Extending the Choquet integral

In decision under uncertainty, the Choquet integral yields the expectation of a random variable with respect to a fuzzy measure (or non-additive probability or capacity). In general, for the discrete setting, this technique allows to integrate functions taking values on a finite n-set with respect to a (fuzzy) measure taking values on subsets of such a set. Yet, the integrand may well be treated as an additive function taking values on subsets itself: the value associated with each subset is simply the sum of the values associated with all the atoms (or 1-cardinal subsets) in that subset. The Choquet technique is here extended to the case where the integrand, just like the measure, is a nonadditive function taking values on subsets itself. The resulting aggregation operator is an extension of the Choquet integral: the former coincides with the latter whenever the integrand is additive. Four such extensions are provided, two of which are obtained by means of the Möbius inversion of the integrand and the (fuzzy) measure with respect to which integration is performed. In all cases, the resulting integral is an extension of the measure: it coincides with this latter on the vertices of the n-dimensional unit hypercube. Yet, one of these extensions also inherits another main feature of the (traditional) Choquet integral: if the fuzzy measure is convex, then this extended Choquet integral equals its minimum over all probabilities in the core of the measure. The general technique applies to both monotone and antitone integrands, and when the integrand is real-valued (i.e., taking both positive and negative values) it allows for both a symmetric and an asymmetric form. Two conceivable applications are provided. One furnishes an expectation of diversity in a random sample of a known population. Here the integrand is a diversity function, which is monotone by construction. The other application furnishes the certainty equivalent for a problem of decision making under uncertainty where the decision maker has some belief about what information (in the form of an event containing the ’true’ state) will be available before taking action. Here the integrand is antitone. MSC2000 classification numbers: 49N30, 62C10.