Equivalence between substitutability and M$$^\natural $$-concavity for set functions under discrete transfers

Abstract

AbstractFujishige and Yang (2003) prove the equivalence between two fundamental conditions of a valuation function in a market model with indivisibilities, i.e., the gross substitutes (GS) condition and M$$^\natural $$ ♮ -concavity. We introduce a weaker variant of the GS condition that concerns discrete price changes rather than continuous price changes. We show that this weaker variant is equivalent to M$$^\natural $$ ♮ -concavity if the valuation function takes integer values and has an M$$^\natural $$ ♮ -convex effective domain containing the empty set. Our result indicates that assuming the weaker GS condition is sufficient for M$$^\natural $$ ♮ -concavity in existing auction models.