Chia-Ling Hsu‡

March 10, 2014

Abstract We study conditions on preferences that guarantee a stable allocation in the setting of many-to-many matching with contracts when each agent can sign at most one contract with the same agent on the other side. The existing literature shows that substitutable preferences on both sides are sufficient for existence of a stable allocation. We show that the sufficient condition can be weakened to strongly bilateral substitutes on one side and unilateral substitutes on the other side. We also provide several applications of our findings. JEL: C78, D47, D71 Keywords: many-to-many matching with contracts, cumulative offer algorithm, substitutes, bilateral substitutes, unilateral substitutes, matching with couples

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Introduction

The matching with contracts model of Hatfield and Milgrom (2005) has had a substantial impact on the theory and practice of matching. The model incorporates matching without money (Gale and Shapley (1962)) and matching ∗

Acknowledgement to be added. Department of Economics, University of Illinois at Urbana-Champaign, 214 David Kinley Hall, 1407 W. Gregory Dr., Urbana, IL, 61801. E-mail: [email protected] ‡ Department of Economics, University of Illinois at Urbana-Champaign, 214 David Kinley Hall, 1407 W. Gregory Dr., Urbana, IL, 61801. E-mail: [email protected] †

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with money (Kelso and Crawford (1982)) as special cases.1 The many-to-one framework in Hatfield and Milgrom (2005) captures several natural economic settings, including as firm-worker matching and school-student matching. The notion of a stable matching is generalized to a stable allocation of contracts. Hatfield and Milgrom (2005) show that a stable allocation exists when preferences are substitutable. Hatfield and Kojima (2010) further show that the condition on preferences can be weakened to be unilaterally substitutable or bilaterally substitutable, each with different properties at stable allocations.2 On the other hand, many economic activities are many-to-many. A couple of doctors may work for two different hospitals, while hospitals have multiple hires. An expert may work in an university and be hired by a consulting firm, while both universities and consulting firms usually hire more than one expert. The many-to-many matching with wages model of Roth (1984) captures such economic activities, when contracts are wages.3 A stable matching exists when preferences are substitutable.4 Ostrovsky (2008) generalizes the model to a supply chain network matching model, where any two layers of upstream firms and downstream firms form many-to-many matching relationship. Hatfield and Kominers (2012) show the importance of an agent’s ability to sign multiple contracts with the same agent on the other side in the setting of many-to-many matching with contracts for designing matching markets.5 Hatfield and Kominers (2012) further show that substitutable preferences are not only sufficient but also necessary (in the maximal domain sense) for the existence of a stable allocation in such settings. In this paper, we study weaker conditions for the existence of a stable 1

See also Adachi (2000), Fleiner (2003) and Echenique and Oviedo (2006). The possibility of weaker conditions is first observed by Hatfield and Kojima (2008). 3 See Blair (1988) for the lattice structure when contracts are wages. For pure many¨ to-many matching, see Sotomayor (1999), Konishi and Unver (2006) and Echenique and Oviedo (2006). 4 There are several notions of a stable allocation in the many-to-many matching with contracts setting. See Klaus and Walzl (2009) for a comparison. In this paper, the notion of stable allocation that we refer to is called weakly setwise stable in Klaus and Walzl (2009). 5 The setting in which agents can only sign at most one contract with the same agent on the other side is call unitary by Kominers (2012). We will refer to the setting in which agents can sign multiple contracts with the same agent on the other side as non-unitary. 2

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allocation in a unitary many-to-many matching with contracts setting: each agent can sign at most one contract with each agent on the other side. We propose a condition on preferences which is stronger than bilaterally substitutable preferences but weaker than unilaterally substitutable preferences. We show that a stable allocation exists when the preferences of the agents on one side are strongly bilaterally substitutable and the preferences of the agents on the other side are unilaterally substitutable. We further show that when agents on both side have bilaterally substitutable preferences, a stable allocation may not exist. The assumption of unitarity is crucial for our results. This assumption is satisfied in many real-world applications. We also argue that the models with and without unitarity have different virtues and have different applications. We design the many-to-many cumulative offer algorithm to derive the main results. Fung and Hsu (2013b) and Fung and Hsu (2013a) apply the findings in this paper to study supply chain networks and platform markets, respectively.6 There are several recent applications of matching with contracts to important real-world market design problems. S¨onmez and Switzer (2013) analyze the cadet-branch matching mechanism used by the U.S. Military Academy (West Point). Each cadet has a score that is used to determine his priority in each branch. Moreover, each cadet has an option to sign a three-additionalyear contract to improve his priority in a branch. S¨onmez and Switzer (2013) propose the Cadet Optimal Stable Mechanism (COSM) that has the properties of stability, fairness, strategy-proofness, and respecting improvement of cadet order-of-merit list standings. A very important aspect of this application is that the branches’ priorities over cadets do not satisfy the substitutes condition. However, the priority structure does satisfy the unilateral substitutes condition. This provides the first application of matching with contracts that does not satisfy the substitutes condition while Hatfield and Kojima (2010)’s weaker unilateral substitutes condition holds. S¨onmez (2013) shows that in the Reserved Officer Training Corp (ROTC), with an additional requirement of redistribution of talents across different branches, the problem is different from S¨onmez and Switzer (2013). S¨onmez (2013) proposes a hybrid mechanism that is composed of the COSM and a 6

