I think in some important sense this is the telling limit of why Coscott is right
Right about what? The hint I give at the beginning of the solution? My solution?
Watch your quantifiers. The strategy you propose for Bob can be responded to by Alice never putting 0 apples in any room. This strategy shows that Bob can force a tie, but this is not an example of Bob doing better than a tie.
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.
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.
What? I do not believe Alice can defeat my strategy. She can get arbitrarily close to 50%, but she cannot reach it.
Right about what? The hint I give at the beginning of the solution? My solution?
Watch your quantifiers. The strategy you propose for Bob can be responded to by Alice never putting 0 apples in any room. This strategy shows that Bob can force a tie, but this is not an example of Bob doing better than a tie.
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.
What? I do not believe Alice can defeat my strategy. She can get arbitrarily close to 50%, but she cannot reach it.