Trembles in extensive games with ambiguity averse players

We introduce and analyze three definitions of equilibrium for finite extensive games with imperfect information and ambiguity averse players. In a setting where players’ preferences are represented by maxmin expected utility, as characterized in Gilboa and Schmeidler (J Math Econ 18(2):141–153, 1989), our definitions capture the intuition that players may consider the possibility of slight arbitrary mistakes. This generalizes the idea leading to trembling-hand perfect equilibrium as introduced in Selten (Int J Game Theory 4(1):25–55, 1975), by allowing for ambiguous trembles characterized by sets of distributions. We prove existence for two of our equilibrium notions and relate our definitions to standard equilibrium concepts with expected utility maximizing players. Our analysis shows that ambiguity aversion can lead to behavioral implications that are distinct from those attained under expected utility maximization, even if ambiguous beliefs only arise from the possibility of slight mistakes in the implementation of unambiguous strategies.     

What do uncertainty-averse decision-makers believe?

Summary. This paper introduces the concept of firm belief, which is proposed as a new epistemic model for a wide class of preferences. In particular, firm beliefs are shown to have the following desirable properties: (i) they are derived from preferences according to a plausible rule of epistemic inference; (ii) they satisfy standard logical properties; and (iii) tractable representations of firm belief are available for all (suitably continuous) biseparable preferences [13, 14], including the Choquet expected utility [30] and maxmin expected utility [16] classes. We also use firm belief to construct a generalization of Nash equilibrium for (two-player) normal form games. Received: December 14, 1999; revised version: February 26, 2001     

Strategic games beyond expected utility

This paper argues that Nash equilibrium is a solution where all strategic uncertainty has been resolved and, therefore, inappropriate to model situations that involve ??ambiguity.?? Instead, to capture what players will do in the presence of some strategic uncertainty, takes a solution concept that is closed under best replies. It is shown that such a solution concept, fixed sets under the best reply correspondence, exists for a class of games significantly wider than those games for which generalizations of Nash equilibrium exist. In particular, this solution can do without the expected utility hypothesis.     

Nash equilibrium and party polarization in an electoral model with mixed motivations

We study the existence problem of Nash equilibrium as well as the patterns of equilibrium policy outcomes in an electoral competition model with mixed motivations. Each party maximizes a sum of party members’ expected utility and office rent. The inclusion of office rent renders the payoff of each party discontinuous. This makes it difficult to apply usually fixed point arguments to prove the existence of Nash equilibria. By using a recently developed concept, multiple restrictional security (MR‐security) we provide conditions under which a pure‐strategy Nash equilibrium exists within fairly general settings, and further the analysis by presenting conditions under which various patterns of policy choices, including polarization, arise in equilibrium.     

The local best response criterion: An epistemic approach to equilibrium refinement

The standard refinement criteria for extensive form games, including subgame perfect, perfect, perfect Bayesian, sequential, and proper, reject important classes of reasonable Nash equilibria and accept many unreasonable Nash equilibria. This paper develops a new refinement criterion, based on epistemic game theory, that captures the concept of a Nash equilibrium that is plausible when players are rational. I call this the local best response (LBR) criterion. This criterion is conceptually simpler than the standard refinement criteria because it does not depend on out-of-equilibrium, counterfactual, or passage to the limit arguments. The LBR is also informationally richer because it clarifies the epistemic conditions that render a Nash equilibrium reasonable. The LBR criterion appears to render the traditional refinement criteria superfluous.     

Pairwise epistemic conditions for Nash equilibrium

We introduce a framework for modeling pairwise interactive beliefs and provide an epistemic foundation for Nash equilibrium in terms of pairwise epistemic conditions locally imposed on only some pairs of players. Our main result considerably weakens not only the standard sufficient conditions by Aumann and Brandenburger (1995), but also the subsequent generalization by Barelli (2009). Surprisingly, our conditions do not require nor imply mutual belief in rationality.     

