Disappearing private reputations in long-run relationships

For games of public reputation with uncertainty over types and imperfect public monitoring, Cripps et al. [Imperfect monitoring and impermanent reputations, Econometrica 72 (2004) 407-432] showed that an informed player facing short-lived uninformed opponents cannot maintain a permanent reputation for playing a strategy that is not part of an equilibrium of the game without uncertainty over types. This paper extends that result to games in which the uninformed player is long-lived and has private beliefs, so that the informed player's reputation is private. The rate at which reputations disappear is uniform across equilibria and reputations also disappear in sufficiently long discounted finitely repeated games.     

Reputation and Perfection in Repeated Common Interest Games

We consider a wide class of repeated common interest games perturbed with one-sided incomplete information: one player (the informed player) might be a commitment type playing the Pareto dominant action. As discounting, which is assumed to be symmetric, and the prior probability of the commitment type go to zero, it is shown that the informed player can be held close to her minmax payoff even when perfection is imposed on the equilibrium.Journal of Economic LiteratureClassification Numbers: C73, D83.     

Minimizing Regret: The General Case

In repeated games with differential information on one side, the labelling “general case” refers to games in which the action of the informed player is not known to the uninformed, who can only observe a signal which is the random outcome of his and his opponent's action. Here we consider the problem of minimizing regret (in the sense first formulated by J. Hannan (1956, in Contributions to the Theory of Games, Vol. III, Annals of Mathematics Studies, Vol. 39, Princeton, NJ: Princeton Univ. Press.) when the information available is of this type. We give a simple condition describing the approachable set. Journal of Economic Literature Classification Number: D81, D82, D83.     

Infinite horizon common interest games with perfect information

We consider infinite horizon common interest games with perfect information. A game is a K-coordination game if each player can decrease other players' payoffs by at most K times his own cost of punishment. The number K represents the degree of commonality of payoffs among the players. The smaller K is, the more interest the players share. A K-coordination game tapers off if the greatest payoff variation conditional on the first t periods of an efficient history converges to 0 at a rate faster than K^−t as t→∞. We show that every subgame perfect equilibrium outcome is efficient in any tapering-off game with perfect information. Applications include asynchronously repeated games, repeated games of extensive form games, asymptotically finite horizon games, and asymptotically pure coordination games.     

The value of information in oligopoly with demand uncertainty

This paper discusses the value of information in supermodular and submodular games, using a simple duopoly model where the level of demand is uncertain. It is shown that the value of information issuperadditive (resp.,subadditive) between players if the game issupermodular (resp.,submodular). For example, in a Bertrand (resp., Cournot) market with (possibly imperfect) substitute products, one firm's information acquisition increases (resp., decreases) the other firm's incentive to acquire the same information. Furthermore, when the game is either supermodular or submodular, the value of information is higher when the player isexpected to be informed according to the opponent's belief than when the player is expected to be uninformed; this result is reversed when the game has asymmetric modularity (i.e., one player's action is substitutional to the other's, and the latter's action is complemental to the former's). These qualitative observations have a potential to be applied to a larger class of games with uncertainty where payoffs are smooth (e.g., twice continuously differentiable) in actions and states.     

A general structure theorem for the Nash equilibrium correspondence

I consider n-person normal form games where the strategy set of each player is a non-empty compact convex subset of an Euclidean space, and the payoff function of player i is continuous in joint strategies and continuously differentiable and concave in the player i's strategy. No further restrictions (such as multilinearity of the payoff functions or the requirement that the strategy sets be polyhedral) are imposed. I demonstrate that the graph of the Nash equilibrium correspondence on this domain is homeomorphic to the space of games. This result generalizes a well-known structure theorem in [Kohlberg, E., Mertens, J.-F., 1986. On the strategic stability of equilibria. Econometrica 54, 1003–1037]. It is supplemented by an extension analogous to the unknottedness theorems in [Demichelis S., Germano, F., 2000. Some consequences of the unknottedness of the Walras correspondence. J. Math. Econ. 34, 537–545; Demichelis S., Germano, F., 2002. On (un)knots and dynamics in games. Games Econ. Behav. 41, 46–60]: the graph of the Nash equilibrium correspondence is ambient isotopic to a trivial copy of the space of games.     

