Mark Fey

Stability and Coordination in Duverger's Law: A Formal Model of Preelection Polls and Strategic Voting

This paper investigates the dynamics of the “wasted vote” phenomenon and Duvergers Law. I construct a theoretical model in order to consider how preelection polls serve to inform the electorate about the relative chances of the candidates and how that information acts over time to decrease the support of the trailing candidate. The results shed light on how public opinion polls can aggregate information in the electorate and coordinate voters on the viable candidates in the election. Specifically, I show that in a Bayesian game model of strategic voting there exist non-Duvergerian equilibria in which all three candidates receive votes (in the limit). These equilibria require extreme coordination, however, and any variation in beliefs leads voters away from them to one of the Duvergerian equilibria. Thus, non-Duvergerian equilibria are unstable, while two-party equilibria are not. In addition, I describe how preelection polls provide information to voters about the viability of candidates and can thus be used by voters to coordinate on a Duvergerian outcome.

An experimental study of constant-sum centipede games

In this paper, we report the results of a series of experiments on a version of the centipede game in which the total payoff to the two players is constant. Standard backward induction arguments lead to a unique Nash equilibrium outcome prediction, which is the same as the prediction made by theories of “fair” or “focal” outcomes.We find that subjects frequently fail to select the unique Nash outcome prediction. While this behavior was also observed in McKelvey and Palfrey (1992) in the “growing pie” version of the game they studied, the Nash outcome was not “fair”, and there was the possibility of Pareto improvement by deviating from Nash play. Their findings could therefore be explained by small amounts of altruistic behavior. There are no Pareto improvements available in the constant-sum games we examine. Hence, explanations based on altruism cannot account for these new data.We examine and compare two classes of models to explain these data. The first class consists of non-equilibrium modifications of the standard “Always Take” model. The other class we investigate, the Quantal Response Equilibrium model, describes an equilibrium in which subjects make mistakes in implementing their best replies and assume other players do so as well. One specification of this model fits the experimental data best, among the models we test, and is able to account for all the main features we observe in the data.

A note on the Condorcet Jury Theorem with supermajority voting rules

1 My thanks to Peter Stone for sparking my interest in this issue. Helpful comments and corrections came from two anonymous referees and the editor.

Repeated Downsian electoral competition

We analyze an infinitely repeated version of the Downsian model of elections. The folk theorem suggests that a wide range of policy paths can be supported by subgame perfect equilibria when parties and voters are sufficiently patient. We go beyond this result by imposing several suitable refinements and by giving separate weak conditions on the patience of voters and the patience of parties under which every policy path can be supported. On the other hand, we show that only majority undominated policy paths can be supported in equilibrium for arbitrarily low voter discount factors: if the core is empty, the generic case in multiple dimensions, then voter impatience leads us back to the problem of non-existence of equilibrium. We extend this result to give conditions under which core equivalence holds for a non-trivial range of voter and party discount factors, providing a game-theoretic version of the Median Voter Theorem in a model of repeated Downsian elections.

Risky but Rational: War as an Institutionally Induced Gamble

We present and process-trace a complete information model of diversionary war. In our model, leaders must retain the support of some fraction of a selectorate whose response in turn depends on the outcome of an international conflict. The need to retain the loyalty of a segment of the selectorate generates institutionally induced risk preferences in leaders. Under specified conditions, this in turn results in the leaders’ choice of risky options and war emerges as a rational gamble. We analyze when the leader prefers to impose such a gamble, what the optimal gamble would be, and its effect on crisis bargaining between two leaders. We find that when leaders have institutionally induced risk preferences, whether leaders rationally choose to initiate or continue a war can depend on which selectorate cares most about the outcome of the conflict. A reexamination of the bargaining over a settlement to end the First World War in December 1916–January 1917 and Germanys contemporaneous consideration of unrestricted submarine warfare allows us to demonstrate the empirical relevance of the model.

May’s Theorem with an infinite population

Abstract.In this paper, we investigate majority rule with an infinite number of voters. We use an axiomatic approach and attempt to extend May’s Theorem characterizing majority rule to an infinite population. The analysis hinges on correctly generalizing the anonymity condition and we consider three different versions. We settle on bounded anonymity as the appropriate form for this condition and are able to use the notion of asymptotic density to measure the size of almost all sets of voters. With this technique, we define density q-rules and show that these rules are characterized by neutrality, monotonicity, and bounded anonymity on almost all sets. Although we are unable to provide a complete characterization applying to all possible sets of voters, we construct an example showing that our result is the best possible. Finally, we show that strengthening monotonicity to density positive responsiveness characterizes density majority rule on almost all sets.

Symmetric games with only asymmetric equilibria

It is known that not every symmetric game has a symmetric equilibrium because there are examples of symmetric games that fail to have any equilibria at all. But this leads to the following question: If a symmetric game has a Nash equilibrium, does it have a symmetric Nash equilibrium? In this Note, we show that the answer to this question is no by providing two examples of symmetric games that have only asymmetric equilibria.

Choosing from a large tournament

A tournament can be viewed as a majority preference relation without ties on a set of alternatives. In this way, voting rules based on majority comparisons are equivalent to methods of choosing from a tournament. We consider the size of several of these tournament solutions in tournaments with a large but finite number of alternatives. Our main result is that with probability approaching one, the top cycle set, the uncovered set, and the Banks set are equal to the entire set of alternatives in a randomly chosen large tournament. That is to say, each of these tournament solutions almost never rules out any of the alternatives under consideration. We also discuss some implications and limitations of this result.

The swing voter's curse with adversarial preferences

Abstract We analyze voting behavior in a large electorate in which voters have adversarial state-contingent preferences with incomplete information about the state of the world. We show that one type of voter can suffer from the swing voters curse a la Feddersen and Pesendorfer [The swing voters curse, Amer. Econ. Rev. 86 (1996) 408–424], and go on to characterize the symmetric Nash equilibria of this model under different parameter values. We prove that unlike settings with nonadversarial preferences, there are equilibria in which in one state of the world, a minority-preferred candidate almost surely wins the election and thus the election may fail to correctly aggregate information. Indeed, we show that the fraction of the electorate dissatisfied with the result can be as large as 66 2 3 % .

The minimal covering set in large tournaments

We prove that in almost all large tournaments, the minimal covering set is the entire set of alternatives. That is, as the number of alternatives gets large, the probability that the minimal covering set of a uniformly chosen random tournament is the entire set of alternatives goes to one. In contrast, it follows from a result of (Fisher and Reeves, Linear Algebra Appl 217:83–85, 1995) that the bipartisan set contains about half of the alternatives in almost all large tournaments.