uses the probability distribution associated with the worst scenario. The DM chooses the act whose worst-case scenario is better than the worst-case scenarios of the other acts. See the Appendix for a numerical example. In practical modeling and implementation, the specification of the set Ψ of the DM becomes another issue. Hansen and Sargent (2008) presents a useful solution in the context of modeling in macroeconomics. We first introduce Kullback’s cross-entropy function (Cover and Thomas, 2012) to quantify the extent to which two probability distributions differ from one another. Definition 1 (Kullback’s Cross-Entropy Function) Suppose that there are two probability distributions p and q with the common support set S. Suppose that q is the prior density over the set S . Then, the Kullback’s cross-entropy function is defined as (2) (Cover and Thomas, 2012). A large value of S ( p, q ) implies that p and q are very different from one another. Hansen and Sargent (2008) use cross-entropy to restrict the set of probability distributions considered by the DM. Given the prior distribution q and a parameter η , the DM’s set Π includes all probability distributions p for which S ( p, q ) ≤ η . As long as the probability distributions are not too different from p , in which case S ( p, q ) > η , the DM considers those probability distributions to be equally acceptable. A large value of the parameter η quantifies the DM’s strong ambiguity aversion. 4 With a larger value of η , the DM regards a larger set of probability distributions as candidates for accurate descriptions of the world. Hence, a large η is equivalent to saying that the DM is more ambiguous about the real world. The CPCAM allows the common market maker to simultaneously handle different types of contin- gent claims as long as the claims are written on the same uncertain event (e.g., the outcome of the $^{4}$Illeditsch (2011) also uses the size of the set of possible models under the DM’s consideration as a proxy for the DM’s level of ambiguity aversion. 3 The Microeconomic Analysis of the Convex Pari-mutuel Call Auction Mechanism (CPCAM) 3.1 The Market Setting 6 S ( p, q ) = ∫ S p ( x ) ln [ p ( x ) q ( x ) ] dx (2)