Spatial evolutionary prisoner's dilemma game with three strategies and external constraints

The emergency of mutual cooperation is studied in a spatially extended evol utionary prisoner's dilemma game in which the players are located on the si tes of cubic lattices for dimensions d = 1, 2, and 3. Each player can choos e one of the three following strategies: cooperation (C), defection (D) or "tit for tat" (T). During the evolutionary process the randomly chosen play ers adopt one of their neighboring strategies if the chosen neighbor has a higher payoff. Moreover, an external constraint imposes that the players al ways cooperate with probability p. The stationary state phase diagram is co mputed by both using generalized mean-held approximations and Monte Carlo s imulations. Nonequilibrium second-order phase transitions associated with t he extinction of one of the possible strategies are found and the correspon ding critical exponents belong to the directed percolation universality cla ss. It is shown that externally forcing the collaboration does not always p roduce the desired result.