Risk-dominance and perfect foresight dynamics in N-player games

In perfect foresight dynamics, an action is linearly stable if expectation that people will always choose the action is self-fulfilling. A symmetric game is a PIM game if an opponent's particular action maximizes the incentive of an action, independently of the rest of the players. This class includes supermodular games, games with linear incentives and so forth. We show that, in PIM games, linear stability is equivalent to u-dominance, a generalization of risk-dominance, and that there is no path escaping a u-dominant equilibrium. Existing results on N-player coordination games, games with linear incentives and two-player games are obtained as corollaries.     

Repeated games with one-memory

We study the extent to which equilibrium payoffs of discounted repeated games can be obtained by 1-memory strategies. We establish the following in games with perfect (rich) action spaces: First, when the players are sufficiently patient, the subgame perfect Folk Theorem holds with 1-memory. Second, for arbitrary level of discounting, all strictly enforceable subgame perfect equilibrium payoffs can be approximately supported with 1-memory if the number of players exceeds two. Furthermore, in this case all subgame perfect equilibrium payoffs can be approximately supported by an ε-equilibrium with 1-memory. In two-player games, the same set of results hold if an additional restriction is assumed: Players must have common punishments. Finally, to illustrate the role of our assumptions, we present robust examples of games in which there is a subgame perfect equilibrium payoff profile that cannot be obtained with 1-memory. Thus, our results are the best that can be hoped for.     

Correlated equilibrium existence for infinite games with type-dependent strategies

Under study are games in which players receive private signals and then simultaneously choose actions from compact sets. Payoffs are measurable in signals and jointly continuous in actions. This paper gives a counter-example to the main step in Cotter?s [K. Cotter, Correlated equilibrium in games with type-dependent strategies, J. Econ. Theory 54 (1991) 48-69] argument for correlated equilibrium existence for this class of games, and supplies an alternative proof.     

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.     

Commitment and observability in a contracting environment

This paper examines leader-follower games where a leader must purchase an essential input from a price-setting supplier in order to take an action. We show that equilibrium outcomes when the followers perfectly observe the leaders' actions cannot be approximated by mixed equilibrium outcomes of the game where followers imperfectly observe the leaders' actions, i.e. they are not accessible. Accessibility fails since in a pure strategy equilibrium, a supplier makes positive profits; however in an equilibrium where a leader randomizes, supplier profits must be zero. Our result follows from a generalized indifference principle that mixed strategies must satisfy in economic environments. While supplier profits cannot be approximated, player action profiles are accessible. Our results also apply to games with costly observation.     

Global dynamics in repeated games with additively separable payoffs

This paper studies the global dynamics of a class of infinitely repeated two-player games in which the action space of each player is an interval, and the one-shot payoff of each player is additively separable in actions. We define an immediately reactive equilibrium (IRE) as a pure-strategy subgame perfect equilibrium such that each player's action is a stationary function of the opponent's last action. We completely characterize IREs and their dynamics in terms of certain indifference curves. Our results are used to show that in a prisoners' dilemma game with mixed strategies, gradual cooperation occurs when the players are sufficiently patient, and that in a certain duopoly game, kinked demand curves emerge naturally.     

On the impossibility of achieving no regrets in repeated games

Regret-minimizing strategies for repeated games have been receiving increasing attention in the literature. These are simple adaptive behavior rules that lead to no regrets and, if followed by all players, exhibit nice convergence properties: the average play converges to correlated equilibrium, or even to Nash equilibrium in certain classes of games. However, the no-regret property relies on a strong assumption that each player treats her opponents as unresponsive and fully ignores the opponents’ possible reactions to her actions. We show that if at least one player is slightly responsive, it is impossible to achieve no regrets, and convergence results for regret minimization with responsive opponents are unknown.     

Professionals Play Minimax

The implications of the Minimax theorem are tested using natural data. The tests use a unique data set from penalty kicks in professional soccer games. In this natural setting experts play a one-shot two-person zero-sum game. The results of the tests are remarkably consistent with equilibrium play in every respect: (i) winning probabilities are statistically identical across strategies for players; (ii) players' choices are serially independent. The tests have substantial power to distinguish equilibrium play from disequilibrium alternatives. These results represent the first time that both implications of von Neumann's Minimax theorem are supported under natural conditions.     

