On Local Prisoner's Dilemma game with Pareto updating rule

Prisoner's Dilemma games with two and three strategies are studied. The corresponding replicator equations, their steady states and their asymptotic stability are discussed. Local Prisoner's Dilemma games are studied using Pareto optimality. As in the case with Nash updating rule, the existence of tit for rat strategy is crucial to imply cooperation even in one dimension. Pareto updating implies less erratic behavior since the steady state configurations are mostly fixed points or at most 2-cycle. Finally, Prisoner's Dilemma game is simulated on small-world networks which are closer to real systems than regular lattices. There are no significant changes compared to the results of the regular lattice.