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

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.