Modeling Replicator Dynamics in Stochastic Games Using Markov Chain Method

In stochastic games, individuals need to make decisions in multiple states and transitions between states influence the dynamics of strategies significantly. In this work, by describing the dynamic process in stochastic game as a Markov chain and utilizing the transition matrix, we introduce a new method, named state-transition replicator dynamics, to obtain the replicator dynamics of a stochastic game. Based on our proposed model, we can gain qualitative and detailed insights into the influence of transition probabilities on the dynamics of strategies. We illustrate that a set of unbalanced transition probabilities can help players to overcome the social dilemmas and lead to mutual cooperation in a cooperation back state, even if the stochastic game has the same social dilemmas in each state. Moreover, we also present that a set of specifically designed transition probabilities can fix the expected payoffs of one player and make him lose the motivation to update his strategies in the stochastic game.