Fung and Hsu (2013b) and Fung and Hsu (2013a) apply the simplified many-to-many cumulative offer algorithm located in the Appendix in this paper to different settings.

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modified priority structure7 . The hybrid mechanism in S¨onmez (2013) has the properties of stability, being weakly preferred by the cadets to any stable allocation, strategy-proofness, fairness and respecting improvements. Kominers and S¨onmez (2013) greatly generalize S¨onmez and Switzer (2013). Each slot in a branch may have a different priority over the agents, depending on the contract that the agents are willing to sign. They show that the branchs’ priority structure can be further weakened to bilaterally substitutable. Important applications include airline upgrade programs and affirmative action in school choice. We make three important observations based on S¨onmez and Switzer (2013), S¨onmez (2013), and Kominers and S¨onmez (2013). First, a branch’s priority structure in these applications does not satisfy the substitues conditions but does satisfy one of the weaker conditions. In other words, substitutes may not be satisfied in real-world applications. Second, unitarity is assumed in all three papers. Some applications may be less meaningful without unitarity.8 Third, the designs of the algorithm are based on the many-to-one cumulative offer algorithm of Hatfield and Milgrom (2005). This suggests that the cumulative offer algorithm has the potential to be a flexible tool for dealing with real-world design applications. Our results have several applications. We provide the weakest preferences in the literature for matching between couples and hospitals with contracts. Our findings generalize the setting in the couple-hospital matching example in Hatfield and Kojima (2010). We also provide an extension to the airplane seat upgrades example in Kominers and S¨onmez (2013). The difference between our example and the setting in Kominers and S¨onmez (2013) is that we allow passengers to have multiple flights. There are several important recent discoveries of the properties in matching with contracts. Echenique (2012) shows that the matching with contracts of Hatfield and Milgrom (2005) can be embedded into Kelso and Crawford (1982) when preferences of agents are substitutable. He also shows that the 7

S¨ onmez (2013) calls it the Bid-for-Your-Career (BfYC) priority in . In S¨ onmez and Switzer (2013) and S¨onmez (2013), the contract terms are providing additional three-year service or not. A cadet cannot sign both contracts. In the application to affirmative action in school choice in Kominers and S¨onmez (2013), the contract terms are majority or minority. A student cannot be both a majority student and a minority student. In the application to the flight upgrade problem in Kominers and S¨onmez (2013), the contract terms are the channels for upgrading to business class. A traveler cannot use more than one channel. 8

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embedding does not hold when preferences are weakened to unilateral substitutes or bilateral substitutes. Kominers (2012) extends the embedding to many-to-many matching with contracts setting. Ayg¨ un and S¨onmez (2012b) show that when choice sets instead of preferences are primitive, an additional assumption, the Irrelevance of Rejected Contracts (IRC) condition, needs to be made along with substitutes in order to guarantee the existence of a stable (many-to-one) allocation. Ayg¨ un and S¨onmez (2012a) extend this finding to the bilaterally substitutable preferences. Both Ayg¨ un and S¨onmez (2012b) and Ayg¨ un and S¨onmez (2012a) show the generality of choice sets being primitive. The rest of the paper is organized as follows. Section 2 presents the model. Section 3 presents the main results. Section 4 concludes. A special case of the main results is contained in Appendix.

