SIAM Journal on Control and Optimization, Volume 58, Issue 1, Page 580-604, January 2020.
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.

中文翻译：

---

带脉冲控制的零和马尔可夫博弈

SIAM控制与优化杂志，第58卷，第1期，第580-604页，2020年1月。
在本文中，我们考虑了具有脉冲策略和无限时域折扣收益的一般状态空间上的零和马尔可夫停止博弈，其中状态动力学是一个弱Feller-Markov过程。关键贡献之一是我们基于“转移”策略对这一问题的分析，从而证明了原始游戏实际上可以被限制在Dynkin的一系列停止游戏中，而不会影响鞍点平衡的最优性，从而完全解决了一些问题。现有文献中存在未解决的问题。在两个非常笼统（弱）的假设下，我们证明了博弈价值的存在以及在转移策略集中的鞍点（最优）均衡形式。此外，我们的方法与现有文献中使用的先前技术不同，并且仅基于概率论。在此过程中，我们建立了潜在的Feller-Markov过程的有趣性质，以及在鞍点策略下在有限时间内不可能无限数量的脉冲的可能性，这对于验证相应的Isaacs-Bellman方程至关重要。