[alanbrown]

I'm not sure I can claim to know you well but I think I know why you didn't like Busto's answer...

Busto's reference to the binomial distribution was unnecessary. It doesn't enter into the logic required here and therefore only served to confuse. Throwing in unnecessary equations changes the landscape of the discussion to Why Is This Equation Unnecessary, rather than What Is The Answer To The Solution. That is what is to be disliked about his answer.

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variance is maximized when your winrate is 50%. so people who think they can lower variance by playing less optimal are mistaken.

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I have no idea what this is trying to communicate.

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play as optimal as you can and you will maximize the probability that you win that leg of the match and the match itself.

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Agreed. Though the question we're trying to answer is does Play As Optimally As You Can change as you accumulate more wins.

The simple key here is 'independent events'. If each game is an independent event with the same starting conditions (chips, who deals, randomness of cards, rules) and the same goal (win the other's chips), then the optimal strategy must not have changed to succeed at it.

Let's imagine that the villain has amnesia and has no memory of previous games we've played or what the current scoreline is. His game is optimized to win This Game. Not a series of games. There is no 'series of games' to him. Just this one. And our strategy that is optimized to win This Game can not be improved upon by knowing we won't need to win another one afterwards.

All the points Busto made about variance and probability are unrelated to this issue. And unrelated points are harder to refute than false ones. I think that's why you hated his answer.