Are “Anti-Folk Theorems” in repeated games nongeneric?

Folk Theorems in repeated games hold fixed the game payoffs, while the discount factor is varied freely. We show that these results may be sensitive to the order of limits in situations where players move asynchronously. Specifically, we show that when moves are asynchronous, then for a fixed discount factor close to one there is an open neighborhood of games which contains a pure coordination game such that every Perfect equilibrium of every game in the neighborhood approximates to an arbitrary degree the unique Pareto dominant payoff of the pure coordination game.     

Negotiation games with unobservable mixed disagreement actions

A negotiation model combines an alternating offers bargaining game with a normal form stage game that determines players’ (interim) disagreement payoffs. Busch and Wen [Econometrica 63 (1995) 545] investigated this negotiation model under the assumption that players’ past mixed disagreement actions are observable. The question arises if this assumption is a substantial restriction. In this paper, we adopt the more plausible assumption that only the realizations of past mixed actions are observable, and find that the set of equilibrium payoffs shrinks, compared to that when mixed disagreement actions are observable. We precisely identify the effects of the unobservability and characterize the limiting set of equilibrium payoffs as the two players become sufficiently patient.     

Generic finiteness of equilibrium payoffs for bimatrix games

It is shown that in any affine space of payoff matrices the equilibrium payoffs of bimatrix games are generically finite.     

Convergence to competitive equilibria and elimination of no-trade (in a strategic market game with limit prices)

The strategic market games literature contains many results that predict Walrasian equilibria in the competitive limit. However, they usually come at the expense of ad hoc assumptions that rule out “pathological” no trade equilibria. This paper studies a strategic market game with limit prices. The set of Nash equilibrium allocations of this game converges to the set containing all competitive equilibria and no-trade, when players are replicated. Moreover, two rounds of iterated deletion of weakly dominated strategies eliminate the no-trade equilibria. Hence, replication paired with two rounds of iterated dominance gives a clean prediction of competitive equilibrium.     

In Support of Trigger Strategies: Experimental Evidence from Two-Person Noncooperative Games

Cooperative equilibria can be supported in a repeated game when players use trigger strategies. This paper tests how well trigger strategies explain behavior in two-person experimental games. Reducing payoffs for choices larger than the Cournot level induces smaller average outputs, behavior generally consistent with trigger strategy models. Reducing payoffs for choices well above the Cournot level will not affect behavior if actions are consistent with a trigger strategy involving longer-lived, less intense punishment phases (the grim-reaper strategy), but would matter for trigger strategies with short-lived but intense punishment phases. Results show that behavior is most consistent with the former.     

Endogenous mechanisms and Nash equilibrium in competitive contracting games

We model strategic competition in a market with asymmetric information as a noncooperative game in which each firm competes for the business of a buyer of unknown type by offering the buyer a catalog of products and prices. The timing in our model is Stackelberg: in the first stage, given the distribution of buyer types known to all firms and the deducible, type-dependent best responses of the agent, firms simultaneously and noncooperatively choose their catalog offers. In the second stage the buyer, knowing his type, chooses a single firm and product-price pair from that firm’s catalog. By backward induction, this Stackelberg game with asymmetric information reduces to a game over catalogs with payoff indeterminacies. In particular, due to ties within catalogs and/or across catalogs, corresponding to any catalog profile offered by firms there may be multiple possible expected firm payoffs, all consistent with the rational optimizing behavior of the agent for each of his types. The resolution of these indeterminacies depends on the tie-breaking mechanism which emerges in the market. Because each tie-breaking mechanism induces a particular game over catalogs, a reasonable candidate would be a tie-breaking mechanism which supports a Nash equilibrium in the corresponding catalog game. We call such a mechanism an endogenous Nash mechanism. The fundamental question we address in this paper is, does there exist an endogenous Nash mechanism—and therefore, does there exist a Nash equilibrium for the catalog game? We show under fairly mild conditions on primitives that catalog games naturally possess tie-breaking mechanisms which support Nash equilibria.     

Repeated two-person zero-sum games with unequal discounting and private monitoring

We consider discounted repeated two-person zero-sum games with private monitoring. We show that even when players have different and time-varying discount factors, each player’s payoff is equal to his stage-game minmax payoff in every sequential equilibrium. Furthermore, we show that: (a) in every history on the equilibrium path, the pair formed by each player’s conjecture about his opponent’s action must be a Nash equilibrium of the stage game, and (b) the distribution of action profiles in every period is a correlated equilibrium of the stage game. In the particular case of public strategies in public monitoring games, players must play a Nash equilibrium after any public history.     

