We consider a variant of the hide-and-seek game in which a seeker inspects multiple hiding locations to find multiple items hidden by a hider. Each hiding location has a maximum hiding capacity and a probability of detecting its hidden items when an inspection by the seeker takes place. The objective of the seeker (resp. hider) is to minimize (resp. maximize) the expected number of undetected items. This model is motivated by strategic inspection problems, where a security agency is tasked with coordinating multiple inspection resources to detect and seize illegal commodities hidden by a criminal organization. To solve this large-scale zero-sum game, we leverage its structure and show that its mixed strategies Nash equilibria can be characterized using their unidimensional marginal distributions, which are Nash equilibria of a lower dimensional continuous zero-sum game. This leads to a two-step approach for efficiently solving our hide-and-seek game: First, we analytically solve the continuous game and compute the equilibrium marginal distributions. Second, we derive a combinatorial algorithm to coordinate the players' resources and compute equilibrium mixed strategies that satisfy the marginal distributions. We show that this solution approach computes a Nash equilibrium of the hide-and-seek game in quadratic time with linear support. Our analysis reveals a complex interplay between the game parameters and allows us to evaluate their impact on the players' behaviors in equilibrium and the criticality of each location.

翻译：我们考虑捉迷藏博弈的一种变体：搜寻者检查多个隐藏位置，以找到隐藏者藏匿的多个物品。每个隐藏位置具有最大隐藏容量，并且当搜寻者检查时，存在检测其隐藏物品的概率。搜寻者（相应地，隐藏者）的目标是最小化（相应地，最大化）未被检测物品的期望数量。该模型受到策略性检查问题的启发，其中安全机构负责协调多个检查资源以检测和查获犯罪组织隐藏的非法商品。为求解此大规模零和博弈，我们利用其结构，证明其混合策略纳什均衡可以通过其一维边际分布来刻画，而这些边际分布是更低维连续零和博弈的纳什均衡。这导出了高效求解捉迷藏博弈的两步方法：首先，我们解析地求解连续博弈并计算均衡边际分布。其次，我们推导一种组合算法来协调参与者的资源，并计算满足边际分布的均衡混合策略。我们证明，该求解方法能在二次时间内计算捉迷藏博弈的纳什均衡，且具有线性支撑。我们分析揭示了博弈参数之间复杂的相互作用，并使我们能够评估这些参数对均衡中参与者行为以及每个位置关键性的影响。