Finite-step Algorithms for Single-controller and Perfect Information Stochastic Games

After a brief survey of iterative algorithms for general stochastic games, we concentrate on finite-step algorithms for two special classes of stochastic games. They are Single-Controller Stochastic Games and Perfect 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 efficiently 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 information stochastic games, a policy improvement algorithm is presented. Many other classes of stochastic games with orderfield property still await efficient finite-step algorithms.