Equilibrium in multicandidate probabilistic spatial voting

This paper presents a multicandidate spatial model of probabilistic voting in which voter utility functions contain a random element specific to each candidate. The model assumes no abstentions, sincere voting, and the maximization of expected vote by each candidate. We derive a sufficient condition for concavity of the candidate expected vote function with which the existence of equilibrium is related to the degree of voter uncertainty. We show that, under concavity, convergent equilibrium exists at a “minimum-sum point” at which total distances from all voter ideal points are minimized. We then discuss the location of convergent equilibrium for various measures of distance. In our examples, computer analysis indicates that non-convergent equilibria are only locally stable and disappear as voter uncertainty increases.     

Voter uncertainty and failure of Duverger’s law: an?empirical analysis

Real-world simple plurality elections rarely bear out the strong Duvergerian prediction that, in equilibrium, only two competitors receive votes. Recent advances in strategic voting theory demonstrate that voter uncertainty about competitors’ true support levels in the constituency may lead to limited strategic voting, and hence incomplete desertion of trailing parties. This is the first attempt to estimate empirically the impact of uncertainty on incentives to vote strategically. Calibrating a model of strategic voting to voting results from simple plurality elections in German constituencies, it is found that strategic voters operate under high levels of uncertainty. These results support the proposition that uncertainty about party support impedes formation of Duvergerian equilibria.     

Selecting the Condorcet Winner: single-stage versus multi-stage voting rules

In this paper, I study elections where voters are strategic. I find that the commonly used voting rules, such as Plurality Rule, Majority Rule, Approval Voting, and Single Transferable Vote, do not always select the Condorcet Winner and suffer from multiple equilibria. Multi-stage voting rules offer a way to get around this problem. I introduce two voting rules—Multi-Stage Runoff and the Nominate-Two Rule—that select the Condorcet Winner as the unique equilibrium outcome under mild conditions. I show that a third class of voting rules—Binary Voting Trees—also select the Condorcet Winner.     

Who will vote quadratically? Voter turnout and votes cast under quadratic voting

Who will vote quadratically in large-N elections under quadratic voting (QV)? First, who will vote? Although the core QV literature assumes that everyone votes, turnout is endogenous. Drawing on other work, we consider the representativeness of endogenously determined turnout under QV. Second, who will vote quadratically? Conditional on turning out, we examine reasons that, in large-N elections, the number of votes that an individual casts may deviate substantially from that under pure, rational QV equilibrium play. Because turnout itself is driven by other factors, the same determinants may influence how voters who do turn out choose the quantity of votes to cast. Independently, the number of votes actually cast may deviate dramatically from pure QV predictions because of the complex and refined nature of equilibrium play. Most plausibly, voting behavior and outcomes would be determined predominately by social and psychological forces, would exhibit few of the features emphasized in the analysis of hyper-rational equilibrium play, and would have consequential properties that require a different research agenda to bring into focus. Some of our analysis also has implications for voting behavior under other procedures, including one person, one vote.     

Endogenous Parties in an Assembly

In this article, I show how members of an assembly form voting blocs strategically to coordinate their votes and affect the policy outcome chosen by the assembly. In a repeated voting game, permanent voting blocs form in equilibrium. These permanent voting blocs act as endogenous political parties that exercise party discipline. In a stylized assembly I prove that the equilibrium parties must be two small polarized voting blocs, one at each side of the ideological divide.     

Computing Nash Equilibria in Probabilistic, Multiparty Spatial Models with Nonpolicy Components

Although there exist extensive results concerning equilibriain spatial models of two-party elections with probabilisticvoting, we know far less about equilibria in multiparty elections—i.e.,under what conditions will equilibria exist, and what are thecharacteristics of equilibrium configurations? We derive conditionsthat guarantee the existence of a unique Nash equilibrium anddevelop an algorithm to compute that equilibrium inmultipartyelections with probabilistic voting, in which voters chooseaccording to the behaviorists' fully specified multivariatevote model. Previously, such computations could only be approximatedby laborious search methods. The algorithm, which assumes aconditional logit choice function, can be applied to spatialcompetition for a variety of party objectives including vote-maximizationand margin-maximization, and can also encompass alternativevoter policy metrics such as quadratic and linear loss functions.We show that our conditions for an equilibrium are plausiblegiven the empirically-estimated parameters that behavioristsreport for voting behavior in historical elections. We alsoshow that parties' equilibrium positions depend not only onthe distribution of voters' policy preferences but also on theirnonpolicy-related attributes such as partisanship and sociodemographicvariables. Empirical applications to data from a recent Frenchelection illustrate the use of the algorithm and suggest thata unique Nash equilibrium existed in that election.     