Non-cooperative games on hyperfinite Loeb spaces

We present a particular class of measure spaces, hyperfinite Loeb spaces, as a model of situations where individual players are strategically negligible, as in large non-anonymous games, or where information is diffused, as in games with imperfect information. We present results on the existence of Nash equilibria in both kinds of games. Our results cover the case when the action sets are taken to be the unit interval, results now known to be false when they are based on more familiar measure spaces such as the Lebesgue unit interval. We also emphasize three criteria for the modelling of such game-theoretic situations—asymptotic implementability, homogeneity and measurability—and argue for games on hyperfinite Loeb spaces on the basis of these criteria. In particular, we show through explicit examples that a sequence of finite games with an increasing number of players or sample points cannot always be represented by a limit game on a Lebesgue space, and even when it can be so represented, the limit of an existing approximate equilibrium may disappear in the limit game. Thus, games on hyperfinite Loeb spaces constitute the ‘right' model even if one is primarily interested in capturing the asymptotic nature of large but finite game-theoretic phenomena.     

Best response dynamics with level-n expectations in two-stage games

This paper analyzes behavior in repeatedly played two-stage games, where players choose actions in both stages according to best replies using level-n expectations about the opponent׳s actions in both stages. Level-n expectations are recursively defined in a way that a player holding level-n expectations correctly predicts the action of an opponent holding level-(n−1) expectations. A general conceptual framework to study such dynamics for two-stage games is developed and it is shown that, contrary to results for single-stage games, the fixed points of the dynamics depend on the level of the expectations. In particular, for level-0 expectation, fixed points correspond to a Nash equilibrium of a simultaneous move version of the game, whereas (under certain conditions) fixed points converge towards the subgame perfect equilibrium of the two-stage game if the level of expectations goes to infinity. The approach is illustrated using a two-stage duopoly game, where firms in the first stage invest in activities reducing their marginal costs and in the second stage engage in Cournot competition. Conditions for local stability of the fixed points are derived for different levels of expectations and it is shown that level-2 expectations are sufficient to move the fixed-point of the dynamics to a close neighborhood of the subgame perfect equilibrium. Furthermore, it is demonstrated that although firms benefit from unilateral increases in the level of expectations, an increase of n by all firms reduces all profits.     

Coordinating under incomplete information

We show that, in a minimum effort game with incomplete information where player types are independently drawn, there is a largest and smallest Bayesian equilibrium, leading to the set of equilibrium payoffs (as evaluated at the interim stage) having a lattice structure. Furthermore, the range of equilibrium payoffs converges to those of the deterministic complete information version of the game, in the limit as the incomplete information vanishes. This entails that such incomplete information alone cannot explain the equilibrium selection suggested by experimental evidence. We thank Atila Abdulkadiroglu, Hans Carlsson, Ani Guerdjikova and an anonymous referee for helpful comments. Part of this work was done while Asheim was visiting Cornell University, which hospitality is gratefully acknowledged.     

MACROECONOMIC POLICY EVALUATION AND OPTIMAL CONTROL THEORY: A CRITICAL REVIEW OF SOME RECENT DEVELOPMENTS

Abstract. This survey addresses the recent literature on the application of optimal control theory and game theory to macroeconomic policy evaluation and design. This literature focuses on strategic interactions between governments and private agents engaged in dynamic non-cooperative games and emphasizes such issues as precommitment, credibility and reputation as important and endemic aspects of optimal policy design. A dominant theme is the problem of dynamic inconsistency and the inferiority of an equilibrium in single-stage full-information games without commitment. This may be alleviated in repeated games involving reputation effects and threat strategies designed to coerce rivals into pursuing particular strategies and to sustain a sequential equilibrium. The type and characteristics of a sequential equilibrium also depend importantly upon the information structure conditioning players'beliefs under uncertainty. Included here are separating and pooling equilibria, involving notions of learning, signalling and information revelation. Additional dimensions exist in stochastic systems which introduce other forms of uncertainty. Optimal control and game theory have also been applied to international macroeconomic policy design, with emphasis on the potential costs and benefits of non-cooperative and cooperative behaviour between countries and problems of international policy coordination.     

A dominance solvable global game with strategic substitutes

Global games emerged as an approach to equilibrium selection. For a general setting with supermodular payoffs, unique selection of equilibrium has been obtained through iterative elimination of strictly dominated strategies. For the case of global games with strategic substitutes, uniqueness of equilibrium has not been proved by iterative elimination of strictly dominated strategies, making the equilibrium less appealing. In this work we provide a condition for dominance solvability in a simple three-player binary-action global game with strategic substitutes. This opens an unexplored research agenda on the study of global games with strategic substitutes.     

