Equimeasurable Rearrangements with Capacities
Mathematics of Operations Research, 40(2):429-445, 2015
30 Pages Posted: 28 Mar 2012 Last revised: 20 Jul 2015
Date Written: September 1, 2013
In the classical theory of monotone equimeasurable rearrangements of functions, "equimeasurability" (i.e., the fact the two functions have the same distribution) is defined relative to a given additive probability measure. These rearrangement tools have been successfully used in many problems in economic theory dealing with uncertainty where the monotonicity of a solution is desired. However, in all of these problems, uncertainty refers to the classical Bayesian understanding of the term, where the idea of ambiguity is absent. Arguably, Knightian uncertainty, or ambiguity is one of the cornerstones of modern decision theory. It is hence natural to seek an extension of these classical tools of equimeasurable rearrangements to situations of ambiguity. This paper introduces the idea of a monotone equimeasurable rearrangement in the context of non-additive probabilities, or capacities that satisfy a property that I call strong diffuseness. The latter is a strengthening of the usual notion of diffuseness, and these two properties coincide for additive measures and for submodular (i.e., concave) capacities. To illustrate the usefulness of these tools in economic theory, I consider an application to a problem arising in the theory of production under uncertainty.
Keywords: Ambiguity, Capacity, Non-Additive Probability, Choquet Integral, Monotone Equimeasurable Rearrangement
JEL Classification: C02, C65, D89
Suggested Citation: Suggested Citation