Majority rule when voters like to win

I analyze symmetric majority rule voting equilibria when voters wish to elect the better candidate and to vote for the winner. When voters care only about the winning candidate (the standard formulation) a unique responsive equilibrium exists. The addition of a desire to win creates multiple equilibria, some with unusual properties. In most of these equilibria information is not aggregated effectively, and I uncover the novel possibility of negative information aggregation in which information aggregated in equilibrium is used to select the worse rather than the better candidate.I then characterize the efficiency of optimal equilibria as the population becomes large and show that a discontinuity arises in the information aggregation capabilities of the majority rule voting mechanism: in elections without a dominant front-running candidate the better candidate is almost surely elected, whereas in races with a front-runner information cannot be effectively aggregated in equilibrium.     

Majority rule when voters like to win

I analyze symmetric majority rule voting equilibria when voters wish to elect the better candidate and to vote for the winner. When voters care only about the winning candidate (the standard formulation) a unique responsive equilibrium exists. The addition of a desire to win creates multiple equilibria, some with unusual properties. In most of these equilibria information is not aggregated effectively, and I uncover the novel possibility of negative information aggregation in which information aggregated in equilibrium is used to select the worse rather than the better candidate.I then characterize the efficiency of optimal equilibria as the population becomes large and show that a discontinuity arises in the information aggregation capabilities of the majority rule voting mechanism: in elections without a dominant front-running candidate the better candidate is almost surely elected, whereas in races with a front-runner information cannot be effectively aggregated in equilibrium.     

Bandwagons and Momentum in Sequential Voting

In this paper I show that an equilibrium exists to the sequential voting game in which a bandwagon begins with probability 1. These bandwagons are driven by a combination of beliefs and the desire of voters to vote for the winning candidate. Significantly, in this equilibrium the pivot probability for each voter is non-zero, even in an infinite population. Consequently, the bandwagons do not always start after one (or at most two) favourable decisions (as do economic cascades) and varying levels of informative voting are observed, consistent with observations from sequential voting in U.S. presidential primaries. Further, voters are exposed to counterintuitive incentives, referred to as "buyers' remorse", that have been attributed to real primary voters.
I also derive equilibrium behaviour in this environment when voting is simultaneous and compare the quality of information aggregation within each mechanism. I relate the conclusions to U.S. presidential primaries and find they are consistent with a common conclusion about the front-loading of the primary process: that in tight elections (with no front-runner) simultaneous voting is preferred, whereas in lopsided elections sequential voting is preferred. The superior performance of sequential voting in lopsided races is precisely because bandwagons occur.     

SEQUENTIAL OR SIMULTANEOUS ELECTIONS? A WELFARE ANALYSIS

Should all voters vote on the same day or should elections be staggered? Using a model of voting and social learning, we illustrate that sequential elections place too much weight on early states but also provide late voters with valuable information. Simultaneous elections equally weigh states but place too much weight on voter priors, providing an inappropriate advantage to front‐runners. Simultaneous elections are thus preferred if the front‐runner advantage is small, but sequential elections are preferred if the advantage is large. Our quantitative welfare analysis of presidential primaries suggests that simultaneous systems slightly outperform sequential systems.     

Learning in Elections and Voter Turnout

Voter turnout in game theoretic models of voting has typically been difficult to predict because of the problem of multiple Nash equilibria ( Palfrey and Rosenthal 1983, 1985 ). Many of these equilibria require an extreme precision of beliefs among voters that is unlikely to be reached in real elections. At the same time, mechanisms like pre‐election polls exist to shape the beliefs of voters about expected turnout. We combine these two features in a model of voter learning in elections and characterize the asymptotically stable equilibria of both complete and incomplete information games in a simple symmetric setting with two candidates. We also show how the model can be used to qualitatively explain several phenomena observed in reality: increases in costs of voting affect turnout adversely but there may be persistence of turnout levels between elections even though costs and other parameters change. Increase in uncertainty increases turnout while increases in the size of the electorate decrease it, in line with intuition.     

Coordinated voting in sequential and simultaneous elections: some experimental evidence

This paper studies a situation wherein a set of voters choose between two alternatives in the presence of a payoff externality. Specifically, regardless of her intrinsic preference, a voter’s payoff is maximized should she vote for the alternative that garners a majority of the votes cast. Are votes coordinated on a single alternative? Using laboratory experiments, we examine voting patterns in sequential voting and simultaneous voting elections. Across both election types, we also vary the amount of information that an individual voter has regarding the intrinsic preferences of the other voters. Our main findings are as follows. In the “low” information treatment, sequential voting elections facilitate coordinated voting. However, in the “high” information treatment, voting patterns are not dependent on how the election is structured.     

Comparison of Scoring Rules in Poisson Voting Games

Scoring rules are compared by their equilibria in simple voting games with Poisson population uncertainty, using new techniques for computing pivot probabilities. Best-rewarding rules like plurality voting can generate discriminatory equilibria where the voters disregard some candidate as not a serious contender, although he may be universally liked, or may be symmetric to other candidates as in the Condorcet cycle. Such discriminatory equilibria are eliminated by worst-punishing rules like negative voting, but then even a universally disliked candidate may have to be taken seriously. In simple bipolar elections, equilibria are always majoritarian and efficient under approval voting, but not other scoring rules. Journal of Economic Literature Classification: D72.     

