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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
I don't know if that applies to this scenario. In this game, the player is always in the lead until they aren't, but I don't see how that works in their favor.
Oh wait you mean the player has to stop if they lose? That's different.
Well, they have to start over with a $1 bet.
And, in real life, the house has a much larger bankroll.