Computability and Robustness of Equilibrium in Finite Games

This paper considers finite games in strategic form. It is shown that, when payoffs are computable, every game has a Nash equilibrium in which each player uses a strategy which is computable. There can, however, exist no algorithm that is guaranteed to find an equilibrium for every instance of a game with computable payoffs. In contrast, an approximate equilibrium is always computable (this is what Scarf type algorithms find). This paper develops the relationship between notions of robustness (or stability) of games and computability. The emphasis here is on notions of evolutionary stability, such as ESS. For finite symmetric games, in a model where players know only their own payoffs and form beliefs about the distribution of play of others, the two concepts are essentially identical.