The success probability in Lionel Levine's hat problem is strictly decreasing with the number of players, and this is related to interesting questions regarding Hamming powers of Kneser graphs and independent sets in random subgraphs

TL;DR

This paper proves that the probability of success in Levine's hat problem decreases as the number of players increases and explores its connections to graph theory, specifically Hamming powers of Kneser graphs and independent sets.

Contribution

It establishes that the success probability is strictly decreasing with more players and links this to new questions in graph theory.

Findings

Success probability decreases with more players

Established a strict decreasing property of success probability

Connected the problem to Hamming powers of Kneser graphs

Abstract

Lionel Levine's hat challenge has $t$ players, each with a (very large, or infinite) stack of hats on their head, each hat independently colored at random black or white. The players are allowed to coordinate before the random colors are chosen, but not after. Each player sees all hats except for those on her own head. They then proceed to simultaneously try and each pick a black hat from their respective stacks. They are proclaimed successful only if they are all correct. Levine's conjecture was the success probability tends to zero when the number of players grows. We prove that this success probability is strictly decreasing in the number of players, and present some connections to questions in graph theory.