# A caching game with infinitely divisible hidden material

Lidbetter, Thomas (2015) A caching game with infinitely divisible hidden material. SIAM Journal on Control and Optimization, 53 (5). pp. 3040-3056. ISSN 0363-0129

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Identification Number: 10.1137/140997075

## Abstract

We consider a caching game in which a unit amount of infinitely divisible material is distributed among $n\geq 2$ 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 $\bar{m} =1-m\leq r$ of the material. If the search effort $y_{i}$ invested by the Searcher in a given location $i$ is at least as great as the amount of material $x_{i}$ located there she finds all of it, otherwise the amount she finds is only $y_{i}$. In other words she finds $\min \left\{ x_{i},y_{i}\right\}$ 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 $r=\bar{m}$ 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, Pitman, Boston, 1983]. 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 $r\geq \bar{m}$ we show that for $n=2$ the game reduces to Ruckle's game

Item Type: Article http://epubs.siam.org/loi/sjcodc © 2015 Society for Industrial and Applied Mathematics Management Q Science > QA Mathematics 23 Nov 2015 15:32 20 Sep 2021 00:57 http://eprints.lse.ac.uk/id/eprint/64478

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