[akoboldfrying]

This should be higher.

The OP's article is interesting, but it assumes a very weak notion of "adversarial", in which Ballmer still commits to some initial choice.

Interestingly it's actually possible for a player to know this is the case if Ballmer uses a commitment scheme [1]. For example, at the start of the game Ballmer could generate 500 random bits, append his chosen number in the range 1-100 to this, hash the result and then send you that hash: At the conclusion of the game, he sends you the 500 random bits, and you can check that concatenating his chosen number (now revealed) to those bits and hashing the result produces the hash he originally sent. (If Ballmer lies and changes his number, he would need to somehow come up with 500 bits that when concatenated with this different number still produce the original hash. This is hard.)

[1]: https://en.wikipedia.org/wiki/Commitment_scheme