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If Ballmer is being adversarial, he won’t pick the number at the start, and always win.

Of course you can set up the game such that Ballmer has to commit on a number at the start of the game (by sealing it in an envelope or whatever), but that wasn’t specified.

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Ballmer opens with "I'm thinking of a number between 1 and 100". If he uses your strategy instead that's a different scenario.

That’s only if you’re willing to trust Ballmer to do what he claims, which I wouldn’t.

Do you trust Ballmer to give you $1 in the cases where he's said he's going to give you $1? If not the EV calculation looks pretty bad...

they’re being adversarial within the framed rules of the game, not breaking the rules of the game?

Without writing the number down, it's up to Ballmer to decide that aspect, because you cannot look into his brain, or prove that he didn't commit to a prior number. Therefore, it's fair game.

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