preference is reflected by the expected utilities of the two gambles. We reach a contradiction in our utility calculations. This contradiction indicates that the participant's preferences are inconsistent with the expected-utility theory. The Generality of the paradox The result holds regardless of the utility function. Indeed, the amount of the payoff is likewise irrelevant. Whichever gamble is selected, the prize for winning it is the same, and the cost of losing it is the same (no cost), so ultimately there are only two outcomes: receive a specific amount of money or nothing. Therefore, it is sufficient to assume that the preference is to receive some money to nothing (this assumption is not necessary: in the mathematical treatment above, it was assumed U($100) > U($0), but a contradiction can still be obtained for U($100) < U($0) and for U($100) = U($0)). In addition, the result holds regardless of risk aversion—all gambles involve risk. By choosing Gamble D, the participant has a 1 in 3 chance of receiving nothing, and by choosing Gamble A, a 2 in 3 chance of receiving nothing. If Gamble A was less risky than Gamble B, it would follow that Gamble C was less risky than Gamble D (and vice versa), so the risk is not averted in this way. However, because the exact chances of winning are known for Gambles A and D and not known for Gambles B and C, this can be taken as evidence for some sort of ambiguity aversion, which cannot be accounted for in expected utility theory. It has been demonstrated that this phenomenon occurs only when the choice set permits the comparison of the ambiguous proposition with a less vague proposition (but not when ambiguous propositions are evaluated in isolation). Possible explanations There have been various attempts to provide decision-theoretic explanations of Ellsberg's observation. Since the probabilistic information available to the decision-maker is incomplete, these attempts sometimes focus on quantifying the non-probabilistic ambiguity that the decision-maker faces – see Knightian uncertainty. That is, these alternative approaches sometimes suppose that the agent formulates a subjective (though not necessarily Bayesian) probability for possible outcomes. One such attempt is based on info-gap decision theory. The agent is told precise probabilities of some outcomes, though the practical meaning of the probability numbers is not entirely clear. For instance, in the gambles discussed above, the probability of a red ball is , which is a