Organizations and overlapping generations games: Memory,communication, and altruism

This paper studies the role of memory and communication in overlapping generations (OLG) games between ongoing organizations. In each organization, each individual, upon entry into the game, replaces his predecessor who has the same preferences and faces the same strategic possibilities. An individual has no prior memory - that is, he does not directly witness the events that occur before his tenure. Instead, each individual relies on information about the past from his predecessor via cheap talk. This paper highlights the role of communication as a surrogate for memory.It has been shown elsewhere that Folk Theorems hold in OLG games with long enough lived individuals who can perfectly observe the past. However, the Folk Theorem fails for many games when individuals have no prior memory. We show that for OLG games without prior memory but with costly communication, a Folk Theorem holds only when there is some altruistic link between cohorts in an organization. Our main result asserts that if communication costs are sufficiently small, or if altruistic weights on successors are sufficiently large, then a strongly stationary Folk Theorem (i.e., equilibrium payoffs are time invariant) obtains if a managers message is public information. The equilibria in this Folk Theorem require a special form of intergenerational sanctions. In these sanctions, punishment is sometimes carried out long after both victim and perpetrator have left the game. Without this special structure, altruism may in fact destroy cooperation when it would otherwise be possible.Received: 18 April 2003, Accepted: 20 October 2003, JEL Classification: C72, C73, D74, D82We have benefitted from helpful comments and conversations with Luca Anderlini, Kirk Bailey, Hans Haller, Matthew Jackson, and Takashi Shimizu, and numerous seminar participants. This research is partially supported by the National Science Foundation, and by grants-in-aid for scientific research of the Ministry of Education of Japan. Of course, any remaining error is our own.     

Bounded memory and permanent reputations

A recent result in repeated incomplete information games is that after an arbitrarily long history, any equilibrium of the continuation game must be an equilibrium of the complete information game. This result implies that reputation is a short-run phenomenon. We study a particular class of reputation games and show that bounded memory may lead to permanent reputations. In fact, for a particular range of parameters, a bounded memory player may never be able to learn anything at all.     

Potential games with incomplete preferences

This paper studies potential games allowing the possibility that players have incomplete preferences and empty best-response sets. We define four notions of potential games, ordinal, generalized ordinal, best-response, and generalized best-response potential games, and characterize them using cycle conditions. We study Nash equilibria of potential games and show that the set of Nash equilibria remains the same when every player’s preferences are replaced with the smallest generalized (best-response) potential relation or a completion of it. Similar results are established about strict Nash equilibria of ordinal and best-response potential games. Lastly, we examine the relations among the four notions of potential games as well as pseudo-potential games.     

The revelation of information in strategic market games: A critique of rational expectations equilibrium

We criticize the R.E.E. approach to asymmetric information general equilibrium because it does not explain how information gets ‘into’ the prices. This leads to well-known paradoxes. We suggest a multiperiod game instead, where the flow of information into and out of prices is explicitly modeled. In our game Nash equilibria (N.E.) (1) generalize Walrasian equilibria to asymmetric information, (2) exist generically, (3) eliminate pure speculation, (4) allow prices to reveal information and markets to become more efficient over time, (5) are consistent with the weak efficient markets hypothesis that tracking past prices is not profitable, (6) yet always lead to higher utility for better informed agents (such as experts). Throughout the paper we use one concrete game. In the last section we prove that there is a broad range of games that would have the same properties.     

The Theory of Socially Embedded Games: Applications and Extensions to Open and Closed Games

An earlier article, drawing on the mathematical theory of rules and rule complexes, extends and generalizes game theory (GGT). The theory has been used to conceptualize and analyze diverse social relationships, roles, and games as particular types of rule complexes.For instance, a social role, as a major basis of a parent's action in a game, consists of at least four key components – which are mathematical objects – in the determination of action: value complex, model of reality (including beliefs and knowledge bases), a repertoire of acts, routines, programs, and strategies, and modality, a role-specific algorithm for determining or generating action in game settings. This article applies and extends GGT in analyses of a market bargaining game (a type of open game) and of the classical game of prisoners' dilemma (a type of closed game). The applications show the concrete effects of social embeddedness on game structuring, game interaction patterns and outcomes, and social equilibria.     

Regularity and robustness in monotone Bayesian games

This paper defines regular and weakly regular equilibria for monotone Bayesian games with one-dimensional actions and types. It analyzes the robustness of equilibria with respect to perturbations. It also proves an index theorem and provides applications to uniqueness of equilibrium.