具有三个盒子和异质成本的双球多视角搜索博弈的精确解
An Exact Solution of the Two-Ball Multi-Look Search Game with Three Boxes and Heterogeneous Costs
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中文总结 AI 辅助
研究具有三个盒子和异质成本的双球多视角搜索博弈，给出其精确解，明确最优策略形式，通过多种方法确立混合策略可行性并重现相关结果。
中文摘要 AI 辅助
隐藏者将两个相同的球分配到三个盒子中，其搜索成本满足a≥b≥c>0。搜索者自适应地打开盒子，直到找到两个球；每次打开会产生对应盒子的成本，且最多只能找回一个球。我们给出了这个异质多视角搜索成本博弈的精确解，其值是三个显式有理函数的最大值。在三个参数区间中，最优隐藏者策略是分别限制在三个、五个或全部六个放置方式上的乘积形式分布。每一种非浪费性的确定性搜索者策略都等价于72个基本决策树中的一个，这些决策树仅产生42种不同的打开次数分布。第一个区间通过解析论证解决，第二个区间通过两种精确的搜索者混合策略解决，第三个区间通过四种解决。参数化混合策略在整个成本区域上的可行性由精确有理伯恩斯坦基证书确立。独立实现重现了策略集、零和线性规划的值以及伯恩斯坦验证器检查的全部154个二进节点。
英文摘要
A Hider distributes two identical balls among three boxes whose search costs satisfy $a\ge b\ge c>0$. A Searcher opens boxes adaptively until both balls are found; every opening incurs the corresponding box cost and recovers at most one ball. We give an exact solution of this heterogeneous multi-look search-cost game. The value is the maximum of three explicit rational functions. In the three parameter regimes, an optimal Hider strategy is the product-form distribution restricted respectively to three, five, or all six placements. Every non-wasteful deterministic Searcher policy is payoff-equivalent to one of $72$ elementary decision trees, which induce only $42$ distinct opening-count profiles. An analytic argument settles the first regime. Two exact Searcher mixtures settle the second, and four settle the third. Feasibility of the parameterized mixtures over the full cost region is established by exact rational Bernstein-basis certificates. Independent implementations reproduce the policy set, the zero-sum linear-program values, and all $154$ dyadic nodes examined by the Bernstein verifier.
