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, among others: r = 2 or n − 2; n ≤ 7; n = 2r − 1; h < 2 + 1/ (n− r − 1) and n ≤ 2r. The last result uses the Erdos-Ko-Rado theorem. We establish a connection between Ruckle’s Conjecture and the difficult Hoeffding problem of bounding tail probabilities of sums of random variables.