The solution to an open problem for a caching game

TL;DR

This paper solves the remaining open cases of a two-object caching game with four locations and provides asymptotic results for the game's value as the number of locations grows large.

Contribution

It completes the analysis of the caching game for two objects in four locations and offers asymptotic insights for large numbers of locations.

Findings

Complete solution for 2 objects in 4 locations.

Asymptotic value of the game approaches h/n for large n.

Geometrical argument used for asymptotic analysis.

Abstract

In a caching game introduced by Alpern et al., a Hider who can dig to a total fixed depth normalized to $1$ buries a fixed number of objects among $n$ discrete locations. A Searcher who can dig to a total depth of $h$ searches the locations with the aim of finding all of the hidden objects. If he does so, he wins, otherwise the Hider wins. This zero-sum game is complicated to analyze even for small values of its parameters, and for the case of $2$ hidden objects has been completely solved only when the game is played in up to $3$ locations. For some values of $h$ the solution of the game with $2$ objects hidden in $4$ locations is known, but the solution in the remaining cases was an open question recently highlighted by Fokkink et al. Here we solve the remaining cases of the game with $2$ objects hidden in $4$ locations. We also give some more general results for the game, in particular using a geometrical argument to show that when there are $2$ objects hidden in $n$ locations and $n \rightarrow \infty$, the value of the game is asymptotically equal to $h/n$ for $h \ge n/2$.