Green, and later Sabourian studied repeated games where a player's payoff depends on his actions and an anonymous aggregate outcome, and show that long-run players behave myopically in any equilibrium of such games. In this paper we extend these results to games where the aggregate outcome is not necessarily an anonymous function of players' actions, and where players' strategies may depend nonanonymously on signals of other players' behavior. Our argument also provides a conceptually simpler proof of Green and Sabourian's results, showing how their analysis is driven by general bounds on the number of pivotal players in a game.

We examine repeated games with incomplete information where the type spaces of the players may be large. It is shown that if the belief of each player, regarding future play of the game, accommodates the true play then a Nash equilibrium of the incomplete information game will evolve, with time, into an equilibrium of the complete information game, i.e., the realized game where the types of all players are common knowledge. We introduce the notion of accommodating beliefs which involves two requirements. The first is that the belief assigns positive probability to neighborhoods of the true distribution and the second is that what lies outside of a neighborhood is separated from the true distribution by sufficient incoming observations this is the separation property defined in the paper.

Society uses the following mechanism to decide on the supply of a public good. Each agent canchoose whether or not to contribute to the good. Contributions are collected, and the good issupplied whenever total contributions exceed a threshold. We study the case where the publicgood is excludable, agents have a common value, and each agent receives a private signal aboutthe common value. We study how such collective decisions perform in terms of informationaggregation, social efficiency, and market traction.

Consider an agent who faces a sequential decision problem. At each stage the agent takes an action and observes a stochastic outcome e.g., daily prices, weather conditions, opponents' actions in a repeated game, etc. The agent's stage-utility depends on his action, the observed outcome and on previous outcomes. We assume the agent is Bayesian and is endowed with a subjective belief over the distribution of outcomes. The agent's initial belief is typically inaccurate. Therefore, his subjectively optimal strategy is initially suboptimal. As time passes information about the true dynamics is accumulated and, depending on the compatibility of the belief with respect to the truth, the agent may eventually learn to optimize. We introduce the notion of relative entropy, which is a natural adaptation of the entropy of a stochastic process to the subjective set-up. We present conditions, expressed in terms of relative entropy, that determine whether the agent will eventually learn to optimize. It is shown that low entropy yields asymptotic optimal behavior. In addition, we present a notion of point wise merging and link it with relative entropy.

This paper addresses the question of multi-party computation in a model with asymmetric information. Each agent has a private value (secret), but in contrast to standard models, the agent incurs a cost when retrieving the secret. There is a social choice function the agents would like to compute and implement. All agents would like to perform a joint computation, which input is their vector of secrets. However, agents would like to free-ride on others' contribution. A mechanism which elicits players' secrets and performs the desired computation defines a game. A mechanism is 'appropriate' if it (weakly) implements the social choice function for all secret vectors. namely, if there exists an equilibrium in which it is able to elicit (sufficiently many) agents' secrets and perform the computation, for all possible secret vectors. We show that 'appropriate' mechanisms approach agents sequentially and that they have low communication complexity.

We provide a sufficient condition for the expected aggregate contribution to a public good to be bounded, independently of the size of the population.