The success probability in Levine's hat problem, and independent sets in graphs

TL;DR

This paper analyzes Levine's hat problem, proving the success probability decreases with more players and exploring its connections to graph theory, including independent sets and bounds on vertex sets intersecting maximum independent sets.

Contribution

It establishes that the success probability in Levine's hat problem is strictly decreasing with the number of players and links the problem to key concepts in graph theory.

Findings

Success probability decreases as number of players increases

Connections established between the problem and independent sets in graphs

Bounds derived for vertex sets intersecting all maximum independent sets

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 is that 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 problems in graph theory: relating the size of the largest independent set in a graph and in a random induced subgraph of it, and bounding the size of a set of vertices intersecting every maximum-size independent set in a graph.