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

Abstract: We study the Levine hat problem, a classic combinatorial puzzle introduced by Lionel Levine in 2010. This problem involves a game in which $n \geq 2$ players, each seeing an infinite stack of hats on each of their teammates' heads but not on their own, must simultaneously guess the index of a black hat on their own stack. If one of the players fails to do so, the team loses collectively. The players must therefore come up with a good strategy before the game starts. While the optimal winning probability $V_{n}$ remains unknown even for $n=2$, we make three key advances. First, we develop a novel geometric framework for representing strategies through measurable functions, providing a new expression of $V_{n}$ and a unified treatment of the game for finite and for infinite stacks via integral formulations. Secondly, we construct a new strategy $K_{5}$ that reaches the conjectured optimal probability of victory : $0.35$. We also show that $K_{5}$ is part of a larger class of strategies that allow us to improve current bounds and resolve conjectured inequalities. Finally, we introduce and entirely solve a continuous generalization of the problem, demonstrating that extending to uncountable hat stacks increases the optimal winning probability to exactly $1/2$. This generalization naturally leads to a broader and smoother strategic framework, within which we also describe how to compute optimal responses to a range of strategies.