On a continuous solution to the Bellman-Poisson equation in stochastic games
Zero-sum semi-Markov games are considered with the set of states \(S=\{s\}\), sets of actions of two players \(A=\{a\}\) and \(B=\{b\}\), weakly continuous transition probabilities and expected average payoff criterion. Assuming the \(V\)-geometric ergodicity of the embedded Markov chain it is proved the existence of a continuous solution to the Bellman-Poisson equation \[ \begin{aligned} \hat{h}(s)&=\max_{a} \min_{b} \left[r(s,a,b)- \hat{\xi} \tau(s,a,b) +\int_S \hat{h}(y) q(dy|s,a,b)\right]\\ &=\min_{b} \max_{a} \left[r(s,a,b)- \hat{\xi} \tau(s,a,b) +\int_S \hat{h}(y) q(dy|s,a,b)\right] \end{aligned} \] for all \(s \in S\). Here \(r(s,a,b)\) and \(\tau(s,a,b)\) are reward of player~1 and expected occupation time of the process in the state \(s\) provided that actions \(a\) and \(b\) were chosen, \(q(\cdot|s,a,b)\) is the transition law of the embedded Markov chain, \(\hat{\xi}\) is the value of the game. Both players possess optimal stationary strategies.
scientific article; zbMATH DE number 5760461