For the two-ball, three-box search game with arbitrary box costs, the exact value is the maximum of three rational functions, with optimal product-form Hider strategies and certified Searcher mixtures.
A review of minimum cost box searching games
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abstract
We consider a class of zero-sum search games in which a Hider hides one or more target among a set of $n$ boxes. The boxes may require differing amount of time to search, and detection may be imperfect, so that there is a certain probability that a target may not be found when a box is searched, even when it is there. A Searcher must choose how to search the boxes sequentially, and wishes to minimize the expected time to find the target(s), whereas the Hider wishes to maximize this payoff. We review some known solutions to different cases of this game.
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An Exact Solution of the Two-Ball Multi-Look Search Game with Three Boxes and Heterogeneous Costs
For the two-ball, three-box search game with arbitrary box costs, the exact value is the maximum of three rational functions, with optimal product-form Hider strategies and certified Searcher mixtures.