UNIQUENESS OF EQUILIBRIUM PAYOFFS IN FINITELY REPEATED GAMES WITH IMPERFECT MONITORING*

This paper examines when a finitely repeated game with imperfect monitoring has a unique equilibrium payoff vector. This problem is nontrivial under imperfect monitoring, because uniqueness of equilibrium (outcome) in the stage game does not extend to finitely repeated games. A (correlated) equilibrium is equilibrium minimaxing if any player's equilibrium payoff is her minimax value when the other players choose a correlated action profile from the actions played in the equilibrium. The uniqueness result holds if all stage game correlated equilibria are equilibrium minimaxing and have the same payoffs. The uniqueness result does not hold under weaker conditions.     

Coordination cycles

Players repeatedly face a coordination problem in a dynamic global game. By choosing a risky action (invest) instead of waiting, players risk instantaneous losses as well as a loss of payoffs from future stages, in which they cannot participate if they go bankrupt. Thus, the total strategic risk associated with investment in a particular stage depends on the expected continuation payoff. High continuation payoff makes investment today more risky and therefore harder to coordinate on, which decreases today's payoff. Thus, expectation of successful coordination tomorrow undermines successful coordination today, which leads to fluctuations of equilibrium behavior even if the underlying economic fundamentals happen to be the same across the rounds. The dynamic game inherits the equilibrium uniqueness of the underlying static global game.     

Kindness,confusion, or … ambiguity?

Other-regarding preferences or decision errors are the main explanations put forward to justify contributions exceeding the non-cooperative optimum in VCM games. An alternative rationale relies on ambiguity aversion. Ambiguity aversion increases the perceived marginal benefit of own contributions, which in equilibrium will exceed the Nash level. We present a series of experiments testing this hypothesis. To control for other-regarding preferences, we run a two-player game in which a human player plays with a virtual agent. Players are assigned either to a risky setting (known probabilities of opponent’s choices) or to an ambiguity setting (probabilities of opponent’s contribution are vague). Results show that ambiguity affects contributions. However, attitude to ambiguity appears to be affected by the location of the aggregate Nash optimum inside the decision space.     

Bargaining and markets: complexity and the competitive outcome

Rubinstein and Wolinsky (Rev. Econ. Stud. 57 (1990) 63-78) consider a simple decentralised market game in which agents meet randomly or voluntarily and bargain over the terms of trade. They show that any individually rational price can be sustained as a sequential equilibrium even though the model has a unique competitive outcome. Here, I consider Rubinstein and Wolinsky's model and show that if complexity costs of implementing strategies enter players’ preferences, together with the standard payoff in the game, then every equilibrium is stationary/Markov and induces the unique competitive price. Thus, I demonstrate that aversion to complexity may provide a justification for the competitive outcome.     

The desire for land: Strategic lending with adverse selection

This paper deals with a setting in which borrowers and lenders place different values on an asset that can be used as collateral. Under adverse selection, lenders may rationally choose credit contracts with the object of attracting a relatively risky group of clients, so raising their chances of gaining possession of the asset through default. Contracts of differing attractiveness to borrowers can also coexist in equilibrium. When an ‘inside’ and an ‘outside’ lender compete, the latter placing a lower value on the collateral, and their loanable funds are sufficiently limited, a separating equilibrium may exist in which the insider offers a contract which attracts risky borrowers, whereas the outsider's contract is aimed at a safer group. If loanable funds are ample, the only equilibrium will involve pooling contracts, but the insider may still offer more attractive contracts in an entry game.     

Reputation and exogenous private learning

Short-run competitors in the chain store game receive noisy signals of the long-run incumbent firm's type. The history of signals, which in the limit is fully revealing, is observable to the competitors but possibly not to the incumbent. As long as there is sufficient noise in the signals, then in any equilibrium a patient weak incumbent obtains a payoff strictly higher than her minmax payoff.     

Value of public information in sender-receiver games

I find in two classes of sender-receiver games that the receiver’s equilibrium payoff is not increasing in the informativeness of a public signal because the sender may transmit less information when the public signal is more informative.     

