On the computational complexity of evolution

TL;DR

This paper proves that recognizing evolutionarily stable strategies (ESS) and asymptotically stable equilibria in doubly symmetric bimatrix games is coNP-complete, highlighting the computational difficulty of these problems.

Contribution

It extends the coNP-completeness result for recognizing ESS from symmetric to doubly symmetric bimatrix games, and shows the same complexity for stable equilibria of the replicator dynamic.

Findings

Recognizing ESS in doubly symmetric games is coNP-complete.

Recognizing asymptotically stable equilibria of the replicator dynamic is coNP-complete.

Complexity results apply to a broader class of symmetric games.

Abstract

It is well-known that the problem of recognizing an ESS in a symmetric bimatrix game is coNP-complete. 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.