A voter model of the spatial prisoner's dilemma

The Prisoner's Dilemma (PD) involves contests between two players, and may naturally be played on a spatial grid using voter model rules. In the model of spatial PD discussed here, the sites of a 2 dimensional lattice are occupied by strategies. At each time-step a site is chosen to play a PD game with one of its neighbors. The strategy of the chosen site then invades its neighbor with a probability which is proportional to the pay-off from the game. Using results from the analysis of voter models, it is shown that with simple linear strategies this scenario results in the long-term survival of only one strategy. If three non-linear strategies have a cyclic dominance relation between one-another, then it is possible for relatively cooperative strategies to persist indefinitely. With the voter model dynamics, however, the average level of cooperation decreases with time if mutation of the strategies is included. Spatial effects are not in themselves sufficient to lead to the maintenance of cooperation.