  The effects of payoffs and noise on the maintenance of cooperative behavior
are studied in an evolutionary Prisoner's Dilemma game with players located on
the sites of different two-dimensional lattices. This system exhibits a phase
transition from a mixed state of cooperators and defectors to a homogeneous one
where only the defectors remain alive. Using systematic Monte Carlo simulations
and different levels of the generalized mean-field approximations we have
determined the phase boundaries (critical points) separating the two phases on
the plane of the temperature (noise) and temptation to choose defection. In the
zero temperature limit this analysis suggests that the cooperation can be
sustained only for those connectivity structures where three-site clique
percolation occurs.

Statistical Mechanics