Pattern Formation, Social Forces, and Diffusion Instability in Games with Success-Driven Motion

TL;DR

This paper explores how success-driven motion influences pattern formation and cooperation in spatial games, showing it can induce local agglomerations and instabilities that support cooperation even when traditional models predict its extinction.

Contribution

It introduces a formulation of success-driven motion based on game payoffs, revealing its role in creating attractive or repulsive forces and pattern instabilities in spatial game dynamics.

Findings

Success-driven motion can induce local cooperation clusters.

Pattern formation instability occurs under specific conditions in spatial games.

Asymmetrical diffusion can destabilize systems that are otherwise stable.

Abstract

A local agglomeration of cooperators can support the survival or spreading of cooperation, even when cooperation is predicted to die out according to the replicator equation, which is often used in evolutionary game theory to study the spreading and disappearance of strategies. In this paper, it is shown that success-driven motion can trigger such local agglomeration and may, therefore, be used to supplement other mechanisms supporting cooperation, like reputation or punishment. Success-driven motion is formulated here as a function of the game-theoretical payoffs. It can change the outcome and dynamics of spatial games dramatically, in particular as it causes attractive or repulsive interaction forces. These forces act when the spatial distributions of strategies are inhomogeneous. However, even when starting with homogeneous initial conditions, small perturbations can trigger large inhomogeneities by a pattern-formation instability, when certain conditions are fulfilled. Here, these instability conditions are studied for the prisoner's dilemma and the snowdrift game. Furthermore, it is demonstrated that asymmetrical diffusion can drive social, economic, and biological systems into the unstable regime, if these would be stable without diffusion.