We study preferences over Savage acts that map states to opportunity sets and satisfy the Savage axioms. Preferences over opportunity sets may exhibit a preference for flexibility due to an implicit uncertainty about future preferences reflecting anticipated unforeseen contingencies. The main result of this paper characterizes maximization of the expected indirect utility in terms of an ‘Indirect Stochastic Dominance’ axiom that expresses a preference for ‘more opportunities in expectation.’ The key technical tool of the paper, a version of Möbius inversion, has been imported from the theory of nonadditive belief functions; it allows an alternative representation using Choquet integration, and yields a simple proof of Kreps' (1979) classic result.

IN AN INTERESTING RECENT PAPER, Sarin and Wakker (1992) (henceforth S-W) have provided a new axiomatization of expected utility maximization with (CEU). Its simplicity is attractive in that it provides a constructive interpretation of the role of capacities and Choquet integration in the main representation theorem. An important issue raised by the paper is the interpretation of the key Axiom P4 (Cumulative Dominance). S-W suggest an appreciation of the axiom as an adaptation of principles to nonadditive-probability contexts, most eloquently on page 1260. If viable, such an interpretation would supply the CEU model with a powerful intuitive foundation that has been lacking so far. This note argues that a interpretation can be maintained only at the price of either an arbitrary choice of specification which undermines its intuitive force or, alternatively, of an unintended restriction of the class of characterized preferences. Two logically independent arguments are presented. The first points out an arbitrariness in the of the more-likely-than relation in terms of preferences (Proposition 1). The second shows a similar arbitrariness in the of a stochastic dominance relation in terms of a more-likely-than relation (Proposition 2). In each case, the invoked symmetry conditions yield a characterization of CEU preferences with symmetric capacities, a nontrivial generalization of the SEU model that has received little attention in the literature. For notation and definitions, the reader is referred to S-W's paper. S-W's concern is to develop an intuitively convincing axiomatization of CEU-representable preference relations. Their key Axiom P4 is formulated in terms of a more-likelythan relation > on the algebra v of events that is defined in terms of the preference relation a on the set of acts Y To facilitate the subsequent discussion, their definition is introduced here formally as a condition on the pair of relations (a, >):

How can diversity be measured? What does it mean to value biodiversity? Can we assist Noah in constructing his preferences? To address these questions, we propose a multi-attribute approach under which the diversity of a set of species is the sum of the values of all attributes possessed by some species in the set. We develop the basic intuitions and requirements for a theory of diversity and show that the multi-attribute approach satisfies them in a flexible yet tractable manner. A natural starting point is to think of the diversity of a set as an aggregate of the pairwise dissimilarities between its elements. The multi-attribute framework allows one to make this program formally precise. It is shown that the program can be realized if and only if the family of relevant attributes is well-ordered (“acyclic”). Moreover, there is a unique functional form aggregating dissimilarity into diversity, the length of a minimum spanning tree. Examples are taxonomic hierarchies and lines representing uni-dimensional qualities. In multi-dimensional settings, pairwise dissimilarity information among elements is insufficient to determine their diversity. By consequence, the qualitative and quantitative behavior of diversity differs fundamentally.