# Existence of Evolutionarily Stable Strategies Remains Hard to Decide for a Wide Range of Payoff Values

## TL;DR

This paper investigates the computational complexity of determining the existence of evolutionarily stable strategies (ESS) in symmetric games, showing it remains hard under payoff perturbations but almost surely exists in large random games.

## Contribution

It introduces a 'reduction robustness' concept demonstrating ESS decision remains coNP-hard despite payoff perturbations, contrasting with the high probability of ESS existence in large random games.

## Key findings

- Deciding ESS existence is coNP-hard even with payoff perturbations.
- ESS almost surely exist in large games with random payoffs.
- The paper extends complexity results to perturbed payoff scenarios.

## 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.