Anonymous yes–no voting with abstention and multiple levels of approval

Symmetric (3,2) simple games serve as models for anonymous voting systems in which each voter may vote “yes,” abstain, or vote “no,” the outcome is “yes” or “no,” and all voters play interchangeable roles. The extension to symmetric (j,2) simple games, in which each voter chooses from among j ordered levels of approval, also models some natural decision rules, such as pass–fail grading systems. Each such game is determined by the set of (anonymous) minimal winning profiles. This makes it possible to count the possible systems, and the counts suggest some interesting patterns. In the (3,2) case, the approach yields a version of May's Theorem, classifying all possible anonymous voting rules with abstention in terms of quota functions. In contrast to the situation for ordinary simple games these results reveal that the class of simple games with 3 or more levels of approval remains large and varied, even after the imposition of symmetry.     

Non-Additive Beliefs and Strategic Equilibria

This paper studies n-player games where players' beliefs about their opponents' behaviour are modelled as non-additive probabilities. The concept of an “equilibrium under uncertainty” which is introduced in this paper extends the equilibrium notion of Dow and Werlang (1994, J. Econom. Theory64, 305–324) to n-player games in strategic form. Existence of such an equilibrium is demonstrated under usual conditions. For low degrees of ambiguity, equilibria under uncertainty approximate Nash equilibria. At the other extreme, with a low degree of confidence, maximin equilibria appear. Finally, robustness against a lack of confidence may be viewed as a refinement for Nash equilibria. Journal of Economic Literature Classification Numbers: C72, D81.     

Learning to Learn, Pattern Recognition, and Nash Equilibrium

The paper studies a large class of bounded-rationality, probabilistic learning models on strategic-form games. The main assumption is that players “recognize” cyclic patterns in the observed history of play. The main result is convergence with probability one to a fixed pattern of pure strategy Nash equilibria, in a large class of “simple games” in which the pure equilibria are nicely spread along the lattice of the game. We also prove that a necessary condition for convergence of behavior to a mixed strategy Nash equilibrium is that the players consider arbitrarily long histories when forming their predictions.Journal of Economic LiteratureClassification Numbers: C72, D83.     

Evolutionary stability and lexicographic preferences

We explore the interaction between evolutionary stability and lexicographic preferences. To do so, we define a limit Nash equilibrium for a lexicographic game as the limit of Nash equilibria of nearby games with continuous preferences. Nash equilibria of lexicographic games are limit Nash equilibria, but not conversely. Modified evolutionarily stable strategies (Binmore and Samuelson, 1992. J. Econ. Theory 57, 278–305) are limit Nash equilibria. Modified evolutionary stability differs from “lexicographic evolutionarily stability” (defined by extending the common characterization of evolutionary stability to lexicographic preferences) in the order in which limits in the payoff space and the space of invasion barriers are taken.     

Bounding the inefficiency of equilibria in nonatomic congestion games

Equilibria in noncooperative games are typically inefficient, as illustrated by the Prisoner's Dilemma. In this paper, we quantify this inefficiency by comparing the payoffs of equilibria to the payoffs of a “best possible” outcome. We study a nonatomic version of the congestion games defined by Rosenthal [Int. J. Game Theory 2 (1973) 65], and identify games in which equilibria are approximately optimal in the sense that no other outcome achieves a significantly larger total payoff to the players—games in which optimization by individuals approximately optimizes the social good, in spite of the lack of coordination between players. Our results extend previous work on traffic routing games.     

Reference Points and Negative Reciprocity in Simple Sequential Games

We investigate experimentally whether preferences over an outcome depend on what other possible outcomes of the situation under consideration are, i.e., whether choices are “menu dependent.” In simple sequential games we analyze whether reactions to a certain benchmark outcome are influenced by changes in the payoffs of another outcome, not attainable at that time, called the “reference point.” Our data provide evidence that is favorable to the notion of menu dependence. Alterations of the reference point can lead to quantitatively significant changes in behavior at the benchmark outcome. The behavior we observe can be interpreted in terms of negative reciprocity. Journal of Economic Literature Classification Numbers: C91, C92, C72.     

