The Knob of the Discord

For (S; ) a measurable space, let C1 and C2 be convex, weak* closed sets of probability measures on . We show that if C1[C2 satis…es the Lyapunov property then there exists a set A 2 such that min 12C1 1(A) > max 22C2 2(A). We give applications to Maxmin Expected Utility and to the core of a lower probability. 1 Main result If 1 and 2 are two probability measures on a -algebra , then (by de…nition) 1 6= 2 means that there exists a set A 2 such that 1(A) > 2(A). Equivalently, the two disjoint sets f 1g and f 2g can be separated by means of a linear functional having an especially simple form, namely one that is de…ned by an indicator function. Here, we are concerned with extending this property to sets of measures which are not singletons. Our motivation stems from some questions arising in the theory of decision making under uncertainty. We o¤er a simple application to this area after we prove our separation result. However, given its nature, we expect our result to be widely applicable in areas di¤erent from the one we consider. Let (S; ) be a measurable space and let ( ) denote the set of all (countably additive) probability measures on . ( ) is a subset of the norm dual of the Banach space B( ) of bounded, -measurable functions. De…nition 1 Let C = f igi2I ( ). We say that C has the Lyapunov property if the range of the vector measure ( i)i2I on E is a convex and compact subset of R (equipped with the product topology), for all E 2 . Notice that if C has the Lyapunov property, then so does any subset of C. Sets of measures with the Lyapunov property have special importance in the theory of decision making under uncertainty. For a decision maker described by a set of priors like in [8] or in [7], the Lyapunov property corresponds to the demand that the class of unambiguous events in the sense of [12] or [7] be “rich” (see Section 2).