On Levine's notorious hat puzzle

TL;DR

This paper reviews strategies for Levine's hat puzzle, analyzes the case with two players, and proves that the maximum success probability decreases as the number of players increases.

Contribution

It provides an overview of known strategies, discusses the two-player case with a conjecture, and proves that success probability declines with more players.

Findings

Optimal success probability decreases as number of players increases

Extended discussion and conjecture for the two-player case

Proved that success probability is a strictly decreasing function of n

Abstract

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 $n$.