Re: allowing your opponent to martingale you
It has an effect on that particular day's variance.
It has no effect weird effect on your EV. If you are equal players, your EV is 0, no matter if you double the stakes after every game or not.
As someone said it becomes more like a tournament, yes that is correct. You have a smaller chance of winning, but if you win, the profit will be proportionally larger.
So in the long run, it matters [censored]. If he thinks it gives him an advantage and that makes him happy, go along and double those stakes.
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