An analytical framework for the Levine hats problem: new strategies, bounds and generalizations

TL;DR

This paper introduces a geometric integral framework for the Levine hats problem, constructs new strategies that improve bounds, and generalizes the problem to infinite stacks, providing exact solutions and better convergence rates.

Contribution

It develops a unified geometric approach, constructs recursive strategies with improved bounds, and solves a generalized infinite-stack version of the Levine hats problem.

Findings

Constructed a recursive strategy $\\mathscr{S}_5$ achieving the conjectured probability $7/20$.

Improved the convergence rate of $V_{2,h}$ to near-optimal levels.

Solved the infinite-stack generalization, showing the winning probability is exactly 1/2.

Abstract

We study the Levine hat problem, a cooperative puzzle introduced by Lionel Levine in 2010, in which $n \geq 2$ players must simultaneously identify a black hat on their own infinite stack, each seeing only their teammates' stacks. While the optimal winning probability $V_n$ remains unknown even for $n=2$, we make three key advances. First, we develop a geometric and integral framework representing strategies as Lebesgue-measurable functions, yielding a new integral expression for $V_n$ and a unified treatment of finite and infinite stacks. Second, we construct a recursive strategy $\mathscr{S}_5$ processing hats in blocks of five, which attains the conjectured optimal probability $7/20$ for two players. Although this bound was already achieved by the known strategy $\mathscr{S}_3$, the existence of $\mathscr{S}_5$ refutes the previously held expectation that recursive strategies with block size greater than three yield no improvement, and produces a strictly better geometric convergence rate for $V_{2,h}$ as well as a new lower bound for $V_2(p)$ which improves known results for $p < 0.312$. Building upon this, we improve the geometric convergence rate of $V_{2,h}$ up to the near-optimal $1/4^{1-\varepsilon}$ for any $\varepsilon > 0$. Third, we introduce and completely solve a generalization of the problem where players are given uncountably infinite stacks of hats, showing that the optimal winning probability in this setting equals exactly $1/2$ for all $n \geq 2$. This new formulation allows to study the original combinatorial problem using tools from analytic optimization, and provides a natural framework for computing optimal responses to fixed strategies.