Unilateral substitutability implies substitutable completability in many-to-one matching with contracts

Abstract

We prove that the unilateral substitutability property introduced in Hatfield and Kojima [2010] implies the substitutable completability property from Hatfield and Kominers [2014]. This paper provides a novel linkage between these two sufficient conditions for the existence of a stable matching in many-to-one matching markets with contracts. A substitutable completion of a preference is a substitutable preference created by adding some sets of contracts to the original preference order. We provide an algorithm which when operated on the unilaterally substitutable preferences produces such a substitutable completion. Thus it provides a constructive proof of the connection between the two properties. * I am deeply indebted to Scott Kominers and Eric Maskin for extremely helpful discussions and detailed comments on this paper. I would also like to thank

On sufficient conditions for the existence of stable matchings with contracts

“…In this note we constructively prove that if a choice function satisfies unilateral substitutability and IRC, it must have a substitutable completion. This result is more general than that of Kadam (2014) since he only proves the result for choice functions derived from preferences.…”

“…In the graph "condition a → condition b" means that condition a implies condition b. Hatfield and Kojima (2010) Unilateral substitutability & IRC Kadam (2014) this paper In Section 2 we define the standard many-to-one matching with contracts model. In Section 3 we define weakly observable substitutability and weakly observable substitutability across workers and prove that they are sufficient conditions.…”

Contracts versus salaries in matching: A general result

“…15 The construction resembles the substitutable completion procedure introduced by Hatfield and Kominers (2015). Preferences that satisfy the unilateral substitutability condition can always be substitutably completed (Kadam, 2015). The two constructions however have significant differences.…”