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arXiv · 1712.05906

Chaos and unpredictability in evolution of cooperation in continuous time

Taekho You·Minji Kwon·Hang-Hyun Jo·Woo-Sung Jung·Seung Ki Baek
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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 $μ\ll 1$, the position of the bifurcation sequence on the $c$ axis is numerically found to scale as $μ^{0.1}$, and such sensitivity to $μ$ 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.

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Taekho You, Minji Kwon, Hang-Hyun Jo, Woo-Sung Jung, Seung Ki Baek. 2017-12-16. Chaos and unpredictability in evolution of cooperation in continuous time. https://doi.org/10.1103/physreve.96.062310

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