Hats: all or nothing
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N distinguishable players are randomly fitted with a white or black hat, where the probabilities of getting a white or black hat may be different, but known and the same to all the players. All players guess simultaneously the color of their own hat observing only the hat colors of the other N-1 players. The team wins if all players guess right. No communication of any sort is allowed, except for an initial strategy session before the game begins. Our goal is to maximize the probability of winning the game and to describe optimal strategies.
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