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Model

Let D be the set of doctors and H the finite set of hospitals.9 Let X ≡ D × H × T be the set of all contracts, where T is a finite set of contract terms. A contract x ∈ X specifies a bilateral relationship between a doctor xD ∈ D and a hospital xH ∈ H at terms xT ∈ T . Each agent may sign multiple contracts, or choose an outside option ∅. Any doctor-hospital pair can sign at most one contract between them. For a given set of available contracts Y ⊆ X, let YD ⊆ D denote the set of doctors associated with Y and let YH ⊆ H denote the set of hospitals associated with Y . For an agent i ∈ D ∪ H, let Yi ⊆ Y denote the (possibly empty) set of contracts in Y associated with i. Each agent i ∈ D ∪ H has a strict preference relation Pi over sets of contracts associated with him. The preference can easily extended to set of contracts. Define i so that Yi Pi Yi0 if and only if Y i Y 0 . A doctor d’s chosen set, Cd (Y ) ⊆ Yd , denotes d’s (possibly empty) most preferred subset of contracts in Y : 0 Cd (Y ) ≡ max{Y 0 ⊆ Yi : yH 6= yH , ∀y 6= y 0 ∈ Y 0 }. Pi

Let Rd (Y ) = Y − Cd (Y ) be d’s rejected set from Y . Let CD (Y ) = ∪d∈D Cd (Y ) and RD (Y ) = Y − CD (Y ). A hospital h’s chosen set and re9

Following a convention of the matching literature, we use doctors and hospitals as the different sets of agents. However, they can be interpreted as workers and firms, or students and schools, etc.

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jected set are analogously defined. Let CH (Y ) = ∪h∈H Ch (Y ) and RH (Y ) = Y − CH (Y ).

2.1

Preference

We will discuss the following preferences in this paper. Note that although we use hospitals in the definitions, all the definitions apply analogously to doctors. Definition 1. Contracts are substitutes for a hospital h ∈ H, if for any two sets of contracts Y, Y 0 ⊆ X such that Y ⊆ Y 0 , we have Rh (Y ) ⊆ Rh (Y 0 ). The following two definitions are given by Hatfield and Kojima (2010). Definition 2. Contracts are bilateral substitutes for a hospital h ∈ H, if there do not exist two contracts x, z ∈ X and a set of contracts Y ⊆ X such that xD , zD ∈ / YD , z ∈ / Ch (Y ∪ {z}) and z ∈ Ch (Y ∪ {x, z}). Definition 3. Contracts are unilateral substitutes for a hospital h ∈ H, if there do not exist two contracts x, z ∈ X and a set of contracts Y ⊆ X such that zD ∈ / YD , z ∈ / Ch (Y ∪ {z}) and z ∈ Ch (Y ∪ {x, z}). In this paper, we propose a new definition on preferences, the strongly bilateral substitutes, as follows. Definition 4. Contracts are strongly bilateral substitutes for a hospital h ∈ H if there do not exist a contract z ∈ X and two sets of contracts Y, Z ⊆ X, such that zD ∈ / YD and ZD \ {zD } 6= YD , z ∈ / Ch (Y ∪ {z}) and z ∈ Ch (Y ∪ Z ∪ {z}). Strongly bilateral substitutes say that if a hospital h does not accept a contract z when he faces Y ∪ {z} and zD ∈ / YD , then when he faces a larger set of contracts Y ∪ Z ∪ {z}, he still will not choose z, as long as the set of doctors in ZD \ {zD } is not equal to YD . Remark 1. The definition of strongly bilateral substitutes is stronger than the definition of bilateral substitutes, but it is weaker than the definition of unilateral substitutes.

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If a preference satisfies bilateral substitutes, then it must be strongly bilateral substitutes. Consider the definition of strongly bilateral substitutes, by letting Z a singleton and ZD * YD , then we have the definition of bilateral substitutes. However, if a preference satisfies strongly bilateral substitutes, it may not satisfy bilateral substitutes. To see this, consider a preference Ph and contracts Z, Y and {z}, such that |Z| = 1, |Y | > 2, zD ∈ / ZD , and ZD ⊆ YD . This preference does not satisfy bilateral substitutes, but it satisfies strongly bilateral substitutes. To compare strongly bilateral substitutes and unilateral substitutes, note that by footnote 17 in Hatfield and Kojima (2010), the definition of unilateral substitutes is equivalent to the following. Suppose Ph satisfies unilateral substitutes, then there do not exist a contract z and two set of contracts Y, Z such that zD ∈ / YD , z ∈ / Ch (Y ∪ {z}) and z ∈ Ch (Y ∪ Z ∪ {z}). The difference between strongly bilateral substitutes and this version of unilateral substitutes is that in strongly bilateral substitutes we have a requirement on Z so that ZD \ {zD } = 6 YD , while in unilateral substitutes we do not have any requirement on Z. The following definition is important when chosen set is primitive. Definition 5. Contracts satisfy the Irrelevance of Rejected Contracts (IRC) condition if for every Y ⊆ X and z 6∈ Y , z 6∈ Ci (Y ∪ {z}) implies Ci (Y ) = Ci (Y ∪ {z}), for all i ∈ D ∪ H. Note that by Ayg¨ un and S¨onmez (2012b), when choice sets are generated by strict preferences, the choice sets satisfy the IRC condition. Therefore, in our setting, the IRC condition is automatically satisfied. We will still point out the places we use the properties of IRC condition in our proofs of the main results.

2.2

Stable Allocation

The central notion of solution concept, the stable allocation, is introduced as follows. Definition 6. An allocation Y ⊆ X is stable, if it satisfies the following two conditions: • Individual rationality: Ci (Y ) = Yi , for all i ∈ D ∪ H. 7

• Having no blocking set of contracts: there does not exist another set of contracts Z 6= Y such that Zi ⊆ Ci (Y ∪ Z) for all i ∈ YD ∪ YH .

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Results

We propose the many-to-many cumulative offer algorithm to derive the main results in this paper. The algorithm produces a stable allocation when the preferences of the agents on one side are strongly bilaterally substitutable and the preferences of the agents on the other side are unilaterally substitutable. A simplified version of the algorithm that produces a stable allocation when all agents’ preferences are substitutable is presented in Appendix.

3.1

The Many-to-Many Cumulative Offer Algorithm

The (doctors-proposing) many-to-many cumulative offer algorithm works as follows. • Step 1: Let Od (1) = Xd for all d ∈ D. An arbitrary doctor d1 proposes Y (1) = Cd1 (Od1 (1)). Hospitals hold Ch (Yh (1)) for all h ∈ H. – Let Oh (1) = Yh (1) for all h ∈ H. – Let OH (1) = ∪h∈H Oh (1) = Y (1). • Step t: Let Od (t) = Xd − RH (OH (t − 1)) for all d ∈ D. An arbitrary doctor dt such that Cdt (Odt (t)) 6⊂ CH (OH (t − 1)) proposes Y (t) = Cdt (Odt (t)). Hospitals hold Ch (Oh (t − 1) ∪ Yh (t)) for all h ∈ H. – Let Oh (t) = Oh (t − 1) ∪ Yh (t) for all h ∈ H. – Let OH (t) = ∪h∈H Oh (t) = OH (t − 1) ∪ Y (t). The algorithm terminates in some step T such that Cd (Od (T )) ⊂ CH (OH (T − 1)) for all d ∈ D. The final allocation is X 0 ≡ CD (OD (T )).

3.2

Weaker Conditions for Existence of a Stable Allocation

For the ease of proving the main theorem, we transform the original definition of a stable allocation to a version of a stable allocation with the modified blocking set defined in the following lemma. 8

Lemma 1. Consider an individually rational allocation Y that has a blocking set Z, i.e., ∃Z 6= Y such that Zi ⊆ Ci (Y ∪ Z), ∀i ∈ ZD ∪ ZH .

(1)

˜ i.e., Then Y has a modified blocking set Z, ˜ ∀i ∈ Z˜D ∪ Z˜H . ∃Z˜ 6= Y such that Z˜i = Ci (Y ∪ Z),

(2)

