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One problem is that, to figure out the long term payoff you want to consider the repeated case where you pick over and over.
But each event needs to be independent, so the amount of $$ you see doesn't give you a hint as to how likely the other hand is to have more money.
To do this, the amount of money in each trial needs to be completely random, with no upper limit because if there was a limit high amounts of money would give you an idea of how much is in the other hand.
Hence infinite amounts of money would have to be possible, and any amount of money from 1 cent to infinity would have to be equally likely -- this sort of setup is impossible.
Hence you cannot think in terms of repeating the trial over and over.
I think.
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