Decentralized learning from failure

We study decentralized learning in organizations. Decentralization is captured through Crawford and Haller's [Learning how to cooperate: optimal play in repeated coordination games, Econometrica 58 (1990) 571-595] attainability constraints on strategies. We analyze a repeated game with imperfectly observable actions. A fixed subset of action profiles are successes and all others are failures. The location of successes is unknown. The game is played until either there is a success or the time horizon is reached. We partially characterize optimal attainable strategies in the infinite horizon game by showing that after any fixed time, agents will occasionally randomize while at the same time mixing probabilities cannot be uniformly bounded away from zero.     

Trading with a common agent under complete information: A characterization of Nash equilibria

We analyze an abstract model of trading where N principals submit quantity-payment schedules that describe the contracts they offer to an agent, and the agent then chooses how much to trade with every principal. This represents a special class of common agency games with complete information. We study all the subgame perfect Nash equilibria of these games, not only truthful ones, providing a complete characterization of equilibrium payoffs. In particular, we show that the equilibrium that is Pareto-dominant for the principals is not truthful when there are more than two of them. We also provide a partial characterization of equilibrium strategies.     

Monotone equilibria in Bayesian games of strategic complementarities

For Bayesian games of strategic complementarities, we provide a constructive proof of the existence of a greatest and a least Bayesian Nash equilibrium, each one in strategies that are monotone in type. Our main assumptions, besides strategic complementarities, are that each player's payoff displays increasing differences in own action and the profile of types and that each player's interim beliefs are increasing in type with respect to first-order stochastic dominance (e.g., types are affiliated). The result holds for general action and type spaces (single-, multi-, or infinite-dimensional; continuous or discrete) and no prior is assumed. We also provide the following comparative statics result: the greatest and least equilibria are higher if there is a first-order stochastic dominant shift in the interim beliefs. We apply this result to strategic information revelation in games of voluntary disclosure.     

Rock-paper-scissors and cycle-based games

The present work characterizes the unique Nash equilibrium for games that are based on a cyclic preference relation. In the Nash equilibrium of these games, each player randomizes between three specific actions. In particular, an alternative way of deriving the unique Nash equilibrium of the rock-paper-scissors game is proposed.     

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.     

Efficiency results in N player games with imperfect private monitoring

We demonstrate that efficiency is achievable in a certain class of N player repeated games with private, almost perfect monitoring. Our equilibrium requires only one period memory and can be implemented by two state automata. Furthermore, we show that this efficiency result holds with any degree of accuracy of monitoring if private signals are hemiindependent. Whereas most existing research focuses on two player cases or only a special example of N player games, our results are applicable to a wide range of N player games of economic relevance, such as trading goods games and price-setting oligopolies.     

Fictitious play in 2×n games

It is known that every discrete-time fictitious play process approaches equilibrium in nondegenerate 2×2 games, and that every continuous-time fictitious play process approaches equilibrium in nondegenerate 2×2 and 2×3 games. It has also been conjectured that convergence to the set of equilibria holds generally for nondegenerate 2×n games. We give a simple geometric proof of this for the continuous-time process, and also extend the result to discrete-time fictitious play.     

Variational convergence: Approximation and existence of equilibria in discontinuous games

We introduce a notion of variational convergence for sequences of games and we show that the Nash equilibrium map is upper semi-continuous with respect to variationally converging sequences. We then show that for a game G with discontinuous payoff, some of the most important existence results of Dasgupta and Maskin, Simon, and Reny are based on constructing approximating sequences of games that variationally converge to G. In fact, this notion of convergence will help simplify these results and make their proofs more transparent. Finally, we use our notion of convergence to establish the existence of a Nash equilibrium for Bertrand-Edgeworth games with very general forms of tie-breaking and residual demand rules.     

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.     

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.     

Essential equilibria in normal-form games

A Nash equilibrium x of a normal-form game G is essential if any perturbation of G has an equilibrium close to x. Using payoff perturbations, we show that for games that are generic in the set of compact, quasiconcave, and generalized payoff secure games with upper semicontinuous sum of payoffs, all equilibria are essential. Some variants of this result are also established.     

Fault tolerance in large games

A Nash equilibrium is an optimal strategy for each player under the assumption that others play according to their respective Nash strategies, but it provides no guarantees in the presence of irrational players or coalitions of colluding players. In fact, no such guarantees exist in general. However, in this paper we show that large games are innately fault tolerant. We quantify the ways in which two subclasses of large games – λ-continuous games and anonymous games – are resilient against Byzantine faults (i.e. irrational behavior), coalitions, and asynchronous play. We also show that general large games have some non-trivial resilience against faults.