On a continuous solution to the Bellman-Poisson equation in stochastic games
type: Item
instance of: scholarly article
title: On a continuous solution to the Bellman-Poisson equation in stochastic games
zbMATH Open document ID: 1198.91034
published in: Journal of Optimization Theory and Applications
publication date: 2010-07-24T00:00:00Z
review text: 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.
reviewed by: Alex V. Kolnogorov
Mathematics Subject Classification ID: 91A15
Mathematics Subject Classification ID: 91A05
zbMATH DE Number: 5760461
zbMATH Keywords: zero-sum semi-Markov games
zbMATH Keywords: optimality equations
zbMATH Keywords: optimal strategies
author: Anna Jaskiewicz
MaRDI profile type: Publication
full work available at URL: s10957-010-9698-x
OpenAlex ID: W2167030417
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DOI: 10.1007/S10957-010-9698-X
description: scientific article; zbMATH DE number 5760461