离散空间中的经典搜索博弈
A Classical Search Game in Discrete Locations
摘要：
考虑一个两人零和搜索博弈，一个是隐藏者，另一个是隐藏者。隐藏者隐藏在$n$离散的位置中，并且搜索者连续地访问各个位置，直到找到隐藏者。两个玩家都知道，在位置$i$的搜索需要$t_i$时间单位，并以概率$q_i$独立地检测隐藏者-如果隐藏在那里-对于$i=1,\ldots,n$。隐藏者的目标是最大化直到检测到的期望时间，而隐藏者的目标是最小化它。我们证明了每个参与者的最优策略的存在性。特别地，隐藏者的最优混合策略以非零概率隐藏在每个位置，并且隐藏者的最优混合策略可以用多达$n$的简单搜索序列来构造。我们开发了一个算法来计算每个玩家的最佳策略，并比较最佳的隐藏策略与简单的隐藏策略，在搜索开始时给出的位置偏好。
英文摘要：
Consider a two-person zero-sum search game between a hider and a searcher. The hider hides among $n$ discrete locations, and the searcher successively visits individual locations until finding the hider. Known to both players, a search at location $i$ takes $t_i$ time units and detects the hider -- if hidden there -- independently with probability $q_i$, for $i=1,\ldots,n$. The hider aims to maximize the expected time until detection, while the searcher aims to minimize it. We prove the existence of an optimal strategy for each player. In particular, the hider's optimal mixed strategy hides in each location with a nonzero probability, and the searcher's optimal mixed strategy can be constructed with up to $n$ simple search sequences. We develop an algorithm to compute an optimal strategy for each player, and compare the optimal hiding strategy with the simple hiding strategy which gives the searcher no location preference at the beginning of the search.
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