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

The n-player game

n players line up, one behind the next, all facing north. Hats are placed on their heads. Thus player j can see the colors of the hats on the heads of players 1 through j-1, but not her own or those of players j+1 through n. In order from player n down to player 1, each player must guess her hat color, out loud. So player j has heard the guesses made by players n down to j+1 when she has to make her guess, but does not know which or how many of those earlier guesses were correct.

At the end, if k of the n guesses were correct, everyone splits k million dollars.

Again, the players agree on a strategy ahead of time but cannot communicate once the game begins.

Obviously if everyone guesses randomly, expected payoff is n/2 million. But with that strategy, in the worst case, everyone might happen to guess wrong, and they get nothing. So can you guarantee that some guesses are correct? Can you guarantee that strictly more than n/2 are correct? How close to all n can you guarantee?

You can guarantee all but the very first guess. That is, you can ensure a win of at least n-1 million dollars.

(a hint for how)

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

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