Phase diagrams for three-strategy evolutionary prisoner's dilemma games on regular graphs

Evolutionary prisoner's dilemma games are studied with players located on square lattice and random regular graphs defining four neighbors for each one. The players follow one of the three strategies: tit-for-tat, unconditional cooperation, and defection. The simplified payoff matrix is characterized by two parameters: the temptation $b$ to choose defection, and the cost $c$ of inspection reducing the income of tit-for-tat. The strategy imitation from one of the neighbors is controlled by pairwise comparison at a fixed level of noise. Using Monte Carlo simulations and the extended versions of pair approximation we have evaluated the $b-c$ phase diagrams indicating a rich plethora of phase transitions between stationary coexistence, absorbing and oscillatory states, including continuous and discontinuous phase transitions. By reasonable costs the tit-for-tat strategy prevents extinction of cooperators across the whole span of $b$ values determining the prisoner's dilemma game, irrespective of the interaction graph structure. We also demonstrate that the system can exhibit a repetitive succession of oscillatory and stationary states upon changing a single payoff value, which highlights the remarkable sensitivity of cyclical interactions on parameters that define the strength of dominance.