Best response dynamics on random graphs

We consider evolutionary games on a population whose underlying topology of interactions is determined by a binomial random graph G(n, p). Our focus is on 2-player symmetric games with 2 strategies played between the incident members of such a population. Players update their strategies synchronously: each player selects the strategy that is the best response to the current set of strategies its neighbours play. We show that such a system reduces to generalised majority and minority dynamics. We show rapid convergence to unanimity for p in a range that depends on a certain characteristic of the payoff matrix. In the presence of a bias among the pure Nash equilibria, we determine a sharp threshold on p above which the largest connected component reaches unanimity with high probability. For p below this critical value, we identify those substructures inside the largest component that block unanimity.