If you accept both properties and you violate independence, you can be money-pumped. Here is how it works, concretely. Suppose your preference between gambles A and B depends on what the common component C is (as the independence axiom says it shouldn't). Before the uncertainty resolves, you evaluate the compound lottery holistically and prefer the plan involving B (because, in combination with the C branch, B produces a better overall distribution). But then the coin comes up heads, the C branch is now off the table, and you find yourself choosing between

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Here is the specific confusion that matters for our purposes. When someone says "a rational agent maximizes expected utility," this sounds, to a casual listener, like it means "a rational agent computes the probability-weighted average of their subjective values across all possible outcomes." In other words, it sounds like the agent takes f1, the function representing how good each outcome feels or how much they value it, and averages it across possible worlds, weighted by probability. This would mean that the agent literally values a gamble at the weighte

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I think the most natural fix within the VNM theory is to just say S' and D' are the events "car is awarded so son/daughter based on a coin toss", which are slightly better than S and D themselves, and that F is really 0.5S' + 0.5D'. Unfortunately, such modifications undermine the applicability of the VNM theorem, which implicitly assumes that the source of probabilities itself is insignificant to the outcomes for the agent.  Luckily, Bolker⁴ has divised an axiomatic theory whose theorems will apply without such assumptions, at the expense of some uniquenes

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