Bill -- How do you deal with collusion? If Dylan and I agree to pool our winnings, surely we want to bracket your guess and box you out entirely, if we can. --Michael On Tue, Jul 12, 2011 at 9:56 PM, Bill Thurston <wpthurston@mac.com> wrote:
One of my cousins asked me this question: Suppose 3 people make secret guesses of a number, uniformly randomly chosen from 1--100. Whoever is closest wins. What is their "best" strategy?
I'd like to modify this to let the unknown number to be uniformly chosen real number in [0,1]. In the event of ties, let's say the winner is chosen uniformly from all best guesses (or the "prize" is distributed equally, I don't care). By "best", I want a Nash optimal probability distribution for the guesses of the the three participants --- that is, if each one knows the probabilities with which the others choose their guess, they can do no better than using the same distribution.
Note that with two guessers, a guess of .5 wins over everything except another guess of .5, when it ties.
For 3 guessers, it's not optimal for all three to pick .5, because .49 then wins against the other two. It's also not optimal to pick a number uniformly at random from the unit interval: if two of the players do that a guess of .5 has an expectation of 5/12.
Bill Thurston
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