Zero-Sum Games for Finite-Horizon Semi-Markov Processes Under the Probability Criterion

This study discusses a two-person zero-sum game for finite-horizon semi-Markov processes, where the concerned criterion is the probability that the total payoff produced by a system during a finite horizon exceeds a prescribed goal. We establish the Shapley equation and the existence of a saddle point under a suitable condition. Moreover, we develop a value iterative algorithm to compute an
$\varepsilon$
-saddle point and approach the value of the game by solving a series of matrix games. The construction of the
$\varepsilon$
-saddle point and the convergence of the algorithm are also shown. Finally, we demonstrate the application of our main results by an example on an inventory system.