Convergence of the iterated prisoner's dilemma game

We consider a stochastic process based on the iterated prisoner's dilemma game. During the game, each of n players has a state, either cooperate or defect. The players are connected by an 'interaction graph', During each step of the process, an edge of the graph is chosen uniformly at random and the states of the players connected by the edge are modified according to the Pavlov strategy. The process converges to a unique absorbing state in which all players cooperate. We prove two conjectures of Kittock: the convergence rate is exponential in n when the interaction graph is a complete graph, and it is polynomial in n when the interaction graph is a cycle. In fact, we show that the rate is O(n log n) in the latter case.