Proof. Suppose (1) holds, and consider i ∈ ZD ∪ ZH . Define Z˜i ≡ Ci (Y ∪ Z), ∀i ∈ ZD ∪ ZH . So Z˜i ⊆ Y ∪ Z, ∀i ∈ ZD ∪ ZH . Let Z˜ = ∪i∈ZD ∪ZH Z˜i . Then ∀i ∈ ZD ∪ ZH , ˜ = Ci (Y ∪ {∪j∈Z ∪Z Z˜j }) = Ci (Y ∪ Z˜i ) = Ci (Y ∪ Ci (Y ∪ Z)) = Z˜i , Ci (Y ∪ Z) D H where the second and third equalities hold by IRC. Lemma 2 presents a property of the many-to-many cumulative offer algorithm. Definition 7. A contract z is a returning contract for a doctor d ∈ D in the algorithm, if there exist t1 , t2 with t1 < t2 ≤ T such that z ∈ / Od (t1 ) and z ∈ Od (t2 ). Lemma 2 (A generalization of Theorem 4 of Hatfield and Kojima (2010)). Suppose doctors’ preferences satisfy unilateral substitutes. Suppose z is held by hospital h at step t of the (doctor-proposing) many-to-many cumulative offer algorithm. At any later step t0 > t, if zD has not been rejected by h since t, then z is still held by h. Proof. The statement of Theorem 4 in Hatfield and Kojima (2010) still holds for the many-to-many cumulative offer algorithm, because (1) when hospitals choose the contracts in each step, their decision does not depend on whether the same doctor also makes proposal to other hospitals, and (2) each doctor can only sign at most one contract with each hospital. Lemma 3 and Lemma 4 present properties of unilaterally and bilaterally substitutable preferences, respectively. The lemmas say that when the contracts are unilaterally substitutable or bilaterally substitutable for an agent, if he considers a set of contracts individually rational, then any subset of that set is also individually rational for him. Note that this result replies on unitarity. We choose the role of doctors or hospitals stated in the lemmas to reduce the amount of notation. 9

Lemma 3. Consider a hospital h ∈ H and a set of contracts Y ⊆ X. Suppose contracts are unilateral substitutes for h. If Ch (Y ) = Y , then Ch (Y 0 ) = Y 0 for any subset of contracts Y 0 ⊆ Y . Proof. Since each hospital can sign at most one contract with each doctor, for any contract y ∈ Y , we have yD ∈ / (Y \{y})D . Suppose the statement is not true. Then there exists a contract y 0 ∈ Y 0 such that y 0 ∈ / Ch (Y 0 ) but y 0 ∈ Ch (Y ). Since yD ∈ / (Y \y 0 )D , this is a violation of unilateral substitutes. Lemma 4. Consider a doctor d ∈ D and a set of contracts Y ⊆ X. Suppose contracts are bilateral substitutes for d. If Cd (Y ) = Y , then Cd (Y 0 ) = Y 0 , for any subset of contracts Y 0 ⊆ Y . Proof. Suppose not. Then there exists z ∈ Z 0 ⊆ Z ⊆ X such that z ∈ / Cd (Z 0 ) and z ∈ Cd (Z) = Z. Let Cd (Z 0 ) = Y 0 . Case 1: Suppose we can pick x ∈ Z such that x ∈ / Cd (Z 0 ) = Y 0 . Let Y = Z\{x, z}. By IRC, z ∈ / Cd (Y 0 ∪ {z}). Repeated application of bilateral substitutes and unitarity implies that, since z ∈ Cd (Y ∪ {x, z}), then we must have xH ∈ YH0 and xH ∈ YH . But Cd (Z) = Z implies this is not possible, since each doctor can only sign at most one contract with each hospital. Case 2: Suppose we cannot pick such x ∈ Z. This implies that Z = 0 Z ∪ {z}, and it further implies that Cd (Z) = Z and Cd (Z 0 ) = Z 0 . Although Lemma 4 implies Lemma 3 in our setting of the model, we still keep both lemmas. This is because in the proof in Lemma 3, we do not use the property of IRC, while in the proof in Lemma 4, we use the property of IRC. The following theorem is the main theorem of this paper. Note that Lemma 4 implies that when a doctor’s preference Pd satisfies strongly bilateral substitutes, then if Cd (Y ) = Y , we have Cd (Y 0 ) = Y 0 for any Y0 ⊆Y. Theorem 1. Suppose contracts are strongly bilateral substitutes for agents on one side and unilateral substitutes for agents on the other side. Then, a stable allocation exists. Proof. Consider the doctors-proposing many-to-many cumulative offer algorithm. Assume that the preferences of doctors are strongly bilaterally substitutable and the preferences of hospitals are unilaterally substitutable. There are three steps in the proof. 10

