New characterizations of M-convex functions and their applications to economic equilibrium models with indivisibilities

Authors:
Kazuo Murota;Akihisa Tamura
Affiliations:
Graduate School of Information Science and Technology, University of Tokyo, Tokyo 113-8656, Japan and Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan;Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
Venue:
Discrete Applied Mathematics - Submodularity
Year:
2003

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Abstract

The concept of M-convex functions plays a central role in "discrete convex analysis", a unified framework of discrete optimization recently developed by Murota and others. This paper gives two new characterizations of M- and M'-convex functions generalizing Gul and Stacchetti's results on the equivalence among the single improvement condition, the gross substitutes condition and the no complementarities condition for set functions (utility functions on {0,1} vectors) as well as Fujishige and Yang's observation on the connection to M-convexity. We also discuss implications of our results in an exchange economy with indivisible goods.