What restrictions do Bayesian games impose on the value of information?

In a Bayesian game players play an unknown game. Before the game starts some players may receive a signal regarding the specific game actually played. Typically, information structures that determine different signals, induce different equilibrium payoffs. In two-person zero-sum games the equilibrium payoff measures the value of the particular information structure which induces it. We pose a question as to what restrictions Bayesian games impose on the value of information. We provide answers for two kinds of information structures: symmetric, where both players are equally informed, and one-sided, where only one player is informed.     

Piecewise closed-loop equilibria in differential games with regime switching strategies

We propose a new methodology exploring piecewise closed-loop equilibrium strategies in differential games with regime switching actions. We develop a general game with two players. Players choose an action that influences the evolution of a state variable, and decide on the switching time from one regime to another. Compared to the optimal control problem with regime switching, necessary optimality conditions are modified for the first player to switch. When choosing her optimal switching strategy, this player considers the impact of her choice on the other player’s actions and consequently on her own payoffs. In order to determine the equilibrium timing of regime changes, we derive conditions that help eliminate candidate equilibrium strategies that do not survive deviations in switching strategies. We then apply this new methodology to an exhaustible resource extraction game.     

Bounds for best response functions in binary games

This paper studies the identification of best response functions in binary games without making strong parametric assumptions about the payoffs. The best response function gives the utility maximizing response to a decision of the other players. This is analogous to the response function in the treatment-response literature, taking the decision of the other players as the treatment, except that the best response function has additional structure implied by the associated utility maximization problem. Further, the relationship between the data and the best response function is not the same as the relationship between the data and the response function in the treatment-response literature. We focus especially on the case of a complete information entry game with two firms. We also discuss the case of an entry game with many firms, non-entry games, and incomplete information. Our analysis of the entry game is based on the observation of realized entry decisions, which we then link to the best response functions under various assumptions including those concerning the level of rationality of the firms, including the assumption of Nash equilibrium play, the symmetry of the payoffs between firms, and whether mixed strategies are admitted.     

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.     

Stopping with congestion and private payoffs

This paper analyses a two-player stopping game with multiarmed bandits in which each player chooses between learning about the quality of her private risky arm and competing for the use of a single shared safe arm. The qualities of the players’ risky arms are independent. A player whose risky arm produces a success no longer competes for the safe arm. We assume that a player observes her opponent’s actions but not his realised payoffs. She is therefore never certain whether her opponent is still competing for the safe arm. When the players’ prior probabilities of success are sufficiently close, there exists no pure strategy equilibrium, and we characterise the unique mixed strategy equilibrium. Otherwise, the unique equilibrium is in pure strategies. The amount of experimentation performed in equilibrium is inefficiently low but, for many priors, higher than if successes are publicly observed.     

Cores of many-player games; nonemptiness and equal treatment

This paper provides sufficient conditions to ensure nonemptiness of approximate cores of many-player games and symmetry of approximate core payoffs (the equal treatment property). The conditions are: (a) essential superadditivity—an option open to a group of players is to partition into smaller groups and realize the worths of these groups and (b) small group effectiveness (SGE)—almost all gains to collective activities can be realized by cooperation only within members of some partition of players into relatively small groups. Another condition, small group negligibility (SGN), is introduced and shown to be equivalent to SGE. SGN dictates that small groups of players cannot have significant effects on average (i.e., per capita) payoffs of large populations; thus, SGN is a analogue, for games with a finite player set, of the condition built into models with a continuum of player that sets of measure zero can be ignored. SGE implies per capita boundedness (PCB), that the supremum of average or per capita payoffs is uniformly bounded above. Further characterization of SGE in terms of its relationship to PCB is provided. It is known that if SGE does not hold, then approximate cores of many-player games may be empty. Examples are developed to show that if SGE does not hold and if there are players of “scarce types” (in other works, players with scarce attributes) then even if there is only a finite number of types of players and approximate cores are non-empty, symmetry may be lost; moreover, even players of abundant types may be treated asymmetrically by the core.     

Purification of Bayesian–Nash equilibria in large games with compact type and action spaces

We present a purification result for incomplete information games with a large but finite number of players that allows compact metric spaces for both actions and types. We then compare our framework and findings to the early purification theorems of Rashid (1983. Equilibrium points of non-atomic games: asymptotic results. Economics Letters 12, 7–10), Cartwright and Wooders (2002 On equilibrium in pure strategies in games with many players. University of Warwick Working Paper 686 (revised 2005)), Kalai (2004. Large robust games. Econometrica 72, 1631–1665) and Wooders, Cartwright and Selten (2006. Behavioral conformity in games with many players. Games and Economic Behavior 57, 347–360). Our proofs are elementary and rely on the Shapley–Folkman theorem.     

