The lifeboat problem

We study an all-pay contest with multiple identical prizes (“lifeboat seats”). Prizes are partitioned into subsets of prizes (“lifeboats”). Players play a two-stage game. First, each player chooses an element of the partition (“a lifeboat”). Then each player competes for a prize in the subset chosen (“a seat”). We characterize and compare the subgame perfect equilibria in which all players employ pure strategies or all players play identical mixed strategies in the first stage. The partitioning of prizes can lead to coordination failure when players employ nondegenerate mixed strategies. In these equilibria some rents are sheltered and rent dissipation is reduced.     

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.     

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.     

Walrasian analysis via two-player games

We associate to any pure exchange economy a game with only two players, regardless of the number of consumers. In this two-player game, each player represents a different role of the society, formed by all the individuals in the economy. Player 1 selects feasible allocations trying to make Pareto improvements. Player 2 chooses an alternative from the wider range of allocations that are feasible in the sense of Aubin. The set of Nash equilibria of our game is non-empty and our main result provides a characterization of Walrasian equilibria allocations as strong Nash equilibria of the associated society game.     

Learning to Learn, Pattern Recognition, and Nash Equilibrium

The paper studies a large class of bounded-rationality, probabilistic learning models on strategic-form games. The main assumption is that players “recognize” cyclic patterns in the observed history of play. The main result is convergence with probability one to a fixed pattern of pure strategy Nash equilibria, in a large class of “simple games” in which the pure equilibria are nicely spread along the lattice of the game. We also prove that a necessary condition for convergence of behavior to a mixed strategy Nash equilibrium is that the players consider arbitrarily long histories when forming their predictions.Journal of Economic LiteratureClassification Numbers: C72, D83.     

Strategic delay and endogenous offers in bargaining games with private information

This paperendogenously determines the order of offers and the duration of delay in reaching agreement between buyers and sellers in a continuous-time bargaining game in which a seller wishes to vend an object of known cost to a buyer, to whom the value of the good is private information, and in which each player can choose to strategically delay a response to a previous offer or to interrupt the delay of his rival. Both buyers and sellers are shown to prefer to move first in a model of bargaining in which: (1) either player can make the first offer; (2) after the minimum time has elapsed from the previous offer, either player can make an offer; and (3) players can choose to strategically delay and refrain from making an offer after the previous offer. When the buyer moves first, the equilibrium response for the seller is to accept the offer immediately. When the seller moves first the equilibrium is characterized by the seller making all but the last offer, with minimal feasible delay between successive offers. Observable endogenous delay in reaching an agreement in such equilibria approaches zero as the minimal feasible delay between offers approaches zero. This indicates that in noncooperative bargaining models with private information, where players can strategically delay their offers, endogenizing the order in which players make offers removes the ability of informational asymmetries to generate equilibria exhibiting endogenous delay in reaching an agreement.     

Coordination and the Provision of Discrete Public Goods by Correlated Equilibria

The strategic analysis of the private provision of a discrete public good has shown the existence of multiple Nash equilibria with the efficient number of players voluntarily contributing. However the coordination issue is left unexplained by this literature. The experimental evidence shows that communication among players is helpful in achieving cooperation. We claim that, from the theoretical point of view, this is equivalent to playing correlated equilibria in an extended public good game with communication, modeled as Chicken. We characterize such equilibria as feasible coordination mechanisms to achieve public goods provision in the general contribution game. We further introduce a second kind of game characterized by payoff externalities that may persist after the minimal threshold of contributors is achieved. While it is easy to show the existence of Pareto efficient correlated equilibria in the first game, in the second one players face incentive problems such that a first best cannot always be an equilibrium. Nevertheless there exist correlated equilibria that can qualify as incentive efficient mechanisms, once free riding is seen as a moral hazard issue. Finally, with an example, we discuss the impact of coalition formation in our framework.     

Congestion games with malicious players

We study the equilibria of non-atomic congestion games in which there are two types of players: rational players, who seek to minimize their own delay, and malicious players, who seek to maximize the average delay experienced by the rational players. We study the existence of pure and mixed Nash equilibria for these games, and we seek to quantify the impact of the malicious players on the equilibrium. One counterintuitive phenomenon which we demonstrate is the “windfall of malice”: paradoxically, when a myopically malicious player gains control of a fraction of the flow, the new equilibrium may be more favorable for the remaining rational players than the previous equilibrium.     

Bayesian learning and convergence to Nash equilibria without common priors

Summary. Consider an infinitely repeated game where each player is characterized by a “type” which may be unknown to the other players in the game. Suppose further that each player's belief about others is independent of that player's type. Impose an absolute continuity condition on the ex ante beliefs of players (weaker than mutual absolute continuity). Then any limit point of beliefs of players about the future of the game conditional on the past lies in the set of Nash or Subjective equilibria. Our assumption does not require common priors so is weaker than Jordan (1991); however our conclusion is weaker, we obtain convergence to subjective and not necessarily Nash equilibria. Our model is a generalization of the Kalai and Lehrer (1993) model. Our assumption is weaker than theirs. However, our conclusion is also weaker, and shows that limit points of beliefs, and not actual play, are subjective equilibria. Received: March 3, 1995; revised version: February 17, 1997     

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.     

