Additive Representation of Non-Additive Measures and the Choquet Integral

This paper studies some new properties of set functions (and, in particular, "non-additive probabilities" or "capacities") and the Choquet integral with respect to such functions, in the case of a finite domain. We use an isomorphism between non-additive measures on the original space (of states of the world) and additive ones on a large space (of events), and embed the space of real-valued functions on the former in the corresponding space on the latter. This embedding gives rise to the following results: the Choquet integral with respect to any totally monotone capacity is an average over minima of the inegrand; the Choquet integral with respect to any capacity is the differences between minima of regular integrals over sets of additive measures; under fairly general conditions one may define a "Radon-Nikodym derivative" of one capacity with respect to another; the "optimistic" pseudo-Bayesian update of a non-additive measure follows from the Bayesian update of the corresponding additive measure on the large space. We also discuss the interpretation o these results and the new light they shed on the theory of expected utility maximization with respect to non-additive measures.