Chaos and unpredictability in evolution of cooperation in continuous time

TL;DR

This paper explores how continuous-time replicator dynamics in the evolution of cooperation can exhibit chaos and unpredictability, influenced by mutation rates and the cost-benefit ratio, revealing complex bifurcation behaviors.

Contribution

It demonstrates that the evolution of cooperation under replicator dynamics can become chaotic through bifurcations, with mutation rates significantly affecting the dynamics.

Findings

Chaos emerges in the replicator dynamics as the cost-benefit ratio varies.

Bifurcation sequences scale with mutation rate as μ^{0.1}.

Mutation can amplify microscopic randomness to macroscopic unpredictability.

Abstract

Cooperators benefit others with paying costs. Evolution of cooperation crucially depends on the cost-benefit ratio of cooperation, denoted as $c$. In this work, we investigate the infinitely repeated prisoner's dilemma for various values of $c$ with four of the representative memory-one strategies, i.e., unconditional cooperation, unconditional defection, tit-for-tat, and win-stay-lose-shift. We consider replicator dynamics which deterministically describes how the fraction of each strategy evolves over time in an infinite-sized well-mixed population in the presence of implementation error and mutation among the four strategies. Our finding is that this three-dimensional continuous-time dynamics exhibits chaos through a bifurcation sequence similar to that of a logistic map as $c$ varies. If mutation occurs with rate $\mu \ll 1$, the position of the bifurcation sequence on the $c$ axis is numerically found to scale as $\mu^{0.1}$, and such sensitivity to $\mu$ suggests that mutation may have non-perturbative effects on evolutionary paths. It demonstrates how the microscopic randomness of the mutation process can be amplified to macroscopic unpredictability by evolutionary dynamics.