[jstanley]

> the win probability will always be >0.5 so long as the "prior" probability distribution has a nonzero probability of being between Alice's two numbers.

Wow. This is the key piece of information that makes the problem interesting, IMO. That's quite unintuitive.

If you know anything at all about how your opponent chooses numbers, you win in the long term.

[kd5bjo]

It also tells us what Alice's optimal strategy is: pick the first number at random, and then select an adjacent integer as the second number. Thus, there's no space between them that your prior can assign any probability to.

[OscarCunningham]

Well your strategy has to have some way of breaking ties, when Alice's number is the same as yours. Lets say that you always say "higher" in that case. Then you always win whenever your number is between Alice's or equal to the smaller of them. Equivalently you could just pick a random half-integer.