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Efficient allocation of indivisible goods is an important problem in mathematical economics and operations research, where the concept of Walrasian equilibrium plays a fundamental role. As a sufficient condition for the existence of a Walrasian equilibrium, the concept of gross substitutes condition for valuation functions is introduced by Kelso and Crawford (1982). Since then, several variants of gross substitutes condition as well as a discrete concavity concept, called M ♮ -concavity, have been introduced to show the existence of an equilibrium in various models. In this paper, we survey the relationship among Kelso and Crawford's gross substitutes condition and its variants, and discuss the connection with M ♮ -concavity. We also review various characterizations and properties of these concepts.

GROSS SUBSTITUTES CONDITION AND DISCRETE CONCAVITY FOR MULTI-UNIT VALUATIONS: A SURVEY