On the Existence of Nash Equilibrium in Games with Resource-Bounded Players

We consider computational games, sequences of games (Formula Presented) where, for all n, (Formula Presented) has the same set of players. Computational games arise in electronic money systems such as Bitcoin, in cryptographic protocols, and in the study of generative adversarial networks in machine learning. Assuming that one-way functions exist, we prove that there is 2-player zero-sum computational game (Formula Presented) such that, for all n, the size of the action space in (Formula Presented) is polynomial in n and the utility function in (Formula Presented) is computable in time polynomial in n, and yet there is no (Formula Presented) -Nash equilibrium if players are restricted to using strategies computable by polynomial-time Turing machines, where we use a notion of Nash equilibrium that is tailored to computational games. We also show that an (Formula Presented) -Nash equilibrium may not exist if players are constrained to perform at most T computational steps in each of the games in the sequence. On the other hand, we show that if players can use arbitrary Turing machines to compute their strategies, then every computational game has an (Formula Presented) -Nash equilibrium. These results may shed light on competitive settings where the availability of more running time or faster algorithms can lead to a “computational arms race”, precluding the existence of equilibrium. They also point to inherent limitations of concepts such as “best response” and Nash equilibrium in games with resource-bounded players.