Design of generalized zero-determinant strategies in time-variant game environment

This paper investigates the control strategies in repeated games with time-variant environment, called generalized zero-determinant (GZD) strategies, which can enforce the linear relationship between collective welfare and environmental quality. First, we give the algebraic model for Markovian profile evolutionary dynamics and then integrate it with Markovian state evolutionary equation into an eco-evolutionary algebraic system. Based on this algebraic form, a simple algebraic formula is provided to design GZD strategies in a repeated three-player game with two different environments, and then some interesting results are presented. It is shown that when the two environments are indistinguishable, the GZD strategies will degenerate to the zero-determinant (ZD) strategies. Finally, to reduce the computation complexity and highlight the importance of coordination to succeed in large groups, the group-based method is proposed to design GZD strategies.