We introduce a discrete-time search game, in which two players compete to find an object first. The object moves according to a time-varying Markov chain on finitely many states. The players know the Markov chain and the initial probability distribution of the object, but do not observe the current state of the object. The players are active in turns. The active player chooses a state, and this choice is observed by the other player. If the object is in the chosen state, this player wins and the game ends. Otherwise, the object moves according to the Markov chain and the game continues at the next period. We show that this game admits a value, and for any error-term $\veps>0$, each player has a pure (subgame-perfect) $\veps$-optimal strategy. Interestingly, a 0-optimal strategy does not always exist. The $\veps$-optimal strategies are robust in the sense that they are $2\veps$-optimal on all finite but sufficiently long horizons, and also $2\veps$-optimal in the discounted version of the game provided that the discount factor is close to 1. We derive results on the analytic and structural properties of the value and the $\veps$-optimal strategies. Moreover, we examine the performance of the finite truncation strategies, which are easy to calculate and to implement. We devote special attention to the important time-homogeneous case, where additional results hold.

中文翻译：

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具有移动目标的竞技搜索游戏

我们引入了一个离散时间搜索游戏，其中两个玩家首先竞争寻找一个对象。对象在有限多个状态上根据时变马尔可夫链移动。玩家知道马尔可夫链和物体的初始概率分布，但不观察物体的当前状态。玩家轮流活跃。主动玩家选择一个状态，这个选择被另一个玩家观察。如果对象处于选定状态，则该玩家获胜并且游戏结束。否则，对象按照马尔可夫链移动，游戏在下一个周期继续。我们证明这个博弈承认一个值，并且对于任何误差项 $\veps>0$，每个玩家都有一个纯（子博弈完美的）$\veps$-最优策略。有趣的是，0 最优策略并不总是存在。$\veps$-最优策略是稳健的，因为它们在所有有限但足够长的范围内都是 $2\veps$-最优，并且在游戏的折扣版本中也是 $2\veps$-最优，前提是折扣因子接近于 1。我们得出了价值和 $\veps$ 最优策略的分析和结构特性的结果。此外，我们检查了易于计算和实施的有限截断策略的性能。我们特别关注重要的时间同质情况，其中额外的结果成立。我们得出了价值的分析和结构特性以及 $\veps$ 最优策略的结果。此外，我们检查了易于计算和实施的有限截断策略的性能。我们特别关注重要的时间同质情况，其中额外的结果成立。我们得出了价值的分析和结构特性以及 $\veps$ 最优策略的结果。此外，我们检查了易于计算和实施的有限截断策略的性能。我们特别关注重要的时间同质情况，其中额外的结果成立。