Discounted Semi-Markov Games and Algorithms for Solving Two Structured Classes

Abstract

Two structured classes of zero-sum two-person finite (state and action spaces) semi-Markov games with discounted payoffs, namely, Additive Reward-Additive Transition and Action Independent Transition Time (AR-AT-AITT) and Additive Reward-Action Independent Transition and Additive Transition Time (AR-AIT-ATT) have been studied. We propose two practical situations of economic competition viz. petroleum game and groundwater game that suitably fit into such classes of games, respectively. Solution (value and pure stationary optimals) to such classes of games can be derived from optimal solution to appropriate bilinear programs with linear constraints. We present a stepwise generalized principal pivoting algorithm for solving the vertical linear complementarity problem (VLCP) obtained from such game problems. Moreover, a neural network dynamics is proposed for solving such structured classes. Examples are worked out to compare the usefulness of the above three algorithms.