Voting for Pareto optimality: a multidimensional analysis

We compare unanimity rule and majority rule in their abilities to produce Pareto superior and Pareto optimal alternatives in fixed number of rounds of voting using a two-dimensional spatial voting model with random proposals, sincere proposals, and strategic proposals. Our findings show that for random or sincere proposals, majority rule is at least as likely to select a Pareto optimal outcome as unanimity rule. For strategic proposals, the subgame perfect equilibrium under unanimity rule is Pareto optimal. For other k-majority rules, the outcome is Pareto optimal or very close to it. For outcomes that are both Pareto optimal and Pareto superior, unanimity rule outperforms majority rule.     

To vote or not to vote: The paradox of nonvoting

One paradox of voting states that, in a general election, in which many citizens vote, the probability that a single voter can affect the outcome is so small that in general citizens have no rational reason for voting. However, if all citizens accept this reasoning, then none will vote, and so each vote has a large probability of affecting the outcome. Hence all should vote after all. The adoption of mixed strategies resolves this paradox: if each citizen adopts a certain (small) probability of voting, then the actual number of citizens voting will be just enough to make it worth those citizens' while to vote. A Nash equilibrium point thus occurs. Here we compute Nash equilibria for the simple case of majority voting; for the more complicated case of composite voting (for example, as in a presidential election), we draw certain qualitative inferences.     

What’s in a poll? Incentives for truthful reporting in pre-election opinion surveys

We examine the ability of pre-election polls to aggregate information about voter preferences. We show that if the electorate is small and voting costs are negligible, then an equilibrium exists in which citizens report their true political preferences. If the electorate is large or voting costs are significant, however, then no such equilibrium exists because poll respondents possess incentives to influence the voting behavior of others by misreporting their true preferences. We find that when a truthful equilibrium does exist, a poll can raise expected welfare by discouraging turnout among members of the minority.     

Sophisticated Behavior under Majoritarian and Non-Majoritarian Voting Procedures

This study addresses the issue of sincere and sophisticated voting under majoritarian and non-majoritarian voting procedures. By conducting experimental voting games, we compared a common majoritarian procedure, Plurality Voting (PV) with a non-majoritarian procedure, Sequential Voting by Veto (SVV). We focused on two different aspects of the subject: the likelihood of sophisticated voting under each one of the voting procedures and the conditions that foster sincere and sophisticated voting under these procedures. The results highlighted the significant differences between majoritarian and non-majoritarian voting procedures as a key factor in determining the tendency of voters to use sincere or sophisticated voting. Clearly, the sincere model was dominant in SVV games while sophisticated voting dominated the PV games. The extent of sophisticated voting ran counter to the group size, a tendency that was stronger under SVV than under PV. By demonstrating the advantages of the minority principle, when voters are small in number, we hope to encourage the development of a solution that will enable the use of SVV in general elections.     

Assessing the citizen – candidate model

Citizen–candidate models postulate a politics without political parties. Any citizen may become a candidate for office. A winner is chosen from among the candidates by voting with ties broken by the flip of a coin. All voters have preferences over a set of policies. The winning candidate adopts his preferred policy. It is proved on certain assumptions that there exists an equilibrium in these models and that the equilibrium is efficient. The significance of the proof is tested here with reference to the paradox of voting, the exploitation problem, the transposition of the Nash equilibrium from markets to politics, and constitutional constraints. The quest for a political equilibrium leads in the end to the recognition of a minimal rock-bottom requirement for cooperation and negotiation in democratic government.     

Random votes under compulsory voting: Evidence from Brazil

Compulsory voting is known for boosting electoral turnout, even when sanctions for abstaining are small or loosely enforced. Much less is known, however, about the consequences of compulsory voting on vote choice, and, in particular, about the quality of electoral decisions. In this paper, we explore the extent to which voters meaningfully engage in the electoral process or simply vote randomly because voting is required by law. We conducted a large online survey in Brazil during the 2018 national elections to assess if voters engage in random voting. We evaluate random voting for low-profile, low-information elected offices (state and federal legislators) and others that receive greater media coverage (governor and president) and evaluate the determinants of random voting for each of them. We find that: 1) random voting does not appear to be affected by social desirability bias; 2) there is substantial random voting under compulsory voting; 3) more voters tend to engage in random voting in low-profile, low-information elections, as compared to elections that receive greater media coverage; and, 4) interest in politics, education, and disposition to vote if voting were to be voluntary reduce random voting. Our findings carry important implications for the study of citizen participation and civic competence under compulsory voting and for democratic representation, more broadly.     

Probabilistic Voting and Equilibrium: An Impossibility Result

The introduction of voters' uncertainty alone is in no way sufficient to guarantee an equilibrium outcome of a voting process. What is needed in addition is the assumption that the voting probabilities depend strictly convex respectively concave of the utility losses caused by the proposed policies of the different parties. This assumption is, however, not compatible with an unrestricted policy space, and it has to be rejected for theoretical as well as empirical reasons. Thus, using models of probabilistic voting we still have to accept that cycles can arise and electoral outcomes are prone to manipulation via agenda setting.     