Strategic approximations of discontinuous games

An infinite game is approximated by restricting the players to finite subsets of their pure strategy spaces. A strategic approximationof an infinite game is a countable subset of pure strategies with the property that limits of all equilibria of all sequences of approximating games whose finite strategy sets eventually include each member of the countable set must be equilibria of the infinite game. We provide conditions under which infinite games admit strategic approximations.     

Sequential communication with ex post participation constraints

This paper examines an informed principal-agent game with ex post participation constraints for the agent. It shows that the players do not lose by communicating in turn among themselves rather than simultaneously if and only if the principal communicates first. It then considers every Bayesian incentive compatible allocation rules that assign non-negative payoffs for one player in a bilateral asymmetric information framework. It provides necessary and sufficient conditions for sequential communication to be as efficient as simultaneous communication in implementing these allocation rules when the player with unbounded payoffs moves first.     

The Envelope Theorem and Comparative Statics of Nash Equilibria

The Envelope Theorem for Nash equilibria shows that the strategic reaction of the other players in the game is important for determining how parameter perturbations affect a given player's indirect objective function. The fundamental comparative statics matrix of Nash equilibria for theithplayer in anN-player static game includes the equilibrium response of the otherN−1players in the game to the parameter perturbation and is symmetric positive semidefinite subject to constraint. This result is fundamental in that it holds for all sufficiently smooth Nash equilibria and is independent of any curvature or stability assumptions imposed on the game.Journal of Economic LiteratureClassification Numbers: C72, C61.     

Partial exposure in large games

In this work we introduce the notion of partial exposure, in which the players of a simultaneous-move Bayesian game are exposed to the realized types and chosen actions of a subset of the other players. We show that in any large simultaneous-move game, each player has very little regret even after being partially exposed to other players. If players are given the opportunity to be exposed to others at the expense of a small decrease in utility, players will decline this opportunity, and the original Nash equilibria of the game will survive.     

ON THE NEUTRALITY OF COALITION FORMATION IN A PURE BARGAINING PROBLEM*

We consider the sequential bargaining game à la Stahl–Binmore–Rubinstein with random proposers, juxtaposing an ex ante coalition formation stage to their bargaining game. On the basis of the expected outcomes in the negotiation over how to split a dollar, players can form coalitions in a sequential manner, within each of which they can redistribute their payoffs. It turns out that the grand coalition does form, and that each player receives his discounted expected payoff, which is obtained by playing as a single player in the negotiation, although there could be many equilibria in the bargaining stage.     

Multiperson Bargaining over Two Alternatives

We present a continuous-time model of multiperson bargaining with incomplete information, where only two agreements are possible. We show that under mild informational assumptions, players behave as if they were representing the other players who prefer the same alternative, individually playing a war-of-attrition against a conglomerate player of the opposite side. We also provide a characterization of the perfect Bayesian equilibria in undominated strategies of the game, which yields a unique outcome in most cases.Journal of Economic LiteratureClassification Numbers: C72, C78, D72, D74.     

Cooperative interim contract and re-contract: Chandler's M-form firm

Summary. At an interim stage players possessing only their private information freely communicate with each other to coordinate their strategies. This results in a core strategy, which is interpreted as an equilibrium set of players' alternative type-contingent contract offers to their fellows. From this set of offers each player then chooses an optimal one and engages in some subsequent action, thus possibly revealing some private information to the others. Now with new information thus obtained from each other, the players play a new game to re-write their contract. In all of the optimization and gaming just described, Bayesian incentive compatibility plays a central role. These ideas are formulated within a model of a profit-center game with incomplete information which formally describes interaction of the asymmetrically informed profit-centers in Chandler's multidivisional firm. Received: May 17, 1996; revised version: January 14, 1997     

Bargaining and efficiency in networks

We study an infinite horizon game in which pairs of players connected in a network are randomly matched to bargain over a unit surplus. Players who reach agreement are removed from the network without replacement. The global logic of efficient matchings and the local nature of bargaining, in combination with the irreversible exit of player pairs following agreements, create severe hurdles to the attainment of efficiency in equilibrium. For many networks all Markov perfect equilibria of the bargaining game are inefficient, even as players become patient. We investigate how incentives need to be structured in order to achieve efficiency via subgame perfect, but non-Markovian, equilibria. The analysis extends to an alternative model in which individual players are selected according to some probability distribution, and a chosen player can select a neighbor with whom to bargain.     

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     

Nash equilibria in ∞-dimensional spaces: an approximation theorem

Summary. We show, by employing a density result for probability measures, that in games with a finite number of players and ∞-dimensional pure strategy spaces Nash equilibria can be approximated by finite mixed strategies. Given ε>0, each player receives an expected utility payoff ε/2 close to his Nash payoff and no player could change his strategy unilaterally and do better than ε. Received: July 15, 1997; revised version: February 6, 1998