3.1. The Ellsberg Paradox
The Ellsberg paradox is a well-known, widely discussed example concerning ambiguity in decision making, which has stimulated development of many alternatives to the classic expected utility theory. We shall present Ellsberg's original decision problems below.
Suppose that there is an urn containing 300 balls, among which 100 balls are known to be red, and the remaining are either blue or green with the exact proportion unknown. More precisely, there are k blue or green balls in the urn, where k can range from 0 to 200. Due to this partial information (ambiguity), the probabilities of the event that a blue ball is drawn and that a green ball is drawn are both indeterminate.
Suppose that an individual is invited to make a decision in each of the choice problems depicted by Table III and Table IV respectively. More specifically, the participant needs to choose between two options in each of the cases. The details of the payoffs associated with the options are given in the tables below. The participant is then asked to answer the following two questions: (i) given the choices a1 and a2 in Table III, which one do you prefer? (ii) given the choices a3 and a4 in Table IV, which one do you prefer?
Table III. First decision problem
| ||r: red||b: blue||g: green|
Table IV. Second decision problem
| ||r: red||b: blue||g: green|
3.2. The Machina Paradoxes
In the same manner as Ellsberg, Machina presents examples of ambiguous choice problems where the typical ambiguity-averse preferences cannot be explained by both the expected utility theory and the CEU theory. After the description of the example, we shall show in detail how the Machina-styled paradoxical choices violate the CEU theory.
Now consider the choice between a pair of options as described in Table V, where means that the ball with number k is drawn (). Which one would you prefer?
Table V. First decision scenario
Then suppose that you are offered another pair of options as depicted in Table VI. Which one would you prefer?
Table VI. Second decision scenario
In Ref. , Machina has convincingly argued that the plausible preferences over these two pairs of options should be like this: and , which exhibit a similar preference pattern as observed in Ellsberg's example. However, both the expected utility theory and the CEU theory preclude such a preference pattern with regard to Machina's choice problem. It is relatively easy to see how this violates the expected utility theory, and so we omit the detailed explanation. Below we demonstrate the paradoxical aspect of Machina's example using the CEU theory.
Let us start with the calculation of the Choquet expected utilities for the options in the first scenario. Let and be a capacity. By formula (1), we obtain the following equations:
With regard to the second scenario, we have the following equations:
On one hand, if you strictly prefer a1 to a2, that is, , the representation theorem requires that , which means that:
This implies the following result:
On the other hand, if you strictly prefer a4 to a3, that is, , the theorem requires that , that is,
which implies the following result:
The inequalities in (4) and (5) together show that according to the CEU theory, your preferences in these decision situations should satisfy that if and only if . However, Machina has argued that the reasonable choice pattern for these problems typically does not obey such a requirement. More precisely, Machina demonstrates that this example indeed violates the comonotonic independence axiom, which is a weaker assumption in comparison to the independence axiom in the expected utility theory. Since the Machina-styled preference relations, namely and contradict this comonotonic independence principle, it immediately follows that these preferences cannot be accommodated within the framework of the expected utility theory, since the independence axiom is one of its fundamental assumptions. Such a phenomenon is also commonly referred to as the 50:51 example in the literature.
Besides the above example, Machina also presents another choice problem known as the reflection example, which poses difficulties to the CEU theory as well. Unlike the previous setting, assume that you have an urn with 100 balls marked with number 1, 2, 3, or 4. You are informed that half of the balls are marked with 1 or 2, and the other half of them are marked with 3 or 4. Similarly, you do not have any further information concerning the exact proportion of the balls. Now consider the decision problem described in Table VII. Which option would you prefer?
Table VII. Reflection example
Intuitively, according to the available information given above, it seems reasonable to say that the events and are equally likely, and there is not enough information for determining which event with a single state occurring is more likely. In this case, we call the states s1 and s2 (respectively, s3 and s4) informationally symmetric. Observe that the option a5 can be obtained from a8 by exchanging the payoffs in those informationally symmetric states. The same is true for the options a6 and a7. In other words, the payoffs of one option are the “mirrored” images of those associated with the other option. For this reason, we say that a5 (respectively, a6) an (informationally symmetric) reflection of a8 (respectively, a7). In view of this, it seems plausible to expect that a decision maker with the preference should have the “reflected” preference ranking, namely , since being indifferent between reflection options seems quite sensible. However, we show below that such a requirement turns out to be inconsistent with the CEU.
Suppose that you strictly prefer a5 to a6. According to the representation theorem, we have that
which implies the following inequality:
Similarly, if you also have a strict preference of , it follows that the exact same inequality as Equation (6) should hold as well. Thus, the CEU theory requires that if and only if .
As a matter of fact, the Machina paradoxes are quite powerful in the sense that several existing decision theories prohibit the commonly observed patterns of preference. It has been shown in Ref.  that the Machina paradoxes raises serious challenges not only to the CEU theory, but also to a number of widely discussed models of decision making under uncertainty.