带容量限制位置与 imperfect 检测的捉迷藏博弈
Hide-and-Seek Game with Capacitated Locations and Imperfect Detection
AI总结：
本文研究带容量限制与 imperfect 检测的大规模零和捉迷藏博弈，提出基于低维连续博弈边际分布的两步求解法，可在二次时间内得到线性支持集的纳什均衡，并分析了参数对博弈行为与位置重要性的影响。
摘要：
我们研究一种捉迷藏博弈的变体：搜寻者检查多个藏匿点，以寻找藏匿者藏起的多件物品。每个藏匿点都有最大藏匿容量，且搜寻者检查该点时有一定概率检测到其中藏匿的物品。搜寻者（对应地，藏匿者）的目标是最小化（对应地，最大化）未被检测到的物品的期望数量。该模型的研究动机来自战略检查问题：安全机构需协调多支检查资源，以检测并查获犯罪组织藏匿的非法物品。为求解这一大规模零和博弈，我们利用其结构证明，该博弈的混合策略纳什均衡可通过其一维边际分布刻画，而这些边际分布是一个更低维度连续零和博弈的纳什均衡。由此我们提出一种高效求解该捉迷藏博弈的两步方法：第一步，我们解析求解该连续博弈并计算均衡边际分布；第二步，我们推导了一种组合算法来协调参与者的资源，并计算满足上述边际分布的均衡混合策略。我们证明该求解方法可在二次时间内计算出捉迷藏博弈的纳什均衡，且支持集规模为线性。我们的分析揭示了博弈参数之间复杂的相互作用，使我们能够评估这些参数对均衡状态下参与者行为以及每个位置重要性的影响。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
