Decomposition - Integral : Unifying Choquet and the Concave

This paper introduces a novel approach to integrals with respect to capacities. Any random variable is decomposed as a combination of indicators. A pre-speci ed set of collections of events indicates which decompositions are allowed and which are not. Each allowable decomposition has a value determined by the capacity. The decomposition-integral of a random variable is de ned as the highest of these values. Thus, di erent sets of collections induce di erent decomposition-integrals. It turns out that this decomposition approach uni es well known integrals, such as Choquet, the concave and Riemann integral. Decomposition-integrals are investigated with respect to a few essential properties that emerge in economic contexts, such as concavity (uncertainty-aversion), monotonicity with respect to stochastic dominance and translation-covariance. The paper characterizes the sets of collections that induce decomposition-integrals which respect each of these properties. JEL Classi cation: C71, D80, D81, D84