A Fourier approach to Levine's hat puzzle

TL;DR

The paper studies Lionel Levine's two-player hat puzzle through a Fourier-analytic lens on the Boolean cube. By expressing the win probability in terms of first-level Fourier coefficients of the players' strategy indicator sets and applying Plancherel's theorem along with Chang's inequality, it derives quantitative upper bounds on the asymptotic optimal success probability $U$. The main contributions are a noncomputer-assisted bound $U\le 0.37406$ and a further improvement to $U\le 0.37193$ using refinements of first-order coefficient bounds, demonstrating how harmonic-analysis techniques yield concrete limits for distributed coordination games with random inputs. These results advance the understanding of how spectral constraints constrain optimal coordination in probabilistic settings.

Abstract

We consider Lionel Levine's notorious hat puzzle with two players. Each player has a stack of hats on their head, and each hat is chosen independently to be either black or white. After observing only the other player's hats, players simultaneously choose one of their own hats. The players win if both chosen hats are black. In this note, we observe an upper bound on the probability of success, using Chang's lemma, a result in Boolean harmonic analysis.

Figures (1)

• Figure 1: The solid line shows the upper bound on squared first order Fourier coefficients from \ref{['eq:chang3']}. The dotted line shows the better (smaller) upper bound from Lemma \ref{['newlemma']}. No improvement to \ref{['eq:chang3']} is possible when $x= { \IfNoValueTF {-NoValue-} {\mathbb{P}\mleft[{A}\mright]} {\mathbb{P}\mleft[{A}\middle\vert{-NoValue-}\mright]} } \in\{1/4,1/2\}$.

Theorems & Definitions (16)

• Lemma 2.1: Chang's Inequality, impa14,odonnell14
• Lemma 2.2
• proof
• Theorem 1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof