Substitutes and stability for many-to-many matching with contracts

Abstract

We examine the roles of (slightly weakened versions of) the observable substitutability condition and the observable substitutability across doctors condition of Hatfield et al. (2021) in many-to-many matching with contracts. We modify the standard cumulative offer algorithm to find stable outcomes and prove new results on the existence of stable outcomes. It is remarkable that size monotonicity at the offer-proposing side is essential for the existence result under observable substitutability across doctors.

Introduction

Since the classical work of Gale and Shapley (1962) and Roth (1984), the two-sided matching theory has attracted increasing attention in past decades for its appeal in both theory and application. Stability, which rules out recontracting and renegotiation after market-clearing, is the key to the success of many designed markets. Achieving stability often requires ruling out complementarity in agents' preferences; that is, if an agent receives more offers, he never accepts an offer that was rejected before. The substitutability condition introduced by Kelso and Crawford (1982) formalizes this requirement, and later it is often believed to be necessary for achieving stability (Hatfield and Milgrom, 2005). Yet, in the model of many-to-one matching with contracts, several weakened substitutability conditions have been found to be sufficient for stability and vital for understanding real-life applications. Since many-to-many relationships abound in real life, in this paper we examine the roles of these weakened conditions in the model of many-to-many matching with contracts, and prove new existence theorems that push forward the frontier of matching theory and motivate potentially new applications.

For convenience, we use the matching between hospitals and doctors to describe our model and results. All weakened substitutability conditions in the literature utilize the fact that there may exist multiple contracts between a doctor and a hospital, and stable outcomes are found through running the doctor-proposing cumulative offer algorithm. Hatfield and Kojima (2010) are the first to propose two weakened substitutability conditions called unilateral substitutability and bilateral substitutability. Under unilateral substitutability, although a hospital's choice function may violate substitutability, it never happens during the cumulative offer algorithm; that is, every hospital never accepts a contract that was rejected before in the procedure of the algorithm. So the outcome of the algorithm is stable for the same argument under substitutability. Bilateral substitutability allows a hospital to accept a contract from a doctor that was rejected before at some step of the algorithm, but this happens only when the hospital has accepted a different contract from the same doctor. In other words, once a hospital rejects all contracts from a doctor, it cannot accept the doctor again if the doctor does not make new offers. Because of this feature, given doctors have unit demands, the outcome of the algorithm is still stable. Interestingly, these seemly technical conditions find useful applications such as the cadet-branch matching (Sönmez, 2013; Sönmez and Switzer, 2013). Hatfield et al. (2021) extend these conditions by proposing observable substitutability and observable substitutability across doctors.¹ The idea is to require Hatfield and Kojima's conditions only to hold on the offer processes that appear in the cumulative offer algorithm. Hatfield et al. (2017) further show that their conditions are not only of theoretical interests, but also of practical relevance: the choice function of a hospital with multiple divisions and flexible allotments is observably substitutable.

On the other hand, Hatfield and Kominers (2016) propose substitutable completability, motivated by the observation that although some choice functions are not substitutable, they can be modified to substitutable ones by allowing multiple contracts with the same doctor. The resulting choice function is called a substitutable completion of the original choice function. This condition has shown its usefulness in several applications (Aygun and Turhan, 2020; Yenmez, 2018). Under this condition, a stable outcome can be found by replacing the original choice function with its substitutable completion that satisfies the irrelevance of rejected contracts condition (IRC; Aygün and Sönmez, 2013). An interesting fact is that, although a choice function has a substitutable completion satisfying IRC, itself may not satisfy IRC. But Zhang (2016) proves that such a choice function must satisfy IRC on offer processes that appear in the cumulative offer algorithm. Zhang calls this property observable IRC and unifies all of the aforementioned conditions by weak observable substitutability (across doctors) (see Fig. 1).^2,3

We apply Zhang's (2016) unified conditions to many-to-many matching with contracts in which a hospital-doctor pair can sign at most one contract; for this sake, choice functions in our model are unitary (Kominers, 2012). Our first theorem proves that, if the choice functions of doctors satisfy substitutability and IRC, and the choice functions of hospitals satisfy weak observable substitutability, then the outcome of the cumulative offer algorithm is stable. The behind rationale is the same as that under unilateral substitutability: during the cumulative offer algorithm every hospital never accepts a contract that was rejected before. This theorem unifies all results of stability in existing applications because, to our best knowledge, all choice functions in existing applications satisfy either substitutable completability or observable substitutability, both stronger than weak observable substitutability.

However, if one hospital has a bilaterally substitutable choice function that does not satisfy weak observable substitutability, whereas the other hospitals have standard choice functions (those satisfying substitutability and IRC), we show through an example that stable outcomes are not ensured to exist. During the cumulative offer algorithm, the hospital may accept a contract from a doctor that was rejected before, and the doctor may therefore wish to withdraw an offer he has made to another hospital. As a result, the outcome of the algorithm is not stable.

Our second theorem proves that, if the choice functions of doctors satisfy substitutability and size monotonicity (also known as law of aggregate demand), and the choice functions of hospitals satisfy weak substitutability across doctors, then stable outcomes exist and can be found by a modified cumulative offer algorithm. The role of size monotonicity is novel in this paper. In many-to-one matching with contracts, size monotonicity is imposed on the hospital side to obtain strategy-proofness for doctors and the so-called rural hospital theorem (Hatfield and Milgrom, 2005; Hatfield et al., 2021).⁴ In many-to-many matching with contracts and its generalization to trading networks, size monotonicity is imposed on agents' choice functions to obtain the generalized rural hospital theorem (Hatfield and Kominers, 2012, Hatfield and Kominers, 2017). To our best knowledge, we are the first to impose size monotonicity only on the doctor side to obtain the existence of stability. To prove our result, we modify the cumulative offer algorithm so that the set of unavailable contracts for doctors keeps expanding during the algorithm and can be different from the set of contracts rejected by hospitals. At the end of the algorithm, the outcome from the perspective of doctors can be different from the outcome from the perspective of hospitals. We choose the outcome from the perspective of hospitals as the final outcome of the algorithm, and prove that it is stable under our conditions imposed on both sides' choice functions.

Hatfield and Kominers (2017) consider many-to-many matching with contracts by allowing a hospital-doctor pair to sign multiple contracts. They state that the weakened substitutability conditions in many-to-one matching with contracts do not carry over to their model. Indeed, unitarity is crucial for our results; this assumption is implicitly made in the definitions of weakened substitutability conditions. Before our paper, Yenmez (2018) makes the first effort to relax substitutability in many-to-many matching with contracts. He generalizes Hatfield and Kominers's (2016) idea by relaxing substitutability to substitutable modification. In the application to college admission, unitarity is satisfied since every student can sign at most one contract with each college. Our paper pushes forward the frontier of this effort and can be potentially applied to new real-life problems.

In Section 2 we present the model and definitions. In Section 3 we present the main results. We conclude in Section 4.