Environmental quality and population welfare in Markovian eco-evolutionary dynamics

Environmental change and human behavior are co-depended. The quality of the environment affects human's welfare, and the human's behavior in turn changes the environment. Yet the co-dependent nature seems to give a single individual few capabilities to change the environment. Intuitively, it is the collective actions that matter. What is a single individual able to do with the population welfare and the environment? We set up a toy model to explicitly address this issue. We take into account the eco-evolutionary nature of the feedback between environment and human behavior. One strategy, termed as Welfare-Time strategy, is found, using which one individual suffices to set a linear relationship between collective welfare and environmental quality, no matter what the opponent does. This linear relationship can be either positively or negatively correlated, which is also unilaterally set by a single individual. It indicates that collective welfare can be higher even if it takes longer in a poor environment. Furthermore, we prove that the Welfare-Time strategy is able to dominate Win-Stay-Lose-Shift strategy, which is evolutionary stable against many strategies in repeated games. Our work reveals a hidden relationship between population welfare and the environment quality, which can be controlled unilaterally by a single individual. In addition, it implies that a single individual is able to control the environmental quality, provided that the rule of the environmental dynamics is known.