wallis Institute of Political Economy at the University of Rochester, where the revision of this paper was conducted. I am grateful to two anonymous referees and the Associate Editor, John Weymark, for their exhaustive and insightful comments. I also thank Dolors Berga, Carmen Beviá, Walter Bossert, Jernej Čopič, Bhaskar Dutta, Matt Jackson, Jordi Massó, Diego Moreno, David Pérez-Castrillo, and Yves Sprumont for helpful conversations and suggestions. Financial support through Research Grant 1998FI00022 from Comissionat per Universitats i Recerca, Generalitat de Catalunya, Research Project PB98-870 from the Ministerio de Ciencia y Tecnología, Fundación Barrié de la Maza, and Consejería de Innovación, Ciencia y Empresa, Junta de Andalucía is gratefully acknowledged. The usual disclaimer applies.     

Endogenous agenda formation in three-person committees

This paper analyzes a 3-person voting game in which two or three players have the ability to choose alternatives to be considered. Once the set of possible alternatives and the structure of the voting procedure are known, the players can solve for the outcome. Thus, the actual choice over outcomes takes place in the choice of alternatives to be voted on, i.e., the agenda. An equilibrium to this agenda-formation game in shown to exist under different assumptions about the information relative to the order of the players in the voting game. Further, this equilibrium is computed and found to possess certain features which are attractive from a normative point of view.Prepared for delivery at the 1985 Annual Meeting of the American Political Science Association, The New Orleans Hilton, August 29–September 1, 1985. We would like to thank Richard McKelvey, Norman Schofield and two anonymous referees for valuable comments and suggestions. We retain responsibility for remaining errors.     

DECEPTION AND POLITICAL PARTICIPATION: THEORY AND LABORATORY EVIDENCE

We model two‐candidate elections in which (1) voters are uncertain about candidates' attributes; and (2) candidates can inform voters of their attributes by sending advertisements. We compare between political campaigns with truthful advertising and campaigns in which there is a small chance of deceptive advertising. Our model predicts that voters should vote in‐line with an advertisement's information. We test our model's predictions using laboratory elections. We find, in the presence of even a small probability that an advertisement is deceptive, voters become substantially more likely to elect a “low‐quality” candidate. We discuss implications of this for existing models of voting decisions. (JEL C92, D72, D82)     

A group majority voting model of public good provision

A model of public good provision by majority rule selection incorporating the behavior of rational group voting is formulated. Two probability maximizing candidates are assumed. If voters are risk averse in public sector preferences, then a unique symmetric (both candidates offering identical platforms) equilibrium exists. If certain additional conditions hold on the group cost functions of political support, this equilibrium will lie at the public good levels which maximizes the sum of voter utility. It is further demonstrated that it is unlikely for those conditions to be satisfied and therefore more realistic asymmetric equilibria with positive voter turnout is predicted.This paper is based on an essay of a dissertation submitted to Tulane University under the supervision and invaluable assistance of Steven Slutsky, Jonathan Hamilton, and William Oakland. The advice of Gerald Whitney, Michaels Johnson, and an anonymous referee is also gratefully acknowledged. All errors remain the authors.     

Sophisticated voting outcomes and agenda control

Necessary and sufficient conditions for an alternative to be a sophisticated voting outcome under an amendment procedure are derived. The uncovered set, as first defined by Miller (1980), is shown to be potentially reducible, and conditions are determined for which this reduction equals the set of sophisticated voting outcomes. In addition, simple methods are given for calculating both the uncovered set and its reduction.Presented at the 1984 Annual Meeting of the Public Choice Society, Phoenix AZ, March 1984. I would like to thank Gerald Kramer, Nicholas Miller, Norman Schofield, two anonymous referees, and especially Richard McKelvey for helpful suggestions and comments.