Asymmetric Equilibria in a Model with Costly Voting

Most of the voting models limit their analysis to the investigation of symmetric equilibria where “similar” voters make “similar” voting decisions. In this paper we examine the validity of this restriction in a model with costly plurality voting. We first show that in any pure strategy equilibrium every two individuals who have the same preferences and participate in elections, would vote for the same candidate. However, this result does not hold for mixed strategies equilibria.     

Voting with abstention

Most voting models in the literature neglect abstention, but is such a simplification justified? I investigate this question in a model with outside pressure on voters. For sequential voting (e.g., roll call votes), with and without an abstention option, there is a unique subgame perfect equilibrium, which implies that true majorities always succeed. Abstention can be an equilibrium strategy for some voters, in particular under complex decision rules (e.g., weighted voting, double majorities). Simultaneous voting often has a unique pure strategy equilibrium but also a plethora of mixed and pure/mixed strategy equilibria. Therefore, only with equilibrium selection, can we evaluate the consequences of neglecting abstention. For equal weight voting, equilibria selected by the procedure of Harsanyi and Selten change completely with an abstention option, even if abstention itself is not or rarely used. With small enough outside pressure, however, the selected equilibrium honors true majorities in both cases.     

Condorcet Jury Theorem or Rational Ignorance

We analyze a symmetric model of an election in which voters are uncertain about which of two alternatives is desirable for them. Each voter must incur some cost to acquire information about the alternatives. We show that by focusing on unbiased voting strategies, general symmetric signal structures can be degenerated to a two‐signal model. In addition, we show that for any sequence of unbiased voting equilibria, if the second‐order derivative of the information cost function at no information is zero, then the probability of electing the desirable alternative converges to one, that is, the Condorcet Jury Theorem is valid. Otherwise, this probability converges to some value less than one; that is, the “rational ignorance” hypothesis is valid.     

The Nonequivalence of Import and Export Quotas in Strategic Equilibrium

The authors use a standard general‐equilibrium trade model to show that export and import policies are not symmetric in the equilibrium of a strategic game with quotas. It is assumed that N (identical) large countries, without cooperation, set their import (or export) quotas to maximize domestic welfare. It is shown that the equilibrium in which all countries use import quotas differs from, and is superior to, the equilibrium in which countries use export quotas. The difference arises because the elasticity of the residual foreign export supply schedule differs between the two equilibria. The authors also study the properties of the sequential equilibrium of the game. In a simultaneous‐move game, each country is indifferent as to whether it uses an import or export quota, given the policy of the other country. However, in a sequential‐move game, the first mover will prefer to use an import quota rather than an export quota.     

Sequential equilibrium in monotone games: A theory-based analysis of experimental data

A monotone game is an extensive-form game with complete information, simultaneous moves and an irreversibility structure on strategies. It captures a variety of situations in which players make partial commitments and allows us to characterize conditions under which equilibria result in socially desirable outcomes. However, since the game has many equilibrium outcomes, the theory lacks predictive power. To produce stronger predictions, one can restrict attention to the set of sequential equilibria, or Markov equilibria, or symmetric equilibria, or pure-strategy equilibria. This paper explores the relationship between equilibrium behavior in a class of monotone games, namely voluntary contribution games, and the behavior of human subjects in an experimental setting. Several key features of the symmetric Markov perfect equilibrium (SMPE) are consistent with the data. To judge how well the SMPE fits the data, we estimate a model of Quantal Response Equilibrium (QRE) [R. McKelvey, T. Palfrey, Quantal response equilibria for normal form games, Games Econ. Behav. 10 (1995) 6-38; R. McKelvey, T. Palfrey, Quantal response equilibria for extensive form games, Exp. Econ. 1 (1998) 9-41] and find that the decision rules of the QRE model are qualitatively very similar to the empirical choice probabilities.     

Voting Equilibria in Multi-candidate Elections

We consider a general plurality voting game with multiple candidates, where voter preferences over candidates are exogenously given. In particular, we allow for arbitrary voter indifferences, as may arise in voting subgames of citizen-candidate or locational models of elections. We prove that the voting game admits pure strategy equilibria in undominated strategies. The proof is constructive: we exhibit an algorithm, the "best winning deviation" algorithm, that produces such an equilibrium in finite time. A byproduct of the algorithm is a simple story for how voters might learn to coordinate on such an equilibrium.     

