On the computational complexity of evolution

It is well-known that the problem of recognizing an ESS in a symmetric bimatrix game is coNPcomplete. In this paper, we show that recognizing an ESS even in doubly symmetric bimatrix games is also coNP-complete. Our result further implies that recognizing asymptotically stable equilibria of the replicator dynamic in this class of games is also a coNP-complete problem. It is also known that a strategy in a doubly symmetric bimatrix game is an ESS if and only if it is locally strictly efficient, that is, if it is a strict local maximum of a corresponding standard quadratic program. Since the latter problem is NP-hard it follows that NP = coNP.