Incorporating social payoff into reinforcement learning promotes cooperation

Reinforcement learning has been demonstrated to be an effective approach to investigate the dynamic of strategy updating and the learning process of agents in game theory. Most studies have shown that [Formula: see text]-learning failed to resolve the problem of cooperation in well-mixed populations or homogeneous networks. To this aim, we investigate the self-regarding [Formula: see text]-learning’s effect on cooperation in spatial prisoner’s dilemma games by incorporating the social payoff. Here, we redefine the reward term of self-regarding [Formula: see text]-learning by involving the social payoff; that is, the reward is defined as a monotonic function of the individual payoff and the social payoff represented by its neighbors’ payoff. Numerical simulations reveal that such a framework can facilitate cooperation remarkably because the social payoff ensures agents learn to cooperate toward socially optimal outcomes. Moreover, we find that self-regarding [Formula: see text]-learning is an innovative rule that ensures cooperators coexist with defectors even at high temptations to defection. The investigation of the emergence and stability of the sublattice-ordered structure shows that such a mechanism tends to generate a checkerboard pattern to increase agents’ payoff. Finally, the effects of [Formula: see text]-learning parameters are also analyzed, and the robustness of this mechanism is verified on different networks.