I am stuck on this question, Three people are going to play a cooperative game. They are allowed to strategize before the game begins, but once it starts they cannot communicate with one another. The game goes as follows. A fair coin is tossed for each player to determine whether that player will receive a red hat or a blue hat, but the color of the hat (and result of the coin toss) is not revealed to the player. Then the three players are allowed to see one another, so each player sees the other two players’ hats, but not her own. Simultaneously, each player must either guess at the color of her own hat or ‘pass’. They win if nobody guesses incorrectly and at least one person guesses correctly (so they can’t all pass). The players would like to maximize their probability of winning, so the question is what should their strategy be? A naive strategy is for them to agree in advance that two people will pass and one person (designated in advance) will guess either red or blue. This strategy gives them a 50% chance of winning (verify this!), but it is not optimal. Devise a strategy that gives the players a greater probability of winning.
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Three people are going to play a cooperative game. They are allowed to strategize before the game begins, but once it starts they cannot communicate with one another. The game goes as follows. A fair coin is tossed for each player to determine whether that player will receive a red hat or a blue hat, but the color of the hat (and result of the coin toss) is not revealed to the player. Then the three players are allowed to see one another, so each player sees the other two players’ hats, but not her own. Simultaneously, each player must either guess at the color of her own hat or ‘pass’. They win if nobody guesses incorrectly and at least one person guesses correctly (so they can’t all pass). The players would like to maximize their
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