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

The emergency of mutual cooperation is studied in a spatially extended evolutionary prisoner’s dilemma game in which the players are located on the sites of cubic lattices for dimensions $d=1,$ 2, and 3. Each player can choose one of the three following strategies: cooperation $\mathrm{}\left(C\right),$ defection (D) or “tit for tat” $\mathrm{}\left(T\right).$ During the evolutionary process the randomly chosen players adopt one of their neighboring strategies if the chosen neighbor has a higher payoff. Moreover, an external constraint imposes that the players always cooperate with probability p. The stationary state phase diagram is computed by both using generalized mean-field approximations and Monte Carlo simulations. Nonequilibrium second-order phase transitions associated with the extinction of one of the possible strategies are found and the corresponding critical exponents belong to the directed percolation universality class. It is shown that externally forcing the collaboration does not always produce the desired result.