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

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.