The von Neumann-Morgenstern theorem states that, assuming a person's preferences under risk satisfy certain rationality axioms, then there exists a utility function u, the von Neumann utility function, such that the person will tend to maximize the expected value of u. For this reason, the hypothesis that people satisfy the von Neumann-Morgenstern rationality axioms is known as expected utility theory. Now one of the major challenges for expected utility theory is the Ellsberg Paradox. It goes as follows.
Suppose you have an urn which has a total of 90 balls, 30 of which are red and each of the other 60 balls is either black or yellow. And suppose a ball is drawn at random from the urn. Then would you rather have lottery A, where you get 100 dollars if a red ball is drawn, or lottery B, where you get 100 dollars if a black ball is drawn? Most people would prefer lottery A. And would you rather have lottery C, where you get 100 dollars if you draw a red or yellow ball, or lottery D, where you get 100 dollars if you draw a black or yellow ball? Most people would prefer lottery D. But the thing is, preferring both lottery A to lottery B and lottery D to lottery C is inconsistent with expected utility theory; see this Wikipedia article for the proof.
I'd like to understand this logic a little better. Consider a new urn that just has 60 balls, and each ball is either black or yellow. Then would you rather have lottery A', where you get 30 dollars guaranteed, or lottery B', where you get a dollar for every black ball in the urn? I think most people would prefer lottery A'. And would you rather have lottery C', where you get 30 dollars plus a dollar for every yellow ball in the urn, or lottery D', where you get 60 dollars guaranteed? I think most people would prefer lottery D'.
So my question is, does it violate expected utility theory to prefer both A' to B' and D' to C'? I think it's consistent with expected utility theory. Assuming I'm right, couldn't you just change the currency from "dollars" to "lottery tickets" everywhere in my example, where a lottery ticket entitles you to a 1/90 chance of getting a 100 dollars? That would transform my example into the example in Ellsburg's paradox. So where's the flaw in my reasoning?