Step 1: The algorithm produces a feasible allocation. By Lemma 2: if a doctor d’s contract x is rejected by hospital xH at some step of the algorithm, then xH does not hold x at any later step. Therefore, no doctor faces a returning contract in the algorithm. This guarantees that there are no proposal cycles, that OD (t) ⊆ OD (t0 ) for all t > t0 , and thus the algorithm terminates at a finite step T . Step 2: The allocation is individually rational. This is shown in the following two claims. Claim 1. Xd0 = Cd (X 0 ), ∀d ∈ D. Proof. By construction of the algorithm, CD (OD (T )) ⊂ CH (OH (T )), and X 0 = CD (OD (T )). By IRC, X 0 = CD (OD (T )) = CD (X 0 ). Claim 2. Xh0 = Ch (X 0 ), ∀h ∈ H. Proof. Note that X 0 ⊂ CH (OH (T )) implies Xh0 ⊂ CH (OH (T )) for all h ∈ H. This means Xh0 ⊂ Ch (Oh (T )) for all h ∈ H. Since Ch (Ch (Oh (T ))) = Ch (Oh (T )), by Lemma 3, for any Y ⊆ Ch (Oh (T )), we have Ch (Y ) = Yh , for all h ∈ H. Step 3: The allocation has no blocking set of contracts. Suppose not. By Lemma 1, there exists a set of contracts Z 6= X 0 such that Zi = Ci (X 0 ∪ Z) for all i ∈ ZD ∪ ZH . Claim 3. There exists d ∈ ZD such that Zd = Zd0 ∪ Zd00 and Zd0 ∩ Zd00 = ∅, where Zd0 ⊂ OD (T ) and Zd00 ⊂ RH (OH (T )). Proof. Suppose Z ⊂ OD (T ). Then X 0 ∪ Z ⊂ OD (T ). Individual rationality for doctors and IRC imply that X 0 = CD (OD (T )) = CD (X 0 ∪ Z) 6= X 0 , a contradiction. So, Z 6⊂ OD (T ), which implies that Z ∩ RH (OH (T )) 6= ∅. On the other hand, if Z ⊂ RH (OH (T )), then Z is not an individually rational deviation for some h ∈ ZH , a contradiction. So, Z 6⊂ RH (OH (T )). Then Z can be partitioned into two sets of contracts Z 0 ⊂ OD (T ) and Z 00 ⊂ RH (OH (T )), 11

such that Z = Z 0 ∪ Z 00 and Z 0 ∩ Z 00 = ∅. If Zd ⊂ OD (T ) for all d ∈ ZD , then Zd = Cd (Xd0 ∪ Zd ) = Cd (OD (T )) = Xd for all d ∈ ZD , contradicting that Z 6= X 0 . Thus there exists some d ∈ ZD such that Zd = Zd0 ∪ Zd00 and Zd0 ∩ Zd00 6= ∅, where Zd0 ⊂ OD (T ) and Zd00 ⊂ RH (OH (T )). Claim 4. Zd0 ∩ (OD (T ) \ CD (OD (T ))) = ∅ for d ∈ ZD such that Zd = Zd0 ∪ Zd00 and Zd0 ∩ Zd00 6= ∅, where Zd0 ⊂ OD (T ) and Zd00 ⊂ RH (OH (T )). Proof. Consider a doctor d ∈ ZD such that Zd = Zd0 ∪ Zd00 and Zd0 ∩ Zd00 6= ∅, where Zd0 ⊂ OD (T ) and Zd00 ⊂ RH (OH (T )). Suppose that Zd0 ∩(OD (T )\CD (OD (T ))) 6= ∅. Then there exists a contract z ∈ Zd0 such that z ∈ Cd (Xd0 ∪ (Zd0 ∪ Zd00 )) but z∈ / Cd (Xd0 ∪ Zd0 ) = Cd (Xd0 ∪ (Zd0 \ {z})), where the last equality follows by IRC. Since each doctor-hospital pair can sign at most one contract, zH ∈ / (Zd0 \ {z})H . Next, we show that zH ∈ / (Xd0 )H . Since z ∈ OD (T ), there are two cases. 0 Case 1: there is no other contract z 0 ∈ Xd0 such that z 0 6= z and zH = zH . 0 Then it is clear that zH ∈ / (Xd )H . 00 Case 2: there is some contract z 00 ∈ Xd0 such that z 00 6= z and zH = 0 0 zH . Cd (Xd ∪ Zd ) implies that Zd Pd Xd . This means that d proposes Zd in the algorithm before the iteration T . By Lemma 2, there is no returning contracts. Then z ∈ OD (T ) implies that zH never rejects z. This contradicts 00 that there is another contract z 00 ∈ Xd0 such that z 00 6= z and zH = zH . 0 Therefore, zH ∈ / (Xