Static cooperator-defector patterns in models of the snowdrift game played on cycle graphs

摘要

Evolutionary graph theory is an extension of evolutionary game theory in which each individual agent,nrepresented by a node, interacts only with a subset of the entire population to which it belongs (i.e., those tonwhich it is connected by edges). In the context of the evolution of cooperation, in which individuals playingnthe cooperator strategy interact with individuals playing the defector strategy and game payoffs are equatednwith fitness, evolutionary games on graphs lead to global standoffs (i.e., static patterns) when all cooperatorsnin a population have the same payoff as any defectors with which they share an edge. I consider the simplestntype of regular-connected graph, the cycle graph, in which every node has exactly two edges (k = 2), for thenprisoner’s dilemma game and the snowdrift game, the two most important pairwise games in cooperation theory.nI show that for simplified payoff structures associated with these games, standoffs are only possible for twonvalid cost-benefit ratios in the snowdrift game. I further show that only the greater of these two cost-benefitnratios is likely to be attracting in most situations (i.e., likely to spontaneously result in a global standoff whennstarting from nonstandoff conditions). Numerical simulations confirm this prediction. This work contributes tonour understanding of the evolution of pattern formation in games played in finite, sparsely connected populations.