Regularity of pure strategy equilibrium points in a class of bargaining games

Summary. We develop an index theory for the Stationary Subgame Perfect (SSP) equilibrium set in a class of n-player sequential bargaining games with probabilistic recognition rules. For games with oligarchic voting rules (a class that includes unanimity rule), we establish conditions on individual utilities that ensure that for almost all discount factors, the number of SSP equilibria is odd and the equilibrium correspondence lower-hemicontinuous. For games with general, monotonic voting rules, we show generic (in discount factors) determinacy of SSP equilibria under the restriction that the agreement space is of dimension one. For non-oligarchic voting rules and agreement spaces of higher finite dimension, we establish generic determinacy for the subset of SSP equilibria in pure strategies. The analysis also extends to the case of fixed delay costs. Lastly, we provide a sufficient condition for uniqueness of SSP equilibrium in oligarchic games.Received: 13 May 2004, Revised: 1 March 2005, JEL Classification Numbers: C62, C72, C78.I thank John Duggan and participants of the 2003 annual meeting of the American Political Science Association, Philadelphia, PA, the Political Economy Seminar at Northwestern University, and the Economic Theory seminar at the University of Rochester for helpful comments.     

Undominated Nash Implementation with Collusion and Renegotiation

This article looks at implementation in economic environments when agents have perfect information about the state of the world, but cannot commit not to renegotiate bad outcomes or to collude against each other. If renegotiation satisfies a weak condition of disagreement point monotonicity, then any Pareto-efficient social choice function can be implemented if there are at least three agents who play undominated Nash equilibria. The mechanism does not use modulo or integer games, has no bad mixed strategy equilibria, and is “bounded.”Journal of Economic LiteratureClassification Number: D71.     

Equilibrium Selection inn-Person Coordination Games

This paper investigates several approaches to equilibrium selection and the relationships between them. The class of games we study aren-person generalized coordination games with multiple Pareto rankable strict Nash equilibria. The main result is that all selection criteria select the same outcome (namely the risk dominant equilibrium) in two-person games, and that most equivalences break for games with more than two players. All criteria select the Pareto efficient equilibrium in voting games, of which pure coordination games are special cases.Journal of Economic LiteratureClassification Numbers: C70, C72, D82.     

Iterated strict dominance in general games

We offer a definition of iterated elimination of strictly dominated strategies (IESDS^*) for games with (in)finite players, (non)compact strategy sets, and (dis)continuous payoff functions. IESDS^* is always a well-defined order independent procedure that can be used to solve Nash equilibrium in dominance-solvable games. We characterize IESDS^* by means of a “stability” criterion, and offer a sufficient and necessary epistemic condition for IESDS^*. We show by an example that IESDS^* may generate spurious Nash equilibria in the class of Reny's better-reply secure games. We provide sufficient/necessary conditions under which IESDS^* preserves the set of Nash equilibria.     

On Renegotiation-Proof Equilibria in Finitely Repeated Games

This paper studies renegotiation-proof equilibria of Beno?̂t and Krishna (1993) in finitely repeated games with more than two players. We provide a simple characterization of the set of average payoffs from renegotiation-proof equilibria: the limiting set of average equilibrium payoffs is either efficient or dimensionally small as the horizon of the repeated game goes to infinity. Two sufficient conditions for the former to occur are provided. An example shows that those conditions cannot be weakened.Journal of Economic LiteratureClassification Numbers: C72, C73.     

Belief-Based Refinements in Signalling Games

There are many economic problems which, when modelled as games of incomplete information, give rise to many sequential equilibria, severely limiting the usefulness of the model. There has recently been a large literature devoted to "refining" the set of equilibria in order to reduce this multiplicity by restricting the set of admissible disequilibrium beliefs. This paper argues that the logical foundations of some refinements and the equilibria they focus on are problematic and, further, proposes an alternative refinement that avoids the difficulties. We also provide an existence theorem covering a broad class of signalling games often studied in economics. Journal of Economic Literature Classification Numbers: C72, C70, D82.     

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.     

Egalitarianism and Efficiency in Repeated Symmetric Games

We analyze the symmetric equilibria of repeated symmetric games where there is a conflict of interests over equilibria—the battle-of-the-sexes or the hawk–dove game are key examples. If one restricts attention to symmetric equilibria, efficient equilibria must be egalitarian. For finitely repeated games, and generic discount factors, there is a unique outcome path which ensures efficiency within the class of symmetric equilibria. This is also true for the infinitely repeated games if the players are sufficiently impatient. Journal of Economic Literature Classification Numbers: C72, C73.     

