Limit Cycles Sparked by Mutation in the Repeated Prisoner's Dilemma

TL;DR

This paper investigates how mutation-driven dynamics in a three-strategy repeated Prisoner's Dilemma can lead to stable oscillations, revealing complex bifurcation structures and robustness of cycles across mutation scenarios.

Contribution

It introduces a replicator-mutator model analyzing limit cycles in the repeated Prisoner's Dilemma with three strategies, highlighting bifurcation mechanisms and robustness of oscillations.

Findings

Limit cycles occur for unidirectional mutations between strategies.

Oscillations are created and destroyed by Hopf and homoclinic bifurcations.

Stable oscillations are robust and independent of mutation implementation details.

Abstract

We explore a replicator-mutator model of the repeated Prisoner's Dilemma involving three strategies: always cooperate (ALLC), always defect (ALLD), and tit-for-tat (TFT). The dynamics resulting from single unidirectional mutations are considered, with detailed results presented for the mutations TFT $\rightarrow$ ALLC and ALLD $\rightarrow$ ALLC. For certain combinations of parameters, given by the mutation rate $\mu$ and the complexity cost $c$ of playing tit-for-tat, we find that the population settles into limit cycle oscillations, with the relative abundance of ALLC, ALLD, and TFT cycling periodically. Surprisingly, these oscillations can occur for unidirectional mutations between any two strategies. In each case, the limit cycles are created and destroyed by supercritical Hopf and homoclinic bifurcations, organized by a Bogdanov-Takens bifurcation. Our results suggest that stable oscillations are a robust aspect of a world of ALLC, ALLD, and costly TFT; the existence of cycles does not depend on the details of assumptions of how mutation is implemented.