Finite-Horizon Partially Observable Semi-Markov Games with Risk Probability Criteria

This paper studies partially observable two-person zero-sum semi-Markov games under a probability criterion, in which the system state may not be completely observed. It focuses on the probability that the accumulated rewards of player 1 (i.e., the incurred costs of player 2) fall short of a specified target at the terminal stage, which represents the risk of player 1 and the capacity of player 2. We study the game model via the technology of augmenting state space with the joint conditional distribution of the current unobserved state and the remaining goal. Under a mild condition, we establish a comparison theorem and derive the Shapley equation for the probability criterion. As a consequence, we prove the existence and the uniqueness of the value function and the existence of a Nash equilibrium.