We study mean field games and corresponding N-player games in continuous time over a finite time horizon where the position of each agent belongs to a finite state space. As opposed to previous works on finite state mean field games, we use a probabilistic representation of the system dynamics in terms of stochastic differential equations driven by Poisson random measures. Under mild assumptions, we prove existence of solutions to the mean field game in relaxed open-loop as well as relaxed feedback controls. Relying on the probabilistic representation and a coupling argument, we show that mean field game solutions provide symmetric \(\varepsilon _N\)-Nash equilibria for the N-player game, both in open-loop and in feedback strategies (not relaxed), with \(\varepsilon _N\le \frac{\text {constant}}{\sqrt{N}}.\) Under stronger assumptions, we also find solutions of the mean field game in ordinary feedback controls and prove uniqueness either in case of a small time horizon or under monotonicity.

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有限状态均值场博弈的概率方法

我们研究了在有限时间范围内连续时间中的平均场游戏和相应的N玩家游戏，其中每个特工的位置属于有限状态空间。与先前关于有限状态均值场博弈的工作相反，我们使用由Poisson随机测度驱动的随机微分方程来表示系统动力学。在温和的假设下，我们证明了在宽松的开环以及宽松的反馈控制下，均值场博弈的解的存在。依靠概率表示和耦合参数，我们证明了均值现场游戏解决方案在开环和反馈策略（不宽松）中为N人游戏提供了对称的\（\ varepsilon _N \）- Nash均衡，与\（\ varepsilon _N \ le \ frac {\ text {constant}} {\ sqrt {N}}。\）在更强的假设下，我们还找到了普通反馈控制中均值场博弈的解，并证明了唯一性时间范围短或单调。