Revealed Preferences for Matching with Contracts

TL;DR

This paper introduces a framework for analyzing many-to-many matching with contracts using revealed preferences, presenting an algorithm for finding stable agreements with desirable properties and exploring their relation to market equilibria.

Contribution

It generalizes existing algorithms to a broader setting, proves the lattice structure of stable agreements, and connects stable agreements to market solutions with prices.

Findings

Algorithm finds stable agreements in linear steps

Stable agreements form a lattice under inverse preferences

Stable agreements encompass and extend competitive equilibria

Abstract

Many-to-many matching with contracts is studied in the framework of revealed preferences. All preferences are described by choice functions that satisfy natural conditions. Under a no-externality assumption individual preferences can be aggregated into a single choice function expressing a collective preference. In this framework, a two-sided matching problem may be described as an agreement problem between two parties: the two parties must find a stable agreement, i.e., a set of contracts from which no party will want to take away any contract and to which the two parties cannot agree to add any contract. On such stable agreements each party's preference relation is a partial order and the two parties have inverse preferences. An algorithm is presented that generalizes algorithms previously proposed in less general situations. This algorithm provides a stable agreement that is preferred to all stable agreements by one of the parties and therefore less preferred than all stable agreements by the other party. The number of steps of the algorithm is linear in the size of the set of contracts, i.e., polynomial in the size of the problem. The algorithm provides a proof that stable agreements form a lattice under the two inverse preference relations. Under additional assumptions on the role of money in preferences, agreement problems can describe general two-sided markets in which goods are exchanged for money. Stable agreements provide a solution concept, including prices, that is more general than competitive equilibria. They satisfy an almost one price law for identical items.