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If both players have infinite bankrolls, but only one of them is allowed to stop the game once they are ahead, the one with the option of stopping has an advantage. They can play until they are in the lead, then stop. The reason this doesn't work in real life is that real bankrolls aren't infinite.
See also: https://en.wikipedia.org/wiki/Gambler%27s_ruin
And, in real life, the house has a much larger bankroll.