Persuasion and learning by countersignaling

We model countersignaling (i.e., very high types refraining from signaling) arising from the tradeoff between persuasion and learning in a signaling game. We assume that the agent has imperfect private information regarding his/her productivity, which the signaling action provides additional verifiable information about. A higher-type agent benefits more from providing such objective, albeit imprecise, “proof” for the market, but may also gain less from learning about his/her productivity. When the latter effect dominates the former for the very high types, the equilibrium exhibits countersignaling: very high and low types pool on refraining from signaling, and only the medium types signal. Under certain conditions, the countersignaling equilibrium is the unique pure-strategy perfect sequential equilibrium.     

Correlated Nash equilibrium

We modify the epistemic conditions for Nash equilibrium only to accommodate Gilboa and Schmeidler's [I. Gilboa, D. Schmeidler, Maxmin expected utility with nonunique prior, J. Math. Econ. 18 (1989) 141-153] maxmin expected utility preferences, and identify the equilibrium concept in n-player strategic games that characterizes the modified epistemic conditions. The epistemic characterization supports the equilibrium concept as a minimal generalization of Nash equilibrium, in the sense that it deviates from Nash equilibrium only in terms of players' attitude towards ambiguity. Consequently, comparing it with Nash equilibrium constitutes a ceteris paribus study of the effects of ambiguity on how a game is played. For example, with ambiguity, (beliefs about) action choices are in general correlated.     

Characterising equilibrium selection in global games with strategic complementarities

Global games are widely used to predict behaviour in games with strategic complementarities and multiple equilibria. We establish two results on the global game selection. First, we show that, for any supermodular complete information game, the global game selection is independent of the payoff functions chosen for the game?s global game embedding. Second, we give a simple sufficient criterion to derive the selection and establish noise independence in many-action games by decomposing them into games with smaller action sets, to which we may often apply simple criteria. We also report in which small games noise independence may be established by counting the number of players or actions.     

Endogenous private information structures

We formally incorporate the option to gather information into a game and thus endogenize the information structure. We ask whether models with exogenous information structures are robust with respect to this endogenization. Any Nash equilibrium of the game with information acquisition induces a Nash equilibrium in the corresponding game with an exogenous structure. We provide sufficient conditions on the structure of the game for which this remains true when ‘Nash’ is replaced by ‘sequential’. We characterize the (sequential) Nash equilibria of games with exogenous information structures that can arise as a (sequential) Nash equilibrium of games with endogenous information acquisition.     

Cycling in a stochastic learning algorithm for normal form games

In this paper we study a stochastic learning model for 2×2 normal form games that are played repeatedly. The main emphasis is put on the emergence of cycles. We assume that the players have neither information about the payoff matrix of their opponent nor about their own. At every round each player can only observe his or her action and the payoff he or she receives. We prove that the learning algorithm, which is modeled by an urn scheme proposed by Arthur (1993), leads with positive probability to a cycling of strategy profiles if the game has a mixed Nash equilibrium. In case there are strict Nash equilibria, the learning process converges a.s. to the set of Nash equilibria.     

Coordination and delay in global games

What is the effect of offering agents an option to delay their choices in a global coordination game? We address this question by considering a canonical binary action global game, and allowing players to delay their irreversible decisions. Those that delay have access to accurate private information at the second stage, but receive lower payoffs. We show that, as noise vanishes, as long as the benefit to taking the risky action early is greater than the benefit of taking the risky action late, the introduction of the option to delay reduces the incidence of coordination failure in equilibrium relative to the standard case where all agents must choose their actions at the same time. We outline the welfare implications of this finding, and probe the robustness of our results from a variety of angles.     

Coordination of mobile labor

We study coordination failures in many simultaneously occurring coordination problems. Players encounter one of the problems but have the outside option of migrating to one of the remaining ones. Drawing on the global games approach, we show that such a mobile game has a unique equilibrium that allows us to examine comparative statics. The endogeneity of the outside option value and of the migration activity leads to non-monotonicity of welfare with respect to mobility friction; high mobility may hurt players. We apply these “general equilibrium’’ findings to the problem of the labor market during industrialization as described by Matsuyama [Increasing returns, industrialization and indeterminacy of equilibrium, Quart. J. Econ. 106 (1991) 617-650].     

