Zero-sum Markov games with impulse controls

In this paper we consider a zero-sum Markov stopping game on a general state space with impulse strategies and infinite time horizon discounted payoff where the state dynamics is a weak Feller--Markov process. One of the key contributions is our analysis of this problem based on “shifted” strategies, thereby proving that the original game can be practically restricted to a sequence of Dynkin's stopping games without affecting the optimalty of the saddle-point equilibria and hence completely solving some open problems in the existing literature. Under two quite general (weak) assumptions, we show the existence of the value of the game and the form of saddle-point (optimal) equilibria in the set of shifted strategies. Moreover, our methodology is different from the previous techniques used in the existing literature and is based on purely probabilistic arguments. In the process, we establish an interesting property of the underlying Feller--Markov process and the impossibility of infinite number of impulses in finite time under saddle-point strategies which is crucial for the verification result of the corresponding Isaacs--Bellman equations.