Gross substitution, discrete convexity, and submodularity

Authors:
V. I. Danilov;G. A. Koshevoy;C. Lang
Affiliations:
Laboratory of Mathematical Economics, Central Institute of Economics and Mathematics, Russian Academy of Sciences, Nakhimovskij Prospekt 47, Moscow 117418, Russia;Laboratory of Mathematical Economics, Central Institute of Economics and Mathematics, Russian Academy of Sciences, Nakhimovskij Prospekt 47, Moscow 117418, Russia;Department of Econometrics, University of Geneva, 40 Bvd du Pont d'Arve, 1211 Genève 4, Switzerland
Venue:
Discrete Applied Mathematics - Submodularity
Year:
2003

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Abstract

We consider a class of functions satisfying the gross-substitutes property (GS-functions). We show that GS-functions are concave functions, whose parquets are constituted by quasi-polymatroids. The class of conjugate functions to GS-functions turns out to be the class of polyhedral supermodular functions. The class of polyhedral GS-functions is a proper subclass of the class of polyhedral submodular functions. PM-functions, concave functions whose parquets are constituted by g-polymatroids, form a proper subclass of the class of GS-functions. We provide an additional characterization of PM-functions.