Chaos and unpredictability in evolution of cooperation in continuous time

Taekho You, Minji Kwon, Hang-Hyun Jo, Woo-Sung Jung, and Seung Ki Baek
Phys. Rev. E 96, 062310 – Published 15 December 2017

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 μ1, the position of the bifurcation sequence on the c axis is numerically found to scale as μ0.1, and such sensitivity to μ suggests that mutation may have nonperturbative 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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  • Received 10 August 2017

DOI:https://doi.org/10.1103/PhysRevE.96.062310

©2017 American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & ThermodynamicsPhysics of Living SystemsInterdisciplinary PhysicsNonlinear Dynamics

Authors & Affiliations

Taekho You1, Minji Kwon1, Hang-Hyun Jo2,3,4,*, Woo-Sung Jung1,2,3,†, and Seung Ki Baek5,‡

  • 1Department of Industrial and Management Engineering, Pohang University of Science and Technology, Pohang 37673, Korea
  • 2Asia Pacific Center for Theoretical Physics, Pohang 37673, Korea
  • 3Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea
  • 4Department of Computer Science, Aalto University, Espoo FI-00076, Finland
  • 5Department of Physics, Pukyong National University, Busan 48513, Korea

  • *hang-hyun.jo@apctp.org
  • wsjung@postech.ac.kr
  • seungki@pknu.ac.kr

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Vol. 96, Iss. 6 — December 2017

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