A Caching Game with Infinitely Divisible Hidden Material

We consider a caching game in which a unit amount of infinitely divisible material is distributed among n ≥ 2 locations. A Searcher chooses how to distribute his search effort r about the locations so as to maximize the probability she will find a given minimum amount m̄ = 1−m ≤ r of the material. If the search effort yi invested by the Searcher in a given location i is at least as great as the amount of material xi located there she finds all of it, otherwise the amount she finds is only yi. In other words she finds min {xi, yi} in location i. We seek the randomized distribution of search effort that maximizes the probability of success for the Searcher in the worst case, hence we model the problem as a zero-sum win-lose game between the Searcher and a malevolent Hider who wishes to keep more than m of the material. We show that in the case r = m̄ the game has a geometric interpretation that for n = 2 corresponds to a problem posed by W. H. Ruckle in his monograph, Geometric Games and Their Applications. We give solutions for the geometric game when n = 3 for certain values of m, and bounds on the value for other values of m. In the more general case r ≥ m̄ we show that for n = 2 the game reduces to Ruckle’s game.