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the house can only make $1 per play, and the bettor can make a functionally unlimited amount.
see the martingale strategy. you are basically sticking the house with a martingale strategy in which you get to decide when they bet.
But the odds of the player managing to do so are proportionate. In theory, if 8 players each decide to go for three rounds, one of them will win, but the losings from the other 7 will pay for that player's winnings.
You're right that the house is performing a Martingale strategy. That's a good insight. That may actually be the source of the house advantage. The scenario is ideal for a Martingale strategy to work.