On Ruckle's Conjecture on Accumulation Games

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≤7; $n=2r-1$; h≤2+1/\,(n-r) and n≤2r. 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.