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:

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.

An optimal strategy wins with 3/4 probability.

Hint: Think about ways of grouping the eight possible hat configurations. And specifically, ways of grouping them that might relate to the 3/4 probability...

Solution: Strategy for each player: if you see two hats of the same color, guess the other color; if you see two different, write "no guess".

This wins with 3/4 probability because in 6 of the 8 configurations, there are two matching hats and one different -- in each of these, the person who does not match will correctly guess that he does not match while the other two will make no guess. In the other 2 configurations, all three hats match, and everyone will guess wrong.

If you make a table of all the possible configurations and the guesses made in each one, you'll find six correct and six incorrect guesses. But the way we obtain a better chance of winning than 1/2 is by concentrating the wrong answers in few configurations, and spreading out the right answers across many.

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

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