Retrospective Voting and Political Representation

This article develops a theoretical model of political representation under the single-member district system. I establish the existence of equilibria in which legislative voting of each legislator depends only on her preference and her electorate's preference and voters sanction badly behaved incumbents and retain well-behaved ones based solely on their own representatives' roll-call records. In equilibrium, voters achieve a partial representation with respect to representatives' behavior in each district. However, with respect to representation of the social majority, my findings are indeterminate. On the one hand, there exists an equilibrium in which the majority-preferred alternative is the outcome guaranteed, except in very special circumstances. On the other hand, this equilibrium is not generally the unique equilibrium, and, for some parameter values, there is an equilibrium in which the majority-preferred alternative is less likely than the alternative preferred only by the minority to be the outcome.     

Proposal Rights and Political Power

In a canonical model of sequential collective bargaining over a divisible good we show that equilibrium expected payoffs are not restricted by players' voting rights or their impatience. For all monotonic voting rules and discount factors, and for all divisions of the good among players, there exists a stationary proposal-making rule such that this division represents players' expected payoffs in a Stationary Subgame Perfect Nash equilibrium in pure strategies . The result highlights the significance of proposal rights in determining political power in collective deliberations.     

On the (mis)measurement of legislator ideology and shirking

In this paper, we show that current statistical measures of legislator's shirking are implicitly based on the electoral concept of a unique majority rule equilibrium point in the policy space where elections are contested. We note that such equilibria do not exist generically and present statistical results showing that cross-sectional regressions where legislators' voting indices are predicted by district average demograhic and economic data are mis-specified. We also discuss a weaker equilibrium construct, the uncovered set, and present statistical evidence showing that differences in voting behavior between Senators from the same state are positively related to the heterogeneity of the electorate. We argue that current evidence alleged to show shirking by Senators is equally consistent with Senators who perfectly represent an idiosyncratic constituency that cannot be represented by district average data.     

Opposition backlash and platform convergence in a spatial voting model with campaign contributions

This paper investigates the effects of campaign contributions on candidate behavior in elections. The particular focus is on how candidates choose their platforms when they know that the positions they take will influence the level of campaign contributions that they (and their opponents) receive from concerned interest groups. The analysis is carried out in the context of a simple one- dimensional spatial voting model with two candidates and two interest groups. Since the earliest Hotelling-Downs formulations, a central issue in the literature on spatial voting has been the degree to which, under various sets of assumptions, the candidates' platforms converge in equilibrium. This paper extends that literature by examining how the introduction of interest groups making campaign contributions affects the degree of platform convergence. The paper shows that when choosing their platforms, candidates face a trade-off between generataing increased support from opponents and provoking a backlash from the opposition. An example is developed to illustrate a surprising result that can occur because of the backlash effect: the introduction of two extremist interest groups may lead the candidates to moderate their platforms, resulting in a greater degree of platform convergence than would be observed in the absence of any campaign contributions.     

Efficacy in small electorates under plurality and approval voting

Assuming independent voting and random tie-breaking, we describe a procedure for computing the efficacy of a voting strategy under approval voting (AV) and plurality voting (PV) in single winner elections. When the number of candidates exceeds three, the voter's efficacy — his or her ability to affect the outcome of an election — is greater under AV than PV. The relative advantage of AV over PV increases as the number of voters or number of candidates increases.This study has been supported by grant 84-00329 of the U.S.A.-Israel Binational Science Foundation (BSF), Jerusalem, Israel, in collaboration with Steven Brams and Zeev Maoz.     

Pareto optimality of policy proposals with probabilistic voting

This paper studies the Pareto optimality properties of policy proposals and electoral outcomes when there is probabilistic voting. Theorem 1 proves that, when the position of one candidate is taken as fixed, the other candidate will propose a Pareto optimal alternative. This implies that whenever there is an electoral equilibrium (in pure strategies) the electoral outcome is Pareto optimal. It also implies that, even if there is no such equilibrium, the electoral outcomes from a sequence of elections will be Pareto optimal (except, possibly, for an initial status quo).     

Bicameral bargaining and federation formation

This paper takes a political economy approach to fiscal centralisation through federation formation. We analyse a two-region model of federalism with inter-regional policy spillovers. Departing from a state of independence with decentralised provision of public policy, we analyse the proposed formation of a federation to internalise the spillovers. We consider equilibrium allocations under alternative institutional environments involving: simple majority voting, a restriction of uniform taxation, and regional bargaining through a bicameral legislature. Bicameralism facilitates federation formation for the largest set of parameter values of the model, which is consistent with the observation that bicameralism and federalism often appear together.