Computability of Nash Equilibrium

Summary

This paper considers the computability of the Nash equilibria of a game, i.e. the possibility of an algorithm to play the game with respect to Nash equilibria. We consider a two-person game, in which both players have countably many feasible actions and their particular payoff functions are computable in the sense that there are algorithms to compute the values of ones for any given action profile. It is proved that there exists no algorithm to decide whether or not a given action profile is a Nash equilibrium. Moreover, we show that the set of the Nash equilibria is not empty, but there exists no algorithm to enumerate it allowing repetitions. These results mean that no Nash equilibrium of the game cannot be computed by any human being in practice, although the existence of the Nash equilibria can be proved in theory. Indeed, it is impossible to supply the players with algorithms regarding how they should decide whether or not a given action profile is a Nash equilibrium of the game or how they should enumerate all Nash equilibria. On the other hand, I show that if players’s payoff functions are any rational number valued polynomials bounded from the above with rational number coefficients, then the Nash equilibria of the game are computable.

The author is grateful to Mamoru Kaneko, Kotaro Suzumura, and especially to Takashi Nagashima. Furthermore, he would like to acknowledge Thomas Brenner, Christian Ewerhart, Peter Hammond, Kaori Hasegawa, Midori Hirokawa, Wiebe van der Hoek, Ryo Kashima, Luchuan A. Liu, Kin Chung Lo, Akihiko Matsui, Shigeo Muto, Mikio Nakayama, John Nash, Jr., Yongsheng Xu, and Itzhak Zilcha for their advice, comments, encouragement, and discussions. Of course, any possible errors and misunderstandings are due to the author.