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Stuart_Armstrong comments on Continuity axiom of vNM - Less Wrong
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At the very minimum, you need to have a distributivity law for lotteries. If 50%(50%A+50%B)+50%(50%A+50%C) is not defined to be the same thing as 50%A+25%B+25%C, then it's easy to find counter-examples...
Yes, I think in the classical conception of lotteries these are regarded as the same. You could reject that, but it seems like it would be similar to how some people think that (A, when B was on the table) is a different outcome from (A, when B was not on the table), and so may be assigned a different utility.