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Stuart_Armstrong comments on Expected utility without the independence axiom - Less Wrong
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It's enough to show that an agent cannot be repeatedly money-pumped. The more opportunities for money pumping, the less chances there are of it succeeding.
Contrast household applicance insurance versus health insurance. Both are a one-shot money-pump, as you get less than your expected utility out of then. An agent following these axioms will probably health-insure, but will not appliance insure.
Can you write out the math on that? To me it looks like the Allais Paradox or a simple variant would still go through. It is easy for the expected variance of a bet to increase as a result of learning additional information - in fact the Allais Paradox describes exactly this. So you could prefer A to B when they are bundled with variance-reducing most probable outcome C, and then after C is ruled out by further evidence, prefer B to A. Thus you'd pay a penny at the start to get A rather than B if not-C, and then after learning not-C, pay another penny to get B rather than A.
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.