Bi-personal stochastic transient Markov games with stopping times and total reward criterion

summary:The article is devoted to a class of Bi-personal (players 1 and 2), zero-sum Markov games evolving in discrete-time on Transient Markov reward chains. At each decision time the second player can stop the system by paying terminal reward to the first player. If the system is not stopped the first player selects a decision and two things will happen: The Markov chain reaches next state according to the known transition law, and the second player must pay a reward to the first player. The first player (resp. the second player) tries to maximize (resp. minimize) his total expected reward (resp. cost). Observe that if the second player is dummy, the problem is reduced to finding optimal policy of a transient Markov reward chain. Contraction properties of the transient model enable to apply the Banach Fixed Point Theorem and establish the Nash Equilibrium. The obtained results are illustrated on two numerical examples