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What color hat am I wearing?   ...   random thoughts , logic , math
tue 2005-may-10 15:07:22 pdt   ...   permalink

The 3-player game

As in the 2-player game, three players sit around a table, and each can see the other players' hats but not her own. Each player may then write down a guess, "red" or "blue", but this time they have the option to instead write "no guess". As before, all players must write down their guess (or lack thereof) without knowing what the other players have written. The guesses are then revealed and:

  • if everyone wrote "no guess", they get nothing;
  • if at least one person made a guess and was wrong, they get nothing;
  • if at least one person made a guess, and everyone who made a guess was right, then they all get a million dollars.
  • As always, they agree on strategy in advance but cannot communicate once the game begins. This time we will assume each hat color is selected uniformly at random. One simple strategy would be to decide that no matter what, player 1 will guess "red" and players 2 and 3 will write "no guess"; this gives a 1/2 chance of winning. Can you do better than 1/2? Can you guarantee a win?

    No, you cannot guarantee a win.

    (likelihood of winning with an optimal strategy, and a hint)

    Please do NOT post solutions in the comments below. Thanks.

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