Discrete and Continuous Caching Games

We investigate a discrete search game called the Multiple Caching Game where the searcher's aim is to find all of a set of $d$ treasures hidden in $n$ locations. Allowed queries are sets of locations of size $k$, and the searcher wins if in all $d$ queries, at least one treasure is hidden in one of the $k$ picked locations. P\'alv\"olgyi showed that the value of the game is at most $\frac{k^d}{\binom{n+d-1}{d}}$, with equality for large enough $n$. We conjecture the exact cases of equality. We also investigate variants of the game and show an example where their values are different, answering a question of P\'alv\"olgyi. This game is closely related to a continuous variant, Alpern's Caching Game, based on which we define other continous variants of the multiple caching game and examine their values.