On Levine's infamous hat puzzle

The Levine hat game asks $n$ players, each with an infinite random stack of black or white hats on their head, to predict the location of a black hat on their own head seeing only the hats on everyone else's heads. They are allowed a strategy session before the game, but no further communication. The players win if and only if all predictions are correct. In addition to giving an overview of the game, we discuss the $n = 2$ case in considerable detail (including a conjecture for an optimal strategy) and prove that the optimal success probability, $V_n$, in the $n$-player game is a strictly decreasing function of $n$.