An axiomatic approach to the definition of the entropy of a discrete Choquet capacity

Authors:
Ivan Kojadinovic;Jean-Luc Marichal;Marc Roubens
Affiliations:
ícole polytechnique de l'Université de Nantes, LINA CNRS FRE 2729, Rue Christian Pauc, 44306 Nantes, France;Faculty of Law, Economics, and Finance, University of Luxembourg, 162A, Avenue de la Faıencerie, L-1511 Luxembourg, Luxembourg;Institute of Mathematics, University of Liège, Grande Traverse 12 (B37), B-4000 Liège, Belgium
Venue:
Information Sciences: an International Journal
Year:
2005

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Abstract

To extend the classical Shannon entropy to nonadditive measures, Marichal recently introduced the concept of generalized entropy for discrete Choquet capacities. We provide a first axiomatization of this new concept on the basis of three axioms: the symmetry property, a boundary condition for which the entropy reduces to the Shannon entropy, and a generalized version of the well-known recursivity property. We also show that this generalized entropy fulfills several properties considered as requisites for defining an entropy-like measure. Lastly, we provide an interpretation of it in the framework of aggregation by the discrete Choquet integral.