Cooperation in two-dimensional mixed-games

TL;DR

This paper explores how mixing two different games in evolutionary cooperation models affects outcomes, showing that under certain conditions, the mixed game behaves like an average game, with deviations when conditions are not met.

Contribution

It introduces a model where two different games are randomly played, analyzing when the mixed game is equivalent to the average game and exploring deviations outside those conditions.

Findings

Mixed games are equivalent to the average game under specific conditions.

Deviations occur when strategies depend on the game distribution or transition rates are nonlinear.

Several key quantities remain similar to the average game even when conditions are not fully met.

Abstract

Evolutionary game theory is a common framework to study the evolution of cooperation, where it is usually assumed that the same game is played in all interactions. Here, we investigate a model where the game that is played by two individuals is uniformly drawn from a sample of two different games. Using the master equation approach we show that the random mixture of two games is equivalent to play the average game when (i) the strategies are statistically independent of the game distribution and (ii) the transition rates are linear functions of the payoffs. We also use Monte-Carlo simulations in a two dimensional lattice and mean-field techniques to investigate the scenario when the two above conditions do not hold. We find that even outside of such conditions, several quantities characterizing the mixed-games are still the same as the ones obtained in the average game when the two games are not very different.