The Spread of Cooperative Strategies on Grids with Random Asynchronous Updating

The Prisoner's Dilemma Process on a graph $G$ is an iterative process where each vertex, with a fixed strategy (cooperate or defect), plays the game with each of its neighbours. At the end of a round each vertex may change its strategy to that of its neighbour with the highest pay-off. Here we study the spread of cooperative and selfish behaviours on a toroidal grid, where each vertex is initially a cooperator with probability $p$. When vertices are permitted to change their strategies via a randomized asynchronous update scheme, we find that for some values of $p$ the limiting density of cooperators may be modelled as a polynomial in $p$. Theoretical bounds for this density are confirmed via simulation.