Hierarchical thinking and learning in rank order contests

We analyze a class of coordination games in which the Kth player to submit an entry wins a contest. These games have an infinite number of symmetric equilibria and the set of equilibria does not change with K. We run experiments with 15 participants and with K=3, 7, and 11. Our experiments show that the value of K affects initial submissions and convergence to equilibrium. When K is small relative to the number of participants, our experiments show that repeated play converges to or near zero. When K is large, an equilibrium is often not reached as a result of repeated play. We seek explanations to these patterns in hierarchical thinking and direction learning.      

Repeated signaling games

I analyze a class of repeated signaling games in which the informed player's type is persistent and the history of actions is perfectly observable. In this context, a large class of possibly complex sequences of signals can be supported as the separating equilibrium actions of the “strong type” of the informed player. I characterize the set of such sequences. I also characterize the sequences of signals in least cost separating equilibria (LCSE) of these games. In doing this, I introduce a state variable that can be interpreted as a measure of reputation. This gives the optimization problem characterizing the LCSE a recursive structure. I show that, in general, the equilibrium path sequences of signals have a simple structure. The shapes of the optimal sequences depend critically on the relative concavities of the payoff functions of different types, which measure the relative preferences towards payoff smoothing.     

Voluntary provision of a pure public good as the game of ‘chicken’

The voluntary exchange model, where the amount of a public good and contributions to its cost are simultaneously determined, is treated as a (2×2) non-cooperative non-constant-sum game. Three conceptually different types of games emerge. One of them is ‘Chicken’; each player can gain by pre-emptively threatening to pay nothing—unless the other player acts likewise. ‘Zero public goods’ is thus a possible outcome of voluntary exchange, even though it is Pareto inferior.     

Evolutionary stability of discrimination under observability

We study the evolution of preferences under perfect and almost perfect observability in symmetric 2-player games. We demonstrate that if nature can choose from a sufficiently general preference space, which includes preferences over outcomes that may depend on the opponent's preference-type, then, in most games, only discriminating preferences (treating different types of opponents differently in the same situation) can be evolutionary stable and some discriminating types are stable in a very strong sense in all games. We use these discriminating types to show that any symmetric outcome which gives players more than their minmax value in material payoffs (fitness) can be seen as equilibrium play of a player population with such strongly stable preferences.     

Fictitious play in ‘one-against-all’ multi-player games

Summary. A compound game is an (n + 1) player game based on n two-person subgames. In each of these subgames player 0 plays against one of the other players. Player 0 is regulated, so that he must choose the same strategy in all n subgames. We show that every fictitious play process approaches the set of equilibria in compound games for which all subgames are either zero-sum games, potential games, or games. Received: July 18, 1997; revised version: December 4, 1998     

Evolutionary stability of discrimination under observability

We study the evolution of preferences under perfect and almost perfect observability in symmetric 2-player games. We demonstrate that if nature can choose from a sufficiently general preference space, which includes preferences over outcomes that may depend on the opponent's preference-type, then, in most games, only discriminating preferences (treating different types of opponents differently in the same situation) can be evolutionary stable and some discriminating types are stable in a very strong sense in all games. We use these discriminating types to show that any symmetric outcome which gives players more than their minmax value in material payoffs (fitness) can be seen as equilibrium play of a player population with such strongly stable preferences.     

