On Levine's notorious hat puzzle

The Levine hat game requires \(n\) players, each wearing an infinite random stack of black and white hats, to guess the location of a black hat on their own head seeing only the hats worn by all the other players. They are allowed a strategy session before the game, but no further communication. The...

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Bibliographic Details
Published in:arXiv.org 2021-06
Main Authors: Buhler, Joe, Freiling, Chris, Graham, Ron, Kariv, Jonathan, Roche, James R, Tiefenbruck, Mark, Clint Van Alten, Yeroshkin, Dmytro
Format: Article
Language:English
Subjects:
Game theory
Lower bounds
Outerwear
Upper bounds
Online Access:Get full text
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Summary:The Levine hat game requires \(n\) players, each wearing an infinite random stack of black and white hats, to guess the location of a black hat on their own head seeing only the hats worn by all the other players. They are allowed a strategy session before the game, but no further communication. The players collectively win if and only if all their guesses are correct. In this paper we give an overview of what is known about strategies for this game, including an extended discussion of the case with \(n = 2\) players (and a conjecture for an optimal strategy in this case). We also prove that \(V_n\), the optimal value of the joint success probability in the \(n\)-player game, is a strictly decreasing function of \(n\).
ISSN:2331-8422