Additive representations 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. The authors use an isomorphism between non-additive measures on the original space (of states of the world) and additive ones on a larger 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: (a) the Choquet integral with respect to any totally monotone capacity is an average over minima of the integrand; (b) the Choquet integral with respect to any capacity is the difference between minima of regular integrals over sets of additive measures; (c) under fairly general conditions one may define a ‘Radon-Nikodym derivative’ of one capacity with respect to another; (d) the ‘optimistic’ pseudo-Bayesian update of a non-additive measure follows from the Bayesian update of the corresponding additive measure on the larger space. The authors also discuss the interpretation of these results and the new light they shed on the theory of expected utility maximization with respect to non-additive measures.