Three players enter a room and a red or blue hat is placed on each
person's head. The color of each hat is determined by a coin toss,
with the outcome of one coin toss having no effect on the others.
Each person can see the other players' hats but not his own.
No communication of any sort is allowed, except for an initial
strategy session before the game begins. Once they have had a
chance to look at the other hats, the players must simultaneously
guess the color of their own hats or pass. The group shares a
hypothetical $3 million prize if at least one player guesses
correctly and no players guess incorrectly. What strategy should they use to maximize their chances of success?
(From - http://www.princeton.edu/~sjmiller/riddles/riddles.html)
A strategy is a mapping of viewed hats to guessed hat. For example,
RR -> B
BB -> R
RB -> X
BR -> X (X = no guess)
or
S = {B, R, X, X}. (Note this is the proposed best strategy) To obtain the best strategy obtain the probabilities for all possible S. This is a total
of 12 strategies or less if you eliminate with
symmetry..
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