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Stability and Optimality of Non-cooperative Stochastic Adaptive Games with Different Estimations

We consider non-cooperative stochastic adaptive games described by a linear discrete-time stochastic system with two players. When the system parameters are unknown, we assume that the players apply the least-square algorithm to estimate the unknowns respectively instead of using the same estimator. According to “certainty equivalence principle”, the players need to design adaptive strategy profile to optimize their one-step-ahead payoff functions. We will show that the resulting adaptive strategy profile can make the closed-loop system globally stable and in some sense, the profile is an asymptotic Nash equilibrium.