Matching and Playing Games

(preliminary and incomplete) Abstract: We examine a new class of games where players not only choose strategies but also choose with whom they play. A group of players who are dissatisfied with the play of their current partners can join together and play a new equilibrium. This imposes interesting new refinements of equilibrium play, where play depends on the relative populations of players in different roles among other things. We also examine finite repetitions of games where players may jointly choose to rematch in any period. Some equilibria of fixed-player repeated games cannot be sustained as equilibria in a repeated matching game. We show that all renegotiation-proof equilibria of fixed-player games have corresponding matching equilibria if all roles have the same size population of players. If roles have different size populations of players then it is possible that no renegotiation proof equilibrium has a corresponding matching equilibrium. The set of matching equilibria also includes some plays that are not part of any equilibrium of the corresponding fixed-player repeated games