Hide-and-Seek Game with Capacitated Locations and Imperfect Detection

IF 2.5 4区 管理学 Q3 MANAGEMENT Decision Analysis Pub Date : 2023-10-25 DOI:10.1287/deca.2023.0012
Bastián Bahamondes, Mathieu Dahan
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Abstract

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 (respectively, hider) is to minimize (respectively, 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-strategy Nash equilibria can be characterized using their unidimensional marginal distributions, which are pure 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 derive closed-form expressions of the equilibrium marginal distributions. Second, we design 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 novel equilibrium behaviors driven by a complex interplay between the game parameters, captured by our closed-form solutions. Funding: This work was supported by the Georgia Tech Stewart Fellowship and the Georgia Tech New Faculty Start Up Grant [for Georgia Tech New Faculty Start Up Grant]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/deca.2023.0012 .
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有容量位置和不完全检测的捉迷藏游戏
我们考虑的是捉迷藏游戏的一种变体,在这种游戏中,寻道者检查多个隐藏地点,以找到寻道者隐藏的多个物品。每个隐藏位置都有一个最大的隐藏容量和一个被搜索者检查时发现其隐藏物品的概率。搜索者(分别是隐藏者)的目标是最小化(分别是最大化)未被探测到的物品的预期数量。这种模式的动机是战略检查问题,其中安全机构的任务是协调多种检查资源,以发现和没收犯罪组织隐藏的非法商品。为了求解这种大规模的零和博弈,我们利用了它的结构,并证明了它的混合策略纳什均衡可以用它们的一维边际分布来表征,这是一个低维连续零和博弈的纯均衡。这导致了有效解决捉迷藏博弈的两步方法:首先,我们解析求解连续博弈并推导出均衡边际分布的封闭形式表达式。其次,我们设计了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。我们证明了这种求解方法在线性支持的二次时间内计算了捉迷藏博弈的纳什均衡。我们的分析揭示了由游戏参数之间复杂的相互作用所驱动的新的平衡行为,并被我们的封闭形式的解决方案所捕获。资金:这项工作得到了乔治亚理工学院斯图尔特奖学金和乔治亚理工学院新教师启动补助金的支持。补充材料:在线附录可在https://doi.org/10.1287/deca.2023.0012上获得。
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来源期刊
Decision Analysis
Decision Analysis MANAGEMENT-
CiteScore
3.10
自引率
21.10%
发文量
19
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