Cooperation in a Multi-Dimensional Local Interaction Model

We consider a local interaction model with a population on an h dimensional torus, in which in each round of play a random player gets a learning draw. This player plays a k+1 action stage game with players in his neighborhood, compares his own average payoff with the average payoff of the neighbors he played against and updates his action based on this comparison. Individuals use the update rule `Win Cooperate, Lose Defect', a multi-player variant of Tit-for-Tat. We prove that there are exactly k+1 stable states and that all of these can be reached with positive probability, for any dimension h of the torus. Furthermore, we prove that when k+1=2, both stable states will be reached with probability 1/2. For k+1>2 we provide some insight in the probability of reaching each of the stable states by presenting simulation results.