Stochastic convergence to recurrent state equilibrium for state-based games

State-based games contain an additional state space by comparing with normal games. Correspondingly, the equilibrium of state-based games is called recurrent state equilibrium (RSE). For state-based games, the stochastic convergence to RSE is investigated in this paper. Firstly, the stochastic convergence of state-based games is defined. Then, a kind of state-based best-response update rule is designed for state-based games. Under this update rule, the state-based games can be converted into the Markovian switching logical networks through the semi-tensor product. Next, based on the results of Markovian switching logical networks and positive systems, the stochastic convergence of state-based games is investigated and a verifiable criterion is derived. Finally, an example is presented to illustrate the validity of the obtained criterion.