One Empty Locker and Two Inspections: An Exact Optimal Team Strategy

Abstract

We study a cooperative search game with $N$ lockers, $N-1$ labelled objects, one empty locker, and $N-1$ players. Player $i$ seeks object $i$ and may inspect at most two lockers; the second inspection may depend on the content of the first. The players may coordinate beforehand but receive no information about the searches of other players after play begins. We prove that the maximum probability that every player finds the assigned object is $I_N/N!$, where $I_N$ is the number of involutions of $N$ elements. An optimal strategy represents the blank by the common fictitious symbol $N$ and follows pointers: player $i$ first opens locker $i$, then opens the locker whose number was observed. The upper bound holds for every deterministic or randomized adaptive strategy and follows from a deletion lemma and two recurrences. We give explicit examples, identify the three-door case with an isomorphic "Return of Monty Hall" game, and report a reproducible mixed-integer verification for $N=2,3,4,5$. The computation is independent of, and not needed for, the proof.

Withdrawn at the request of the co-author due to an unresolved authorship issue. The withdrawal is not related to an error in the mathematical results