Evolutionary games and spatial chaos

Abstract

MUCH attention has been given to the Prisoners' Dilemma as a metaphor for the problems surrounding the evolution of coopera-tive behaviour^1–6. This work has dealt with the relative merits of various strategies (such as tit-for-tat) when players who recognize each other meet repeatedly, and more recently with ensembles of strategies and with the effects of occasional errors. Here we neglect all strategical niceties or memories of past encounters, considering only two simple kinds of players: those who always cooperate and those who always defect. We explore the consequences of placing these players in a two-dimensional spatial array: in each round, every individual 'plays the game' with the immediate neighbours; after this, each site is occupied either by its original owner or by one of the neighbours, depending on who scores the highest total in that round; and so to the next round of the game. This simple, and purely deterministic, spatial version of the Prisoners' Dilemma, with no memories among players and no strategical elaboration, can generate chaotically changing spatial patterns, in which cooperators and defectors both persist indefinitely (in fluctuating proportions about predictable long-term averages). If the starting configurations are sufficiently symmetrical, these ever-changing sequences of spatial patterns—dynamic fractals—can be extraordinarily beautiful, and have interesting mathematical properties. There are potential implications for the dynamics of a wide variety of spatially extended systems in physics and biology.