Learning games

This paper presents a model of learning about a game. Players initially have little knowledge about the game. Through playing the same game repeatedly, each player not only learns which action to choose but also constructs a personal view of the game. The model is studied using a hybrid payoff matrix of the prisoner's dilemma and coordination games. Results of computer simulations show that (1) when all the players are slow at learning the game, they have only a partial understanding of the game, but might enjoy higher payoffs than in cases with full or no understanding of the game; (2) when one player is quick in learning the game, that player obtains a higher payoff than the others. However, all can receive lower payoffs than in the case in which all players are slow learners.     

Is having a unique equilibrium robust?

We investigate whether having a unique equilibrium (or a given number of equilibria) is robust to perturbation of the payoffs, both for Nash equilibrium and correlated equilibrium. We show that the set of n  -player finite games with a unique correlated equilibrium is open, while this is not true of Nash equilibrium for n>2$n>2$. The crucial lemma is that a unique correlated equilibrium is a quasi-strict Nash equilibrium. Related results are studied. For instance, we show that generic two-person zero-sum games have a unique correlated equilibrium and that, while the set of symmetric bimatrix games with a unique symmetric Nash equilibrium is not open, the set of symmetric bimatrix games with a unique and quasi-strict symmetric Nash equilibrium is.     

Solutions for cooperative games with r alternatives

We consider games with n players and r alternatives. In these games each player must choose one and only one alternative, reaching an ordered partition of the set of players. An extension of the Shapley value to this framework is studied. Received: 1 November 1997 / Accepted: 24 January 1999     

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.     

Cost-sharing in directed networks: Experimental study of equilibrium choice and system dynamics

This study reports the results of an experiment on directed networks with positive externalities induced by cost-sharing. Subjects participated in a network game in which they had to choose between private and public transportations. If a player chose public transportation, then she shared the travel cost equally with other players making the same choice, whereas if she chose private transportation, then her travel cost was fixed. Travel costs on the private route were manipulated across the two experimental conditions. In one condition, these costs were homogeneous among players; in the other condition, they were heterogeneous among players and only privately known. We found that half (none) of the player groups in the homogeneous (heterogeneous) condition converged toward the efficient equilibrium. Examination of the system dynamics shows that convergence toward efficiency was facilitated by: (1) the existence of an intermediate equilibrium choice; and (2) strategic teaching by which a farsighted player chooses strategies with poor short-term payoff in order to shift group decisions to the efficient equilibrium and thereby increase her own long-term benefit.     

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.     

Large games with countable characteristics

We consider anonymous games with an atomless probability space of players in which players’ characteristics are countable. Our main result shows that the set of equilibrium distributions coincides with the set of distributions induced by equilibrium strategies together with the function assigning characteristics to players. This result implies the existence of Nash equilibria in continuous large games with countable characteristics.     

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.     

Network formation with sequential demands

This paper introduces a non-cooperative game-theoretic model of sequential network formation, in which players propose links and demand payoffs. Payoff division is therefore endogenous. We show that if the value of networks satisfies size monotonicity, then each and every equilibrium network is efficient. The result holds not only when players make absolute participation demands, but also when they are allowed to make link-specific demands.     

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.     

Asymptotic interpretations for equilibria of nonatomic games

We show that a mixed equilibrium of a semi-anonymous nonatomic game can be used to generate pure-strategy profiles for finite games randomly generated from the type distribution of the nonatomic game. As the numbers of players involved in the finite games increase, the generated profiles will be asymptotically equilibrium. The converse of this result is also true, i.e., a mixed-strategy profile that is not an equilibrium for the nonatomic game will not be able to achieve the above asymptotic rationality for large finite games. The combined finding can be specialized to situations where the nonatomic game is anonymous and where the given equilibrium is pure. Besides their practical values, these results offer yet one more justification for the study of nonatomic games. They also suggest that efforts may be better spent on searching for mixed rather than pure equilibria of nonatomic games.     

Revealed preferences of individual players in sequential games

This paper studies rational choice behavior of a player in sequential games of perfect and complete information without an assumption that the other players who join the same games are rational. The model of individually rational choice is defined through a decomposition of the behavioral norm assumed in the subgame perfect equilibria, and we propose a set of axioms on collective choice behavior that characterize the individual rationality obtained as such. As the choice of subgame perfect equilibrium paths is a special case where all players involved in the choice environment are each individually rational, the paper offers testable characterizations of both individual rationality and collective rationality in sequential games.