Decompositions and potentials for normal form games

A basic model of commitment is to convert a two-player game in strategic form to a “leadership game” with the same payoffs, where one player, the leader, commits to a strategy, to which the second player always chooses a best reply. This paper studies such leadership games for games with convex strategy sets. We apply them to mixed extensions of finite games, which we analyze completely, including nongeneric games. The main result is that leadership is advantageous in the sense that, as a set, the leader's payoffs in equilibrium are at least as high as his Nash and correlated equilibrium payoffs in the simultaneous game. We also consider leadership games with three or more players, where most conclusions no longer hold.     

On the folk theorem with one-dimensional payoffs and different discount factors

Proving the folk theorem in a game with three or more players usually requires imposing restrictions on the dimensionality of the stage-game payoffs. Fudenberg and Maskin (1986) assume full dimensionality of payoffs, while Abreu et al. (1994) assume the weaker NEU condition (“nonequivalent utilities”). In this note, we consider a class of n-player games where each player receives the same stage-game payoff, either zero or one. The stage-game payoffs therefore constitute a one-dimensional set, violating NEU. We show that if all players have different discount factors, then for discount factors sufficiently close to one, any strictly individually rational payoff profile can be obtained as the outcome of a subgame-perfect equilibrium with public correlation.     

Communication, correlation and cheap-talk in games with public information

This paper studies extensive form games with public information where all players have the same information at each point in time. We prove that when there are at least three players, all communication equilibrium payoffs can be obtained by unmediated cheap-talk procedures. The result encompasses repeated games and stochastic games.     

Balance and discontinuities in 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. Stinchcombe (2011) [19] proves the existence of correlated equilibria for this class of games. This paper is a study of the information structures for these games, the discontinuous expected utility functions they give rise to, and the notion of a balanced approximation to an infinite game with discontinuous payoffs.     

The folk theorem for irreducible stochastic games with imperfect public monitoring

This paper introduces stochastic games with imperfect public signals. It provides a sufficient condition for the folk theorem when the game is irreducible, thus generalizing the results of Dutta (1995) [5] and Fudenberg, Levine, and Maskin (1994) [9]. To do this, the paper extends the concept of self-generation (Abreu, Pearce, and Stacchetti, 1990 [1]) to “return generation,” which explicitly tracks actions and incentives until the next time the state returns to its current value, and asks that players not wish to deviate given the way their continuation payoffs from the time of this return depend on the public signals that have been observed.     

Transferable utility games with uncertainty

We introduce the concept of a TUU-game, a transferable utility game with uncertainty. In a TUU-game there is uncertainty regarding the payoffs of coalitions. One out of a finite number of states of nature materializes and conditional on the state, the players are involved in a particular transferable utility game. We consider the case without ex ante commitment possibilities and propose the Weak Sequential Core as a solution concept. We characterize the Weak Sequential Core and show that it is non-empty if all ex post TU-games are convex.     

Between-game rule learning in dissimilar symmetric normal-form games

Rule learning posits that decision makers, rather than choosing over actions, choose over behavioral rules with different levels of sophistication. Rules are reinforced over time based on their historically observed payoffs in a given game. Past works on rule learning have shown that when playing a single game over a number of rounds, players can learn to form sophisticated beliefs about others. Here we are interested in learning that occurs between games where the set of actions is not directly comparable from one game to the next. We study a sequence of ten thrice-played dissimilar games. Using experimental data, we find that our rule learning model captures the ability of players to learn to reason across games. However, this learning appears different from within-game rule learning as previously documented. The main adjustment in sophistication occurs by switching from non-belief-based strategies to belief-based strategies. The sophistication of the beliefs themselves increases only slightly over time.     

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.     

The target projection dynamic

We study the target projection dynamic, a model of learning in normal form games. The dynamic is given a microeconomic foundation in terms of myopic optimization under control costs due to a certain status-quo bias. We establish a number of desirable properties of the dynamic: existence, uniqueness and continuity of solution trajectories, Nash stationarity, positive correlation with payoffs, and innovation. Sufficient conditions are provided under which strictly dominated strategies are wiped out. Finally, some stability results are provided for special classes of games.     

Belief-free equilibria in games with incomplete information: Characterization and existence

We generalize the results of Hörner and Lovo (2009) [15] to N-player games with arbitrary information structure. First, we characterize the set of belief-free equilibrium payoffs under low discounting as the set of feasible payoffs that are individually rational, jointly rational, and incentive compatible. Second, we provide necessary and sufficient conditions on the information structure for this set to be non-empty.     

