Existence of evolutionarily stable strategies remains hard to decide for a wide range of payoff values

Abstract

The concept of an evolutionarily stable strategy (ESS), intro-duced by Smith and Price [4], is a refinement of Nash equilibrium in 2-player symmetric games in order to explain counter-intuitive natural phenomena, whose existence is not guaranteed in every game. The prob-lem of deciding whether a game possesses an ESS has been shown to be Σ2P-complete by Conitzer [1] using the preceding important work by Etessami and Lochbihler [2]. The latter, among other results, proved that deciding the existence of ESS is both NP-hard and coNP-hard. In this paper we introduce a reduction robustness notion and we show that deciding the existence of an ESS remains coNP-hard for a wide range of games even if we arbitrarily perturb within some intervals the pay-off values of the game under consideration. In contrast, ESS exist almost surely for large games with random and independent payoffs chosen from the same distribution [11].