An efficient solution to the informed principal problem

In this paper I study mechanism design by an informed principal. I show that generically this problem has an ex-post efficient solution. In the equilibrium mechanism, the informed principal appropriates all expected social surplus, with each type of her getting all expected social surplus conditional on that type. This outcome is supported as a perfect sequential equilibrium of the informed principal game when the joint probability distribution from which the agents’ types are drawn satisfies two conditions: the well-known condition of Cremer and McLean and Identifiability condition introduced by Kosenok and Severinov [Individually rational, budget-balanced mechanisms and allocation of surplus, J. Econ. Theory (2002), forthcoming]. Conversely, these conditions are necessary for an ex-post efficient outcome to be attainable in an equilibrium of the informed principal game. Under these conditions only our equilibrium outcome constitutes a neutral optimum, i.e. cannot be eliminated by any reasonable concept of blocking [R. Myerson, Mechanism design by an informed principal, Econometrica 51 (1983) 1767-1797]. Identifiability and Cremer-McLean conditions are generic when there are at least three agents, and none of them has more types than the number of type profiles of the other agents.     

Connecting the cooperative and competitive structures of the multiple-partners assignment game

Multiple-partners assignment game is the name used by Sotomayor [The multiple partners game, in: M. Majumdar (Ed.), Equilibrium and Dynamics: Essays in Honor of David Gale, The Macmillan Press Ltd., New York, 1992; The lattice structure of the set of stable outcomes of the multiple partners assignment game, Int. J. Game Theory 28 (1999) 567-583] to describe the cooperative structure of the many-to-many matching market with additively separable utilities. Stability concept is proved to be different from the core concept. An economic structure is proposed where the concept of competitive equilibrium payoff is introduced in connection to the equilibrium concept from standard microeconomic theory. The paper examines how this equilibrium concept compares with the cooperative equilibrium concept. Properties of interest to the cooperative and competitive markets are derived.     

Repeated games with present-biased preferences

We study infinitely repeated games with perfect monitoring, where players have β-δ preferences. We compute the continuation payoff set using recursive techniques and then characterize equilibrium payoffs. We then explore the cost of the present-time bias, producing comparative statics. Unless the minimax outcome is a Nash equilibrium of the stage game, the equilibrium payoff set is not monotonic in β or δ. Finally, we show how the equilibrium payoff set is contained in that of a repeated game with smaller discount factor.     

Uniform payoff security and Nash equilibrium in compact games

We introduce a condition, uniform payoff security, for games with compact Hausdorff strategy spaces and payoffs bounded and measurable in players’ strategies. We show that if any such compact game G is uniformly payoff secure, then its mixed extension is payoff secure. We also establish that if a uniformly payoff secure compact game G has a mixed extension with reciprocally upper semicontinuous payoffs, then G has a Nash equilibrium in mixed strategies. We provide several economic examples of compact games satisfying uniform payoff security.     

Robust equilibria under non-common priors

This paper considers the robustness of equilibria to a small amount of incomplete information, where players are allowed to have heterogeneous priors. An equilibrium of a complete information game is robust to incomplete information under non-common priors if for every incomplete information game where each player's prior assigns high probability on the event that the players know at arbitrarily high order that the payoffs are given by the complete information game, there exists a Bayesian Nash equilibrium that generates behavior close to the equilibrium in consideration. It is shown that for generic games, an equilibrium is robust under non-common priors if and only if it is the unique rationalizable action profile. Set-valued concepts are also introduced, and for generic games, a smallest robust set is shown to exist and coincide with the set of a posteriori equilibria.     

Testing the TASP: An experimental investigation of learning in games with unstable equilibria

We report experiments studying mixed strategy Nash equilibria that are theoretically stable or unstable under learning. The Time Average Shapley Polygon (TASP) predicts behavior in the unstable case. We study two versions of Rock-Paper-Scissors that include a fourth strategy, Dumb. The unique Nash equilibrium is identical in the two games, but the predicted frequency of Dumb is much higher in the game where the NE is stable. Consistent with TASP, the observed frequency of Dumb is lower and play is further from Nash in the high payoff unstable treatment. However, Dumb is played too frequently in all treatments.