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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 onlyifalltheirguessesarecorrect. Inthispaper, wegiveanoverviewofwhatisknown aboutstrategiesforthisgame, includinganextendeddiscussionofthecasewithn= 2 players (and a conjecture for an optimal strategy in this case). We also prove that Vn, the optimal value of the joint success probability in the n-player game, is a strictly decreasing function of n.