arXiv:1407.4711v3 [math.CO] 29 Sep 2015 ON A CERTAIN COOPERATIVE HAT GAME JONATHAN KARIV, CLINT VAN ALTEN, AND DMYTRO YEROSHKIN Abstract. In 2010, Lionel Levine introduced a cooperative game, and posed the question of finding the optimal strategy. We provide an overview of prior work on this question, and describe several strategies that give the best lower bound on the probability of victory. We further generalize the problem to consider the case of arbitrary color distribution. 1. Introduction Lionel Levine introduced the following cooperative game (his initial version was 100 players, modified to 2 players by Tania Khovanova [Kho11]): Two players each have countably many hats put on their heads. Each hat is either black or white with equal probability. Further- more, the players are only able to see the hats on the other person’s head. Simultaneously each player points to a hat on their own head. They win if both players pick out a white hat. The question is what is the optimal strategy for this game? It should be noted that each individual player will pick a white hat with probability one half regardless of the strategy employed. The challenge then is to correlate their choices. At a first glance, it may seem that there is no strategy with better than random (0.25) chance of winning. This can be quickly discounted by the following simple strategy: Each player looks for the first white hat on the partner’s head and chooses the corresponding hat on his or her own head. Observe that if both players have a white hat first (0.25 chance) then they win, if the two first hats are different ( 1 2 chance) then they lose, and if the two first hats are both black (0.25 chance) then effectively they replay the game with the first hat removed. So, we see that the chance of winning is x =0.25 + 0.25 · x, so x =1/3. In this paper we consider the more general problem in which the probability of a white hat in any place is some fixed p, where p ∈ [0, 1]. As is usual we set q := 1 − p. Using the strategy above gives a probability of winning as p 2−p , which comes from the equation x = p 2 + (1 − p) 2 x. Similarly, we can obtain a probability of winning of 2p 2 1+p by having each player choose the hat on their head corresponding to the first black hat on the other player’s head. It is easy to check that this strategy is as good as the above one for p = 1 2 , but does better for 1 2 <p< 1 and worse for 0 <p< 1 2 . Given a strategy S, we denote by V S (p) the probability of winning for the strategy S in the game with probability p of any hat being white. Then, define V (p) := sup{V S (p): S is any strategy}. Intuitively, this is the probability of winning under optimal play. 1