Chaos and unpredictability in evolution of cooperation in continuous time

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≪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 nonperturbative effects on evolutionary paths. It demonstrates how the microscopic randomness of the mutation process can be amplified to macroscopic unpredictability by evolutionary dynamics.

Weight of fitness deviation governs strict physical chaos in replicator dynamics

“…In absence of any mutation process, as is the case with replicator equation, constant fitness functions for the types doesn't give any nontrivial dynamics. However, even with the simplest type of complication, viz., fitnesses are linear functions of the frequencies, replicator equation becomes nonlinear and consequently arXiv:1703.10767v3 [nlin.CD] 17 Feb 2018 exhibits dynamically rich behaviour including chaos [7][8][9][10][11][12] .…”

Bursts and Regularization in Financial Markets: Deterministic or Stochastic?

“…This indicates that even when the system is deterministic, its appearance may look arbitrary. A similar result stating that "the microscopic randomness of the mutation process can be amplified to macroscopic unpredictability by evolutionary dynamics" was recently obtained from an infinitely repeated prisoner's dilemma 18 , and a discrete-time replicator map (a prototype of an evolutionary selection game) was shown to yield a fixed-point solution the dynamics that could be interpreted as the Nash equilibrium of an evolutionarily stable strategy 19 . Besides this relation to game theory, financial markets have an affinity to neural networks behavior.…”

Advanced Tailored Randomness: A Novel Approach for Improving the Efficacy of Biological Systems

“…Chaos in evolution arises under frequency-dependent selection, caused by competitive interactions mediated by various traits (Rego-Costa et al, 2018). Environmental forces affect the outcome of evolution in systems prone to chaos (You et al, 2017). A changing environment produces fluctuations of a phenotype, interacting with the internal dynamics of a chaotic system.…”

“…The microscopic randomness of a mutation process can be amplified to produce macroscopic unpredictability by evolutionary dynamics (You et al, 2017). Replicator dynamics deterministically defines the evolution of an infinite-sized population in the presence of implementation errors and mutations, and exhibits chaos through a bifurcation sequence.…”

“…Replicator dynamics deterministically defines the evolution of an infinite-sized population in the presence of implementation errors and mutations, and exhibits chaos through a bifurcation sequence. This suggests that mutations have nonperturbative effects on evolutionary paths (You et al, 2017). Randomness (due to genetic drift or environmental stochasticity) and chaos (characterized by a dependence on initial conditions) are the key factors that reduce the predictability of evolution.…”