Setting Population Payoff via Transition Function in Stochastic Game

Stochastic game is a crucial framework for studying strategic interactions and environmental uncertainty where payoffs change due to variations in the environment. Exploring the relationship between payoffs and the environmental transition functions is crucial for understanding stochastic games. This paper explores a method to set population long-term payoff through designing transition function in a stochastic game with two environments. Based on the zero-determinant theory, we analyze the feasible region for the equalizer property of transition function, which can set opponents’ payoff via transferring between favorable and unfavorable environment. It is validated in the evolutionary dynamics of finite populations with Fermi dynamics that the population’s payoff is pinned at a expected value. Under the established transition function, the population’s payoff is unaffected by the selection intensity, but the selection intensity can alter the frequency of cooperation in the population. Our results extend the zero-determinant theory in stochastic games and further reveal that setting payoff can, independent of selection intensity, anchor the average payoff of the population at a theoretical value. This deepens our understanding of the control of payoffs in stochastic games, and offers a new perspective on the evolution of population payoff.