Cournot–Nash equilibria in continuum games with non-ordered preferences

In the usual framework of continuum games with externalities, we substantially generalize Cournot–Nash existence results [Balder, A unifying approach to existence of Nash equilibria, Int. J.Game Theory 24 (1995) 79–94; On the existence of Cournot–Nash equilibria in continuum games, J. Math. Econ. 32 (1999) 207–223; A unifying pair of Cournot–Nash equilibrium existence results, J. Econ. Theory 102 (2002) 437–470] to games with possibly non-ordered preferences, providing a continuum analogue of the seminal existence results by Mas-Colell [An equilibrium existence theorem without complete or transitive preferences, J. Math. Econ. 1 (1974) 237–246], Gale and Mas-Colell [An equilibrium existence theorem for a general model without ordered preferences, J. Math. Econ. 2 (1975) 9–15], Shafer and Sonnenschein [Equilibrium in abstract economies without ordered preferences, J. Math. Econ. 2 (1975) 345–348], Borglin and Keiding [Existence of equilibrium actions and of equilibrium: a note on the “new” existence theorems, J. Math. Econ. 3 (1976) 313–316] and Yannelis and Prabhakar [Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econ. 12 (1983) 233–245].     

Robust rational turnout

We show that equilibria of a class of participation games (Palfrey and Rosenthal in Public Choice 41(1):7–53, 1983; Journal of Public Economics 24(2):171–193, 1984) exhibit minimal heterogeneity of behavior so that players’ mixed strategies are summarized by at most two probabilities. We then establish that, except for a finite set of common costs of participation, these games are regular. Thus, equilibria of these voting games are robust to general payoff perturbations and survive in nearby games of incomplete information. Thanks to participants of the 2006 MPSA conference for comments on an early version.     

Mechanism design with partial state verifiability

We study implementation in environments where agents have limited ability to imitate others. Agents are randomly and privately endowed with type-dependent sets of messages. So sending a message becomes a partial proof regarding type. For environments where agents can send any combination of available messages, we develop an Extended Revelation Principle and characterize the incentive constraints which implementable allocations must satisfy. When not all message combinations are feasible, static mechanisms no longer suffice. If a ‘punishment’ allocation exists for each agent, then implementable allocations can be characterized as equilibria of a “Revelation Game,” in which agents first select from the menus of allocation rules, then the mediator requests each agent to send some verifying messages. When a punishment allocation fails to exist for some agent, dynamic games in which agents gradually reveal their evidence implement a larger set of outcomes. The latter result provides a foundation for a theory of debate.     

Beyond chopsticks: Symmetric equilibria in majority auction games

Majority auction games are simultaneous sealed-bid auctions of identical objects among identical bidders who each want to win a specified fraction (more than a half) of the objects. Each bidder receives no benefit from winning less than the specified fraction and no additional benefit from winning more than it. Symmetric equilibria having simple, intuitive forms are shown to exist in first-price, second-price and all-pay versions of such games when the number of bidders is sufficiently large. This contrasts with earlier results for the two-bidder “pure chopstick” majority auction games where the only known equilibria are more complicated.     

On ‘Informationally Robust Equilibria’ for Bimatrix Games

Informationally robust equilibria (IRE) are introduced in Robson (Games Econ Behav 7: 233–245, 1994) as a refinement of Nash equilibria for strategic games. Such equilibria are limits of a sequence of (subgame perfect) Nash equilibria in perturbed games where with small probability information about the strategic behavior is revealed to other players (information leakage). Focusing on bimatrix games, we consider a type of informationally robust equilibria and derive a number of properties they form a non-empty and closed subset of the Nash equilibria. Moreover, IRE is a strict concept in the sense that the IRE are independent of the exact sequence of probabilities with which information is leaked. The set of IRE, like the set of Nash equilibria, is the finite union of polytopes. In potential games, there is an IRE in pure strategies. In zero-sum games, the set of IRE has a product structure and its elements can be computed efficiently by using linear programming. We also discuss extensions to games with infinite strategy spaces and more than two players. The authors would like to thank Marieke Quant for her helpful comments.     

Noncooperative foundations of bargaining power in committees and the Shapley–Shubik index

In this paper we explore the noncooperative foundations of the bargaining power that a voting rule gives to each member of a committee that bargains in search of consensus over a set of feasible agreements under a voting rule. Assuming complete information, we model a variety of bargaining protocols and investigate their stationary subgame perfect equilibria. We show how the Shapley–Shubik index and other power indices can be interpreted as measures of ‘bargaining power’ that appear in this light as limit cases.