Optimal portfolio positioning under ambiguity

This paper analyzes the optimality of financial portfolios when the investor has a utility with ambiguity aversion. It provides a general result about the optimal portfolio profile under ambiguity, in the Anscombe–Aumann framework, using the Maccheroni et al. (2006) approach which includes Gilboa and Schmeidler's (1989) multiple prior preferences and Hansen and Sargent's (2011) multiplier preferences. The paper then details the CRRA case with an ambiguity index based on relative entropy. Such findings have practical applications in structured portfolio management. Indeed, it is important to take account of uncertainty about the true values of financial parameters when determining the best portfolio profile.     

The Nash bargaining solution in general n-person cooperative games

We present a noncooperative foundation for the Nash bargaining solution for an n-person cooperative game in strategic form. The Nash bargaining solution should be immune to any coalitional deviations. Our noncooperative approach yields a new core concept, called the Nash core, for a cooperative game based on a consistency principle. We prove that the Nash bargaining solution can be supported (in every subgame) by a stationary subgame perfect equilibrium of the bargaining game if and only if the Nash bargaining solution belongs to the Nash core.     

Extensive Form Games with Uncertainty Averse Players

Nash equilibrium presumes that the beliefs of a player are represented by a probability measure. Motivated by the Ellsberg Paradox and relevant experimental findings demonstrating that this representation of beliefs may be unrealistic, this paper generalizes Nash equilibrium in finite extensive form games to allow for preferences conforming to the multiple priors model developed by Gilboa and Schmeidler [Journal of Mathematical Economics, 18 (1989), 141–153]. The implications of this generalization for strategy choices and welfare are studied. Journal of Economic Literature Classification Numbers: C72, D81.     

Ambiguity aversion and the absence of indexed debt

Summary. Following the seminal works of Schmeidler (1989), Gilboa and Schmeidler (1989), roughly put, an agents subjective beliefs are said to be ambiguous if the beliefs may not be represented by a unique probability distribution, in the standard Bayesian fashion, but instead by a set of probabilities. An ambiguity averse decision maker evaluates an act by the minimum expected value that may be associated with it. In spite of wide and long-standing support among economists for indexation of loan contracts there has been relatively little use of indexation, except in situations of extremely high inflation. The object of this paper is to provide a (theoretical) explanation for this puzzling phenomenon based on the hypothesis that economic agents are ambiguity averse. The paper considers a general equilibrium model based on Magill and Quinzii (1997) with ambiguity averse agents, where both nominal and indexed bond contracts are available for trade and all relevant prices are determined endogenously. We obtain conditions which prompt an endogenous cessation of trade in indexed bonds: i.e., conditions under which there is no trade in indexed bonds in any equilibrium; only nominal bonds are traded.Received: 7 April 2003, Revised: 8 March 2004, JEL Classification Numbers: D81, E31, D52, E44.Correspondence to: Sujoy MukerjiWe thank seminar members at Birkbeck, Oxford, Paris I, Southampton and Tel Aviv, the audience at the 00 European Workshop on General Equilibrium Theory, and especially, E.Dekel, I. Gilboa, D. Schmeidler and A. Pauzner for helpful comments. The first author acknowledges financial assistance from an Economic and Social Research Council of U.K. Research Fellowship (# R000 27 1065). The second author thanks financial support from the French Ministry of Research (Action Concertée Incitative).     

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.     

Achieving efficiency with manipulative bargainers

Two agents bargain over the allocation of a bundle of divisible commodities. After strategically reporting utility functions to a neutral arbitrator, the outcome is decided by using a bargaining solution concept chosen from a family that includes the Nash and the Raiffa–Kalai–Smorodinsky solutions. When reports are restricted to be continuous, strictly increasing and concave, it has been shown that this kind of “distortion game” leads to inefficient outcomes. We study the distortion game originated when agents are also allowed to claim non-concave utility functions. Contrasting with the previous literature, any interior equilibrium outcome is efficient and any efficient allocation can be supported as an equilibrium outcome of the distortion game. In a similar fashion to the Nash demand game we consider some uncertainty about the opponent's features to virtually implement the Nash bargaining solution.     

