On Arbitrarily Long Periodic Orbits of Evolutionary Games on Graphs

TL;DR

This paper demonstrates that evolutionary games on graphs can exhibit arbitrarily long periodic behaviors, including on acyclic graphs, by constructing specific graphs and initial conditions, revealing complex dynamics in social-dilemma scenarios.

Contribution

It introduces methods to create graphs and initial conditions that produce arbitrary long periodic orbits in evolutionary games, including on acyclic structures.

Findings

Periodic orbits of arbitrary length can be constructed for evolutionary games on graphs.

Acyclic (tree) graphs can also support arbitrarily long periodic behaviors.

The results apply to social-dilemma type games with imitation dynamics.

Abstract

A periodic behavior is a well observed phenomena in biological and economical systems. We show that evolutionary games on graphs with imitation dynamics can display periodic behavior for an arbitrary choice of game theoretical parameters describing social-dilemma games. We construct graphs and corresponding initial conditions whose trajectories are periodic with an arbitrary minimal period length. We also examine a periodic behavior of evolutionary games on graphs with the underlying graph being an acyclic (tree) graph. Astonishingly, even this acyclic structure allows for arbitrary long periodic behavior.