Equilibrium selection in global games with strategic complementarities

We study games with strategic complementarities, arbitrary numbers of players and actions, and slightly noisy payoff signals. We prove limit uniqueness: as the signal noise vanishes, the game has a unique strategy profile that survives iterative dominance. This generalizes a result of Carlsson and van Damme (Econometrica 61 (1993) 989-1018) for two-player, two-action games. The surviving profile, however, may depend on fine details of the structure of the noise. We provide sufficient conditions on payoffs for there to be noise-independent selection.     

Evolutionary selection of ‘chivalrous’ conventions in coordination games without common expectations

We study an evolutionary game-theoretic model where players have to choose within a predetermined set of mixed strategies in a coordination game. Players are of two different kinds, male and female. No common expectations assumption is made; players tend therefore to adopt the strategy that yields larger than average expected payoffs for their kind. In this framework, every stable stationary point of the population dynamics can be interpreted as the emergence of a particular convention. A classification of the possible conventions is provided; conditions for their emergence are determined.     

Bounding the inefficiency of equilibria in nonatomic congestion games

Equilibria in noncooperative games are typically inefficient, as illustrated by the Prisoner's Dilemma. In this paper, we quantify this inefficiency by comparing the payoffs of equilibria to the payoffs of a “best possible” outcome. We study a nonatomic version of the congestion games defined by Rosenthal [Int. J. Game Theory 2 (1973) 65], and identify games in which equilibria are approximately optimal in the sense that no other outcome achieves a significantly larger total payoff to the players—games in which optimization by individuals approximately optimizes the social good, in spite of the lack of coordination between players. Our results extend previous work on traffic routing games.     

How noise matters

Recent advances in evolutionary game theory have introduced noise into decisionmaking to select in favor of certain equilibria in coordination games. Noisy decisionmaking is justified on bounded rationality grounds, and consequently the sources of noise are left unmodelled. This methodological approach can only be successful if the results do not depend too much on the nature of the noise process. This paper investigates invariance to noise of these results, both for the random matching paradigm that has characterized much of the recent literature and for a larger class of two-strategy population games where payoffs may vary non-linearly with the distribution of strategies among the population. Several parametrizations of noise reduction are investigated. The results show that a symmetry property of the noise process and, in the case of non-linear payoffs, bounds on the asymmetry of the payoff functions suffice to preserve the known stochastic stability results.     

Dynamic coordination with individual learning

We study coordination in dynamic global games with private learning. Players choose whether and when to invest irreversibly in a project whose success depends on its quality and the timing of investment. Players gradually learn about project quality. We identify conditions on temporal incentives under which, in sufficiently long games, players coordinate on investing whenever doing so is not dominated. Roughly speaking, this outcome occurs whenever players? payoffs are sufficiently tolerant of non-simultaneous coordination. We also identify conditions under which players coordinate on the risk-dominant action. We provide foundations for these results in terms of higher order beliefs.     

Measuring agents’ reaction to private and public information in games with strategic complementarities

In games with strategic complementarities, public information about the state of the world has a larger impact on equilibrium actions than private information of the same precision, because public signals are more informative about the likely behavior of others. We present an experiment in which agents’ optimal actions are a weighted average of the fundamental state and their expectations of other agents’ actions. We measure the responses to public and private signals. We find that, on average, subjects put a larger weight on the public signal. In line with theoretical predictions, as the relative weight of the coordination component in a player’s utility increases, players put more weight on the public signal when making their choices. However, the weight is smaller than in equilibrium, which indicates that subjects underestimate the information contained in public signals about other players’ beliefs.     

Group play in games and the role of consent in network formation

We study games played between groups of players, where a given group decides which strategy it will play through a vote by its members. When groups consist of two voting players, our games can also be interpreted as network-formation games. In experiments on Stag Hunt games, we find a stark contrast between how groups and individuals play, with payoffs playing a primary role in equilibrium selection when individuals play, but the structure of the voting rule playing the primary role when groups play. We develop a new solution concept, robust-belief equilibrium, which explains the data that we observe. We provide results showing that this solution concept has application beyond the particular games in our experiments.     

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.     

A Property of Nash Equilibria in Repeated Games with Incomplete Information

We consider two-person non-zero-sum infinitely repeated games with lack of information on one side. The characterization of Nash equilibrium payoffs obtained by Hart allows for complex strategies, which are actually required by some equilibrium payoffs in some games. We show that appropriate one-shot public communication mechanisms make Nash equilibrium payoffs achievable by means of simple strategies. Furthermore, these mechanisms satisfy a notion of self-fulfillment.Journal of Economic LiteratureClassification Numbers: D82, C72.