The Levine hat game requires $n$ players, each wearing an infinite random stack of black and white hats, to guess the location of a black hat on their own head seeing only the hats worn by all the other players. They are allowed a strategy session before the game, but no further communication. The players collectively win if and only if all their guesses are correct. In this paper we give an overview of what is known about strategies for this game, including an extended discussion of the case with $n = 2$ players (and a conjecture for an optimal strategy in this case). We also prove that $V_n$, the optimal value of the joint success probability in the $n$-player game, is a strictly decreasing function of $n$.
翻译:莱文帽游戏需要一美元球员,每人身着一串无穷无尽的黑白黑帽子,以猜测黑帽子在自己头上的位置,只看到所有其他球员戴的帽子。 他们可以在比赛之前有一个策略会议,但不能进行进一步的交流。 球员集体获胜, 前提是他们所有的猜想都是正确的。 本文概述了这场比赛的策略, 包括延长与一美元=两美元的球员对案子的讨论( 以及这次最佳策略的猜测 ) 。 我们还证明, $V_n美元是玩家玩家游戏中联合成功概率的最佳值, 美元是绝对下降的一美元。