标题： Levine帽子问题的分析框架：新策略、界和推广 显示英文标题

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

摘要： 我们研究Levine帽子问题，这是一个由Lionel Levine于2010年提出的经典组合谜题。这个问题涉及一个游戏，其中$n \geq 2$名玩家，每个人可以看到队友头上无限堆帽子，但看不到自己头上的帽子，必须同时猜测自己帽子堆中黑色帽子的索引。如果其中一名玩家未能做到这一点，团队将集体失败。因此，玩家必须在游戏开始前制定一个好的策略。尽管对于$n=2$来说，最优获胜概率$V_{n}$仍然未知，但我们取得了三个关键进展。首先，我们开发了一个新的几何框架，通过可测函数表示策略，提供了$V_{n}$的新表达方式，并通过积分公式对有限和无限帽子堆的游戏进行了统一处理。其次，我们构建了一个新的策略$K_{5}$，达到了预期的最优获胜概率：$0.35$。我们还证明$K_{5}$属于一个更大的策略类，使我们能够改进当前的界限并解决预期的不等式。 最后，我们引入并完全解决了该问题的一个连续推广，证明将帽子堆扩展到不可数集会使最优获胜概率提高到恰好$1/2$。 这一推广自然地导致了一个更广泛且更平滑的战略框架，在其中我们还描述了如何计算对一系列策略的最优响应。 显示英文摘要

摘要： 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.