Congestion games with load-dependent failures: Identical resources

We define a new class of games, congestion games with load-dependent failures (CGLFs). In a CGLF each player can choose a subset of a set of available resources in order to try and perform his task. We assume that the resources are identical but that players' benefits from successful completion of their tasks may differ. Each resource is associated with a cost of use and a failure probability which are load-dependent. Although CGLFs in general do not have a pure strategy Nash equilibrium, we prove the existence of a pure strategy Nash equilibrium in every CGLF with nondecreasing cost functions. Moreover, we present a polynomial time algorithm for computing such an equilibrium.     

Imitation and selective matching in reputational games

This paper investigates imitation and selective matching in reputational games with an outside option. We identify two classes of such games, ultimatum and trust games. By selective matching we mean that short-run players have the possibility of selecting the long-run player they play against. We find that selective matching (unlike random matching) favors the equilibrium associated to reputation in the ultimatum game, but not in the trust game.     

Extending Invariant Solutions

In this paper we show that an invariant solution for normal form games can be extended uniquely to an invariant solution for strategic form games. This result has the following consequence for the reduction of a (normal form) game. Suppose that the pure strategies are removed that are payoff equivalent with some (possibly mixed) strategy. Then, if one is concerned with an invariant solution, a further reduction by identifying for each player arbitrary payoff-equivalent strategies is not necessary.Journal of Economic LiteratureClassification Number: C72.     

Learning from Personal Experience: One Rational Guy and the Justification of Myopia

The paper examines a large population analog of fictitious play in which players learn from personal experience, focusing on what happens when a single rational player is added to the population. Because the learning process naturally generates contagion dynamics, the rational player at times has an incentive to act nonmyopically. In 2 × 2 games the dynamics are asymmetric and favor risk dominant equilibria. A variety of other examples are presented.Journal of Economic LiteratureClassification Number: C7.     

Games played in a contracting environment

We analyze normal form games where a player has to pay a price to a supplier in order to play a specific action. Our focus is on supplier competition, arising from the fact that distinct suppliers supply different players, and possibly different actions of the same player. With private contracts, where a player only observes the prices quoted by his own suppliers, the set of equilibrium distributions over player actions coincides with the set of equilibrium distributions when all actions are supplied competitively, at cost. With public contracts, the two distributions differ dramatically even in simple games.     

Games played in a contracting environment

We analyze normal form games where a player has to pay a price to a supplier in order to play a specific action. Our focus is on supplier competition, arising from the fact that distinct suppliers supply different players, and possibly different actions of the same player. With private contracts, where a player only observes the prices quoted by his own suppliers, the set of equilibrium distributions over player actions coincides with the set of equilibrium distributions when all actions are supplied competitively, at cost. With public contracts, the two distributions differ dramatically even in simple games.     

Beliefs and rationalizability in games with complementarities

We propose two characteristics of beliefs and study their role in shaping the set of rationalizable strategy profiles in games with incomplete information. The first characteristic, type-sensitivity, is related to how informative a player thinks his type is. The second characteristic, optimism, is related to how “favorable” a player expects the outcome of the game to be. The paper has two main results: the first result provides an upper bound on the size of the set of rationalizable strategy profiles; the second gives a lower bound on the change of location of this set. These bounds are explicit expressions that involve type-sensitivity, optimism, and payoff characteristics. Our results generalize and clarify the well-known uniqueness result of global games (Carlsson and van Damme, 1993). They also imply new uniqueness results and allow us to study rationalizability in new environments. We provide applications to supermodular mechanism design (Mathevet, 2010b) and information processing errors.     

Fending off one means fending off all: evolutionary stability in quasi-submodular aggregative games

The implications of evolutionarily stable behavior in finite populations have recently been explored for a variety of aggregative games. This note proves an intimate relationship between quasi-submodularity and global evolutionary stability of strategies for these games, which – apart from being of independent interest – accounts for a number of results obtained in the recent literature: we show that any evolutionarily stable strategy of a quasi-submodular aggregative game must also be globally stable. That is, if one mutant cannot successfully invade a population, any number of mutants can do so even lessThe author is grateful to Ana Ania and Carlos Alōs-Ferrer for comments on an earlier version