p-Dominance and Equilibrium Selection under Perfect Foresight Dynamics

This paper studies equilibrium selection based on a class of perfect foresight dynamics and relates it to the notion of p-dominance. A continuum of rational players is repeatedly and randomly matched to play a symmetric n×n game. There are frictions: opportunities to revise actions follow independent Poisson processes. The dynamics has stationary states, each of which corresponds to a Nash equilibrium of the static game. A strict Nash equilibrium is linearly stable under the perfect foresight dynamics if, independent of the current action distribution, there exists a consistent belief that any player necessarily plays the Nash equilibrium action at every revision opportunity. It is shown that a strict Nash equilibrium is linearly stable under the perfect foresight dynamics with a small degree of friction if and only if it is the p-dominant equilibrium with p<1/2. It is also shown that if a strict Nash equilibrium is the p-dominant equilibrium with p<1/2, then it is uniquely absorbing (and globally accessible) for a small friction (but not vice versa). Set-valued stability concepts are introduced and their existence is shown. Journal of Economic Literature Classification Numbers: C72, C73.     

Equilibrium play in matches: Binary Markov games

We study two-person extensive form games, or “matches,” in which the only possible outcomes (if the game terminates) are that one player or the other is declared the winner. The winner of the match is determined by the winning of points, in “point games.” We call these matches binary Markov games. We show that if a simple monotonicity condition is satisfied, then (a) it is a Nash equilibrium of the match for the players, at each point, to play a Nash equilibrium of the point game; (b) it is a minimax behavior strategy in the match for a player to play minimax in each point game; and (c) when the point games all have unique Nash equilibria, the only Nash equilibrium of the binary Markov game consists of minimax play at each point. An application to tennis is provided.     

Epistemic Conditions for Equilibrium in Beliefs without Independence

Aumann and Brandenburger [Econometrica63(1995), 1161–1180.] provide sufficient conditions on the knowledge of the players in a game for their beliefs to constitute a Nash equilibrium. They assume, among other things, mutual knowledge of rationality. By rationality of a player, it is meant that the action chosen by him maximizes his expected utility, given his beliefs. There is, however, no need to restrict the notion of rationality to expected utility maximization. This paper shows that their result can be generalized to the case where players' preferences over uncertain outcomes belong to a large class of non-expected utility preferences.Journal of Economic LiteratureClassification Numbers: C72, D81.     

Consistency of beliefs and epistemic conditions for Nash and correlated equilibria

A consistency condition (action-consistency) on the interim beliefs of players in a game is introduced. Action-consistency is weaker than common priors and, unlike common priors, is characterized by a “no-bets” condition on verifiable events. Using action-consistency, we provide epistemic conditions to Nash and correlated equilibria weakening the common knowledge restrictions in Aumann and Brandenburger [Aumann, R., Brandenburger, A., 1995. Epistemic conditions for Nash equilibrium. Econometrica 63, 1161–1180] and Aumann [Aumann, R., 1987. Correlated equilibrium as an expression of Bayesian rationality. Econometrica 55, 1–18].     

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.     

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.     

Mixed-Strategy Nash Equilibrium Based upon Expected Utility and Quadratic Utility

The notion of a mixed-strategy Nash equilibrium suffers from three inherent difficulties. First, given the equilibrium strategies of other players, there are many best replies. Second, the equilibrium is unstable. Third, comparative statics results are counterintuitive. We demonstrate that these difficulties all have their origin in von Neumann and Morgenstern′s expected utility. In contrast, players with "quadratic utility" have unique best replies and the Nash equilibrium appears to yield intuitive comparative statics results. Journal of Economic Literature Classification Number: C72.     

Testing threats in repeated games

Under most game-theoretic solution concepts, equilibrium beliefs are justified by off-equilibrium events. I propose an equilibrium concept for infinitely repeated games, called “Nash Equilibrium with Tests” (NEWT), according to which players can only justify their equilibrium beliefs with events that take place on the equilibrium path itself. In NEWT, players test every threat that rationalizes a future non-myopic action that they take. The tests are an integral part of equilibrium behavior. Characterization of equilibrium outcomes departs from the classical “folk theorems”. The concept provides new insights into the impact of self-enforcement norms, such as reciprocity, on long-run cooperation.