The way forward is to make Ballmer pay with time for screwing with you, which gets us into geopolitics, and then using the resulting MAD dynamics to make the game fair again. That's how adults with keys to the nukes do it :).

And then everyone gets nukes, or at least anti-matter mined in some vacuum chamber copperstatue configuration.

The ultimate conclusion of which is likely that both parties will decay to picking the secret value/first guess randomly (although I'm not sure if the optimal distribution is perfectly flat?), which is also something that we can model.

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.

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.

Never go in against a Sicilian when death is on the line!

I have not really studied this but maybe choosing the guess randomly when the number of possibilities is even is already enough to counter an adversarial opponent. Note that 50 is not the only 'optimal' guess in the beginning. 51 is just as good.

Any number between 36 and 64 should be as good!

I’d pay $5 to watch a short film of Ballmer asking this question to Wallace Shawn’s Vizzini.

So the actual problem here is to find Nash equilibrium.

Yes, this is a Nash equilibrium question.