The theory of repeated games explores how mutual help and cooperation are sustained through repeated interaction, even when economic agents are completely self-interested beings. This thesis analyzes two models that involve repeated interaction in an environment where some information is private.
In the first chapter, we characterize the equilibrium set of the following game. Two players interact repeatedly over an infinite horizon and occasionally, one of the players has an opportunity to do a favor to the other player. The ability to do a favor is private information and only one of the players is in a position to do a favor at a time. The cost of doing a favor is less than the benefit to the receiver so that, always doing a favor is the socially optimal outcome. Intuitively, a player who develops the ability to do a favor in some period might have an incentive to reveal this information and do a favor if she has reason to expect future favors in return.
We show that the equilibrium set expands monotonically in the likelihood that someone is in a position to do a favor. It also expands with the discount factor. However, there are no fully efficient equilibria for any discount factor less than unity. We find sufficient conditions under which equilibria on the Pareto frontier of the equilibrium set are supported by efficient payoffs. We also provide a partial characterization of payoffs on the frontier in terms of the action profiles that support them.
In the second chapter, we use numerical methods to compute the equilibrium value set of the exchanging favors game. We use techniques from Judd, Yeltekin and Conklin (2003) to find inner and outer approximations of the equilibrium value set which, together, provide bounds on it. Any point contained in the inner approximation is certainly an equilibrium payoff. Any point not in the outer approximation is certainly not in the value set.
These inner and outer monotone approximations are found by looking for boundary points of the relevant sets and then connecting these to form convex sets. Working with eight boundary points gives us estimates that are coarse but still capture the comparative statics of the equilibrium set with respect to the discount factor and the other parameters. By increasing the number of boundary points from eight to twelve, we obtain very precise estimates of the equilibrium set. With this tightly approximated equilibrium set, the properties of its inner approximation provide good indications of the properties of the equilibrium set itself. We find a very specific shape of the equilibrium set and see that payoffs on the Pareto frontier of the equilibrium set are supported by current actions of full favors. This is true so long as there is room for full favors, that is, away from the two ends of the frontier.
The third chapter extends the concept of Quantal Response equilibrium, a statistical version of Nash equilibrium, to repeated games. We prove a limit Folk Theorem for a two person finite action repeated game with private information, the very specific additive kind introduced by the Quantal Response model. If the information is almost complete and the discount factor is high enough, we can construct Quantal Response Equilibria very close to any feasible individually rational payoffs. This is illustrated numerically for the repeated Prisoners' Dilemma game.