The evolutionary spatial snowdrift game on a cycle: An asymptotic analysis

The temporal dynamics of games have been studied widely in evolutionary spatial game theory using simulation. Each player is usually represented by a vertex of a graph and plays a particular game against every adjacent player independently. These games result in payoffs to the players which affect their relative fitness. The fitness of a player, in turn, affects its ability to reproduce. In this paper, we analyse the temporal dynamics of the evolutionary 2-person, 2-strategy snowdrift game in which players are arranged along a cycle of arbitrary length. In this game, each player has the option of adopting one of two strategies, namely cooperation or defection, during each game round. We compute the probability of retaining persistent cooperation over time from a random initial assignment of strategies to players. We also establish bounds on the probability that a small number of players of a particular mutant strategy introduced randomly into a cycle of players which have established the opposite strategy leads to the situation where all players eventually adopt the mutant strategy. We adopt an analytic approach throughout as opposed to a simulation approach clarifying the underlying dynamics intrinsic to the entire class of evolutionary spatial snowdrift games.