Private provision of a discrete public good: efficient equilibria in the private-information contribution game

We study the voluntary provision of a discrete public good via the contribution game. Players independently and simultaneously make nonrefundable contributions to fund a discrete public good, which is provided if and only if contributions cover the cost of production. We characterize nonconstant continuous symmetric equilibria, giving sufficient conditions for their existence. We show the common normalization by which players’ values are distributed over [0, 1] is not without loss of generality: if the distribution over this interval has continuous density f with f(0) >  0, then no (nonconstant) continuous symmetric equilibrium exists. We study in detail the case in which players’ private values are uniformly distributed, showing that, generically, when one continuous equilibrium exists, a continuum of continuous equilibria exists. For any given cost of the good, multiple continuous equilibria cannot be Pareto ranked. Nevertheless, not all continuous equilibria are interim incentive efficient. The set of interim incentive efficient equilibria is exactly determined. The authors thank Manfred Dix, George Mailath, Andrew Postlewaite, and an anonymous referee for their comments.     

Primaries and the New Hampshire Effect

Candidates for U.S. presidential elections are determined through sequential elections in single states, the primaries. We develop a model in which candidates can influence their winning probability in electoral districts by spending money on campaigning. The equilibrium replicates several stylized facts very well: Campaigning is very intensive in the first district. The outcome of the first election then creates an asymmetry in the candidates' incentives to campaign in the next district, which endogenously increases the equilibrium probability that the first winner wins in further districts.On the normative side, our model offers a possible explanation for the sequential organization: It leads (in expectation) to a lower level of advertising expenditures than simultaneous elections. Moreover, if one of the candidates is the more effective campaigner, sequential elections also perform better with regard to the selection of the best candidate.     

Endogenous timing,strategic tariff game and bilateral trade in vertical oligopoly

This paper builds an observable delay game in endogenous timing to study the possible occurrence of trade wars in a vertical, bilateral trade model. It examines the effects of production cost differences and order of moves on optimal tariffs, market equilibria, dumping margin and social welfare in both fixed timing and endogenous timing games. In a fixed timing game, it shows that price dumping in the intermediate good market arises from differences in country-specific final good production costs. Different from Bernhofen (1995; Journal of International Economics), trade costs resulting from reciprocal tariffs in upstream markets can reverse the price dumping under certain conditions. In an endogenous timing game, this paper finds that the magnitude of cost differences significantly influences countries’ decisions on the order of moves in a strategic tariff-making game. Both countries want to be the first movers under a small cost difference and consequently reach a simultaneous equilibrium result. This demonstrates that the second-best equilibrium proceeds under certain conditions. Under a large cost difference, sequential policy-making is the subgame perfect Nash equilibrium. If the cost difference is sufficiently large, both countries have an incentive to launch a trade war as a multiple equilibrium game.     

Let the experts decide? Asymmetric information,abstention, and coordination in standing committees

We examine abstention when voters in standing committees are asymmetrically informed and there are multiple pure-strategy equilibria – swing voterʼs curse (SVC) equilibria where voters with low-quality information abstain and equilibria when all participants vote their information. When the asymmetry in information quality is large, we find that voting groups largely coordinate on the SVC equilibrium which is also Pareto optimal. However, we find that when the asymmetry in information quality is not large and the Pareto optimal equilibrium is for all to participate, significant numbers of voters with low-quality information abstain. Furthermore, we find that information asymmetry induces voters with low-quality information to coordinate on a non-equilibrium outcome. This suggests that coordination on “letting the experts” decide is a likely voting norm that sometimes validates SVC equilibrium predictions but other times does not.     

The power of ESS: An experimental study

Abstract. Our experimental design mimics a traditional evolutionary game framework where players are matched pairwise to play a symmetric 33 bimatrix game that has two Nash equilibria. One equilibrium is an evolutionary stable state, or ESS; the other is an equilibrium in dominated strategies. Our primary experimental result is the observation that the ESS becomes extremely attractive when subjects have minimal information about the payoff functions, although the dominated equilibrium assures the highest equilibrium payoff. The attractiveness of the ESS is only moderate when players are completely informed about the 33 payoff matrix. Correspondence to: S.K. Berninghaus     

A Bayesian Model of Voting in Juries

We model voting in juries as a game of incomplete information, allowing jurors to receive a continuum of signals. We characterize the unique symmetric equilibrium of the game, and give a condition under which no asymmetric equilibria exist under unanimity rule. We offer a condition under which unanimity rule exhibits a bias toward convicting the innocent, regardless of the size of the jury, and give an example showing that this bias can be reversed. We prove a “jury theorem” for our general model: As the size of the jury increases, the probability of a mistaken judgment goes to zero for every voting rule except unanimity rule. For unanimity rule, the probability of making a mistake is bounded strictly above zero if and only if there do not exist arbitrarily strong signals of innocence. Our results explain the asymptotic inefficiency of unanimity rule in finite models and establishes the possibility of asymptotic efficiency, a property that could emerge only in a continuous model. Journal of Economic Literature Classification Numbers: C72, D72.     

Voting in small networks with cross-pressure

We present a model of participation in elections in small networks, in which citizens suffer from cross-pressures if voting against the alternative preferred by some of their social contacts. We analyze how the existence of cross-pressures may shape voting decisions, and so, political outcomes; and how parties may exploit this effect to their interest. We characterize the strong perfect equilibria of the game and show that, in equilibrium, the social network determines which party wins the election. We also show that to dispose of the citizens better connected in the network with the other faction is not a guarantee to win the election.