Person:

Fudenberg, Drew

This paper studies the set of equilibrium payoffs in repeated games with long- and short-run players and little discounting. Because the short-run players are unconcerned about the future, each equilibrium outcome is constrained to lie on their static reaction (best-response) curves. The natural extension of the folk theorem to games of this sort would simply include this constraint in the definitions of the feasible payoffs and minmax values. In fact, this extension does obtain under the assumption that each player's choice of a mixed strategy for the stage game is publicly observable but, in contrast to standard repeated games, the set of equilibrium payoffs is different if players can observe only their opponents' realized actions.

We examine games played by a single large player and a large number of opponents who are small, but not anonymous. If the play of the small players is observed with noise, and if the number of actions the large player controls is bounded as the number of small players grows, the equilibrium set converges to that of the game where there is a continuum of small players. This paper extends previous work on the negligibility of small players by dropping the assumption that small players' actions are “anonymous.” That is, we allow each small player's actions to be observed separately, instead of supposing that the small players' actions are only observed through their effect on an aggregate statistic.

We study a variation of fictitious play, in which the probability of each action is an exponential function of that action's utility against the historical frequency of opponents' play. Regardless of the opponents' strategies, the utility received by an agent using this rule is nearly the best that could be achieved against the historical frequency. Such rules are approximately optimal in i.i.d. environments, and guarantee nearly the minmax regardless of opponents' behavior. Fictitious play shares these properties provided it switches 'infrequently' between actions. We also study the long-run outcomes when all players use consistent and cautious rules.

In some experiments rational players who understand the structure of the game could improve their payoff. We bound the size of the observed losses in several such experiments. To do this, we suppose that observed play resembles an equilibrium because players learn about their opponents' play. Consequently, in an extensive-form game, some actions that are not optimal given the true distribution of opponents' play could be optimal given available information. We find that average losses are small: $0.03 to $0.64 per player with stakes between $2 and $30. In one of the three experiments we examine, this also implies a narrow range of outcome.

This paper studies the way that word-of-mouth communication aggregates the information of individual agents. We find that the structure of the communication process determines whether all agents end up making identical choices, with less communication making this conformity more likely. Despite the players' naive decision rules and the stochastic decision environment, word-of-mouth communication may lead all players to adopt the action that is on average superior. These socially efficient outcomes tend to occur when each agent samples only a few others.

We examine the rational expectations equilibrium paths of the model of search and barter in Diamond's "Aggregate Demand Management in Search Equilibrium." For some initial positions, there are two equilibrium paths converging to different steady states, with the high-activity path Pareto-dominating the low-activity path, With some parameters there is also a continuum of equilibrium paths converging to another steady state. Moreover, there can be cycles that are equilibrium' paths, even though the environment is stationary.

In a self-confirming equilibrium, each player correctly forecasts the actions that opponents will take along the equilibrium path, but may be mistaken about the way that opponents would respond to deviations. This paper develops a refinement of self-confirming equilibrium in which players use information about opponents' payoffs in forming beliefs about the way that opponents play off of the equilibrium path. We show that this concept is robust to payoff uncertainty. We also discuss its relationship to other concepts and show that it is closely related to assuming almost common certainty of payoffs in an epistemic model with independent beliefs.

Forecasts are said to be calibrated if the frequency predictions are approximately correct. This is a refinement of an idea first introduced by David Blackwell in 1955. We show that “K-initialized myopic strategies” are approximately calibrated when K is large. These strategies first “initialize” by making each forecast exactly K times, and thereafter play, in each period t, the minmax strategy in a static game.

The philosophy of equilibrium refinements is that the analyst, if he knows things about the structure of the game, can reject some Nash equilibria as unreasonable. The word “know” in the preceding sentence deserves special emphasis. If in a fixed game the analyst can reject a particular equilibrium outcome, but he cannot do so for games arbitrarily “close by,” then he may have second thoughts about rejecting the outcome. We consider several notions of distance between games, and we characterize their implications for the robustness of equilibrium refinements.

We consider two kinds of ‘outside opportunity’ that a seller of an indivisible good might have: selling to a different buyer and consuming the good herself. In both models the seller is uncertain about the buyer's valuation, and becomes more pessimistic over time. When the seller becomes sufficiently pessimistic, she prefers the outside opportunity, so she will not bargain indefinitely with the current buyer. Despite the resulting finite-horizon nature of negotiations, the link between the buyer's willingness to accept an offer and the seller's eagerness to go "outside" generates multiple equilibria.