Alpern, Steve, Fokkink, Robbert and Kikuta, Ken. (2010) On Ruckle's conjecture on accumulation games. SIAM Journal on Control and Optimization, Vol.48 (No.8). pp. 5073-5083. ISSN 0363-0129

Abstract

In an accumulation game, the Hider secretly distributes his given total wealth h among n locations, while the Searcher picks r locations and confiscates the material placed there. The Hider wins if what is left at the remaining $n-r$ locations is at least 1; otherwise the Searcher wins. Ruckle's conjecture says that an optimal Hider strategy is to put an equal amount $h/k$ at k randomly chosen locations for some k. We extend the work of Kikuta and Ruckle by proving the conjecture for several cases, e.g., $r=2$ or $n-2$; $n\leq7$; $n=2r-1$; $h\leq2+1/\,(n-r)$ and $n\leq2r$. The last result uses the Erdős–Ko–Rado theorem. We establish a connection between Ruckle's conjecture and the Hoeffding problem of bounding tail probabilities of sums of random variables.

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