Spoiler for Still no improvement..

To arrive at a working strategy, the information gathered (for example, the number of apples in the first room) must influence the probability of "finding more apples in the other room". For example, thea priorichance that "room 1 has more apples than room 2" is 0.5. After seeing the number of apples in room 1, this number (0.5) has to change in a predictable way to a posteriori chance that "room 1 has more apples than room 2". Indeed, that would happen if Bob knew the maximum number of apples ('MAX') that Alice is going to keep in the rooms. A strategy can then be formulated..

A change in probability is what happens in case of Monty Hall problem - the information that a third door has a goat changes the probability of finding a car behind doors 1 and 2..

However, in case of OP, Bob doesn't know 'MAX'. So I think what Bob needs to do is:

a. Open room 1 and accept it always. Use the number of apples in room 1 to make an estimate of 'MAX'. I think 2*x is the best estimate for 'MAX', where x = number of apples in room 1.

b. Open room 2. Accept it if the number of apples in room 2 is greater than half of the MAX estimate; else reject it.

..but this turns out to be the same approach as k-man.

I think you guys need a little something.

Spoiler for small hint

The strategy isn't as complicated as you might think, and it makes sense intuitively (unlike Monty Hall which is pretty counterintuitive), although it does take some math to prove.

Spoiler for medium hint

Only one step in k-man's solution must be changed/augmented to arrive at the superior strategy.

Spoiler for big hint

There is no "optimal" strategy. The answer isn't a fixed solution but rather a set of solutions that work better than 50% whose actual effectiveness varies based on the distribution of apples. Thus, the final winning percentage is not fixed either.

Spoiler for giant hint

I will save this one for later for if you guys really need it.