The Rationality / Computability Trade-off in Finite Games

The computability of Nash equilibrium points of finite games is examined. When payoffs are computable there always exists an equilibrium in which all players use computable strategies. However, there is a computable sequence of games for which the equilibrium points do not constitute a computable sequence. For this reason, there can be no algorithm that, given arbitrary payoffs, computes a Nash equilibrium point for the game. Even for games with computable equilibrium points, best responses of the players may not be computable. In contrast, approximate equilibria, and error-prone responses are computable.