Statistics of the number of equilibria in random social dilemma evolutionary games with mutation

TL;DR

This paper analytically derives the probability distributions of the number of equilibria in random social dilemma evolutionary games with mutation, highlighting how mutation probability influences system diversity.

Contribution

It provides explicit formulas for equilibrium distributions in social dilemma games with random payoffs, incorporating mutation effects, which was previously unexplored.

Findings

Mutation probability significantly affects the distribution of equilibria.

Explicit formulas enable better understanding of diversity in evolutionary systems.

Analysis applies to various social dilemma types like cooperation and coordination.

Abstract

In this paper, we study analytically the statistics of the number of equilibria in pairwise social dilemma evolutionary games with mutation where a game's payoff entries are random variables. Using the replicator-mutator equations, we provide explicit formulas for the probability distributions of the number of equilibria as well as other statistical quantities. This analysis is highly relevant assuming that one might know the nature of a social dilemma game at hand (e.g., cooperation vs coordination vs anti-coordination), but measuring the exact values of its payoff entries is difficult. Our delicate analysis shows clearly the influence of the mutation probability on these probability distributions, providing insights into how varying this important factor impacts the overall behavioural or biological diversity of the underlying evolutionary systems.