A Note on Kelso and Crawford's Gross Substitutes Condition

Abstract

In their 1982 article, Kelso and Crawford proposed a gross substitutes condition for the existence of core (and equilibrium) in a two-sided matching model. Since then, this condition has often been used in the literature on matching models and equilibrium models in the presence of indivisibilities. In this paper we prove that a reservation value (or utility) function satisfies the gross substitutes condition if and only if it is an M♮-concave function defined on the unit-hypercube, which is a discrete concave function recently introduced by Murota and Shioura (1999).

Identifying combinatorial valuations from aggregate demand

“…, n, u j : R → R is increasing and concave with u j (0) = 0. Valuations of the form (59) generalize the concave transformations of additive valuations discussed in Section 8 Fujishige and Yang (2003) and Danilov, Koshevoy, and Lang (2003) presented related results. 9 It follows from Theorems 6.42 and 11.4 of Murota (2003) and Theorem 13 of Milgrom and Strulovici (2009). 3.2.…”

“…11 See Theorems 6.19, 6.42 and 11.5 of Murota (2003), originally proven by Murota and Shioura (2001), Murota (1996), Danilov, Koshevoy, and Lang (2003), Fujishige and Yang (2003), and Murota and Tamura (2003). V * g is convex, and hence has nonempty relative interior (i.e., nonempty interior relative to its affine hull aff V * g ), and any measurable set with nonempty interior in a Euclidean space has nonzero Lebesgue measure in that space, V * g has nonzero Lebesgue measure in aff V * g .…”

Gross substitutability: An algorithmic survey

“…Duality theorem for gross substitutes. The duality between gross substitutes and submodular functions was observed in many places, as in Fujishige and Yang [16] and Murota [31], Gul and Stacchetti [20] and Ausubel and Milgrom [2]. Given a valuation function v, they consider the utility function u : R n → R which maps a set of prices p ∈ R n to the optimal utility that can be obtained under such prices u(p) = max S v p (S).…”

“…All characterizations given so far involve prices, i.e., they are of the form: a valuation v satisfied the gross substitutes property if for all price vectors p, the pair (v, p) has some given property. The question of giving an explicit characterization of gross substitutes was resolved simultaneously by Fujishige and Yang [16] and Reijnierse, Gellekom and Potters [39]. The first paper provides a powerful connection to the theory of Discrete Convex Analysis, which we discuss in more detail in Section 7.…”

Two-Sided Matching With Externalities: A Survey

“…These substitutability conditions have also been connected to the literature on discrete convexity. Fujishige and Yang [24] first connects gross substitutability of Kelso and Crawford [31] to M ♮ -convexity of Murota and Shioura [45], which is an equivalent variant of Murota [43,44]. Subsequent matching models that have used discrete convexity include (and are not confined to) Danilov et al [13], Fujishige and Tamura [23], Murota and Yokoi [46], and Kojima et al [37].…”