All or Nothing Caching Games with Bounded Queries

Abstract

We determine the value of some search games where our goal is to find all of some hidden treasures using queries of bounded size. The answer to a query is either empty, in which case we lose, or a location, which contains a treasure. We prove that if we need to find $d$ treasures at $n$ possible locations with queries of size at most $k$, then our chance of winning is $\frac{k^d}{\binom nd}$ if each treasure is at a different location and $\frac{k^d}{\binom{n+d-1}d}$ if each location might hide several treasures for large enough $n$. Our work builds on some results by Csóka who has studied a continuous version of this problem, known as Alpern's Caching Game; we also prove that the value of Alpern's Caching Game is $\frac{k^d}{\binom{n+d-1}d}$ for integer $k$ and large enough $n$.