On Zero-Sum Two Person Perfect Information Stochastic Games

A zero-sum two person Perfect Information Stochastic game (PISG) under limiting average payoff has a value and both the maximiser and the minimiser have optimal pure stationary strategies. Firstly we form the matrix of undiscounted payoffs corresponding to each pair of pure stationary strategies (for each initial state) of the two players and prove that this matrix has a pure saddle point. Then by using the results by Derman [1] we prove the existence of optimal pure stationary strategy pair of the players. A crude but finite step algorithm is given to compute such an optimal pure stationary strategy pair of the players.Comment: arXiv admin note: text overlap with arXiv:2201.0017

Finite-Step Algorithms for Single-Controller and Perfect Information Stochastic Games

Abstract. After a brief survey of iterative algorithms for general stochas-tic games, we concentrate on finite-step algorithms for two special classes of stochastic games. They are Single-Controller Stochastic Games and Per-fect Information Stochastic Games. In the case of single-controller games, the transition probabilities depend on the actions of the same player in all states. In perfect information stochastic games, one of the players has exactly one action in each state. Single-controller zero-sum games are effi-ciently solved by linear programming. Non-zero-sum single-controller stochastic games are reducible to linear complementary problems (LCP). In the discounted case they can be modified to fit into the so-called LCPs of Eave’s class L. In the undiscounted case the LCP’s are reducible to Lemke’s copositive plus class. In either case Lemke’s algorithm can be used to find a Nash equilibrium. In the case of discounted zero-sum perfect informa-tion stochastic games, a policy improvement algorithm is presented. Many other classes of stochastic games with orderfield property still await efficient finite-step algorithms. 1

A Nested Family of kk-total Effective Rewards for Positional Games

We consider Gillette's two-person zero-sum stochastic games with perfect information. For each k \in \ZZ_+ we introduce an effective reward function, called $k$-total. For $k = 0$ and $1$ this function is known as {\it mean payoff} and {\it total reward}, respectively. We restrict our attention to the deterministic case. For all $k$, we prove the existence of a saddle point which can be realized by uniformly optimal pure stationary strategies. We also demonstrate that $k$-total reward games can be embedded into $(k+1)$-total reward games