This project continues research on methods that are extremely useful for rigorously analyzing economic dynamics, strategic behavior and social choice. The specific research topics are (1) the nature of equilibrium in repeated games where actions are not observable or where there is a mixture of long- and short-run players; (2) renegotiation- proof equilibria in repeated games; (3) Markov equilibrium; and (4) implementation theory. The research on repeated games is important because the Folk Theorem, one of the most widely used methods for finding mathematical solutions to repeated games, depends on the assumption that players observe a relationship between the actions of the other players and the outcomes of the games. The purpose of the project is to study the nature of equilibrium in repeated games when the outcomes observed by the players are imperfectly correlated. In a repeated game, cooperation derives from the potential of players to punish non-cooperative behavior. This threat of punishment breaks down, if agents can avoid the punishment by renegotiation. The project continues to study ways of finding equilibria in repeated games that are renegotiation-proof. This line of research is very important because imposing a requirement that equilibria be renegotiation-proof reduces the number of possible outcomes to repeated games in this way enabling economic theorists to make more precise predictions about the outcomes of dynamic economic processes. The project's research on Markov strategies offers another way of reducing the number of possible outcomes of dynamic economic models. The research is based on the idea that strategies depend on only a small set of relevant past variables rather than on the entire history of the game. This project formulates for the first time a general definition of these restricted or Markov strategies and studies their characteristics. Implementation theory is concerned with the question of when it is possible to design a game or "mechanism" whose equilibria have properties that are desirable with respect to a given criterion of social welfare. Traditional implementation theory assumes that the outcome can be enforced by a central authority. This project extends implementation theory to situations where some individuals' actions cannot be observed by anyone else. This project should be supported because it makes fundamental contributions to economic theory. Economic analysis has been hindered by the fact that many dynamic economic models have huge sets of equilibria. The results of this research could enable theorists to make more discriminating forecasts of the outcomes of these models.
