Right about it not being a fair game. My first thought was that it really is a fair game and that by comparing only the cases where fixed numbers a, b, and c are distributed you get the slight advantage for Bob that you claimed. That if you considered ALL possibilities you would have not advantage for Bob.

Then I thought you have a vanishingly small advantage for Bob if you consider Alice using ALL numbers, including very very VERY high numbers, where the probability of ever taking the first room becomes vanishingly small.

And then by thinking of my strategy, of only picking the first room when you were absolutely sure it was correct, i.e. it had in it as low a number of apples as a room can have, I convinced myself that there really is a net advantage to Bob, and that Alice can defeat that advantage if she knows Bob's strategy, but Alice can't find a way to win herself.

So yes, I'm aware that Alice can defeat my 0 apple strategy if she knows about it, just as you are aware that Alice can defeat your 2^-n strategy if she knows about that.