It also doesn't mention if Ballmer is able to change the number during the game. If he is only "thinking" about a number without writing it down beforehand, nothing stops him from doing it, as long as it is consistent with all the statements he made.

This trick actually makes the problem easier, you always need 7 tries, and the payoff, and not just the expected value is $940. Of course, if these are not whole numbers, the payoff is $0.

Now you can continue the interview whether you would cheat if you are guaranteed not to get caught, as it is the assumption you made for Ballmer here ;)