Nonlinear Dynamics of the Rock-Paper-Scissors Game with Mutations

TL;DR

This paper investigates how small mutations affect the stability of coexistence in the Rock-Paper-Scissors game, revealing that even minimal mutations can destabilize equilibrium and lead to stable oscillations.

Contribution

It provides a comprehensive analysis of mutation effects on the dynamics of Rock-Paper-Scissors, including various mutation patterns and bifurcation phenomena.

Findings

Small mutations destabilize coexistence equilibrium.

A supercritical Hopf bifurcation creates stable limit cycles.

Periodic solutions persist near zero mutation rates.

Abstract

We analyze the replicator-mutator equations for the Rock-Paper-Scissors game. Various graph-theoretic patterns of mutation are considered, ranging from a single unidirectional mutation pathway between two of the species, to global bidirectional mutation among all the species. Our main result is that the coexistence state, in which all three species exist in equilibrium, can be destabilized by arbitrarily small mutation rates. After it loses stability, the coexistence state gives birth to a stable limit cycle solution created in a supercritical Hopf bifurcation. This attracting periodic solution exists for all the mutation patterns considered, and persists arbitrarily close to the limit of zero mutation rate and a zero-sum game.