On the number of equilibria of the replicator-mutator dynamics for noisy social dilemmas

TL;DR

This paper investigates how randomness in payoffs affects the number of equilibria in replicator-mutator dynamics across various social dilemmas, revealing the influence of mutation rate and payoff uncertainty on system behavior.

Contribution

It provides the first analytical and numerical characterization of equilibrium probabilities under payoff uncertainty in social dilemma games.

Findings

Probability of equilibria varies with mutation rate.

Uncertainty in payoffs significantly impacts the number of equilibria.

Results enhance understanding of behavioral diversity in complex systems.

Abstract

In this paper, we consider the replicator-mutator dynamics for pairwise social dilemmas where the payoff entries are random variables. The randomness is incorporated to take into account the uncertainty, which is inevitable in practical applications and may arise from different sources such as lack of data for measuring the outcomes, noisy and rapidly changing environments, as well as unavoidable human estimate errors. We analytically and numerically compute the probability that the replicator-mutator dynamics has a given number of equilibria for four classes of pairwise social dilemmas (Prisoner's Dilemma, Snow-Drift Game, Stag-Hunt Game and Harmony Game). As a result, we characterise the qualitative behaviour of such probabilities as a function of the mutation rate. Our results clearly show the influence of the mutation rate and the uncertainty in the payoff matrix definition on the number of equilibria in these games. Overall, our analysis has provided novel theoretical contributions to the understanding of the impact of uncertainty on the behavioural diversity in a complex dynamical system.