Title: 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), introduced by Smith
and Price, 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 problem of deciding whether a game possesses an
ESS has been shown to be $\Sigma_{2}^{P}$-complete by Conitzer using the
preceding important work by Etessami and Lochbihler. 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 payoff 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.