A Caching Game with Infinitely Divisible Hidden Material

Abstract

This document is the author's final accepted version of the journal article. There may be differences between this version and the published version. You are advised to consult the publisher's version if you wish to cite from it. AbstractWe consider a caching game in which a unit amount of in…nitely divisible material is distributed among n 2 locations. A Searcher chooses how to distribute his search e¤ort r about the locations so as to maximize the probability she will …nd a given minimum amount m = 1 m r of the material. If the search e¤ort yi invested by the Searcher in a given location i is at least as great as the amount of material xi located there she …nds all of it, otherwise the amount she …nds is only yi. In other words she …nds min fxi; yig in location i. We seek the randomized distribution of search e¤ort 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 = m 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 m we show that for n = 2 the game reduces to Ruckle's game.