Matrix Approach to Finding Recurrent State Equilibrium of State-based Games

State-based games are an extension of normal games since an underlying state space is introduced, and the equilibrium in state-based games is called recurrent state equilibrium (RSE). How to find RSE of state-based games is a challenging issue. The process of finding RSE includes two steps. Firstly, find the pure Nash equilibria of state-invariant games for each state in the state space. In this step, the basic tool is the semi-tensor product of matrices. Secondly, for each pure Nash equilibrium of state-invariant game, calculate the recurrent classes of state transition process. Furthermore, a necessary and sufficient condition is presented for the existence of RSE, and an algorithm is developed for finding RSE.