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Seems like the distribution definitely won’t be flat since the guesser can randomly choose any of the numbers from 37 to 64 as a first guess without losing anything on the large side, so Ballmer starting with any of those increases his chance of having to pay out the $5. Likewise for other numbers there are nuances to what can be guessed.

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But if you assume that the opponent knows that they shouldn't pick between 37-64, doesn't that change your odds?

The game theory here is similar to another quiz "Guess 1/3 of the average".

You are really trying to guess how deep the other has thought about the problem, so you can tell which strategy they settled on, and then you adapt your strategy based on that. Of course, it's a loop.

Yep, definitely not just what I said. I was saying that guaranteed it won't work out to an even distribution, not what the distribution would be.

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