Stuart_Armstrong comments on Expected utility without the independence axiom - Less Wrong
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I'll try and do the maths. This is somewhat complex without independence, as you have to estimate what the total results of following a certain strategy is, over all the bets you are likely to face. Obviously you can't money pump me if I know you are going to do it; I just combine all the bets and see it's a money pump, and so don't follow it.
So if you tried to money pump me repeatedly, I'd estimate it was likely that I'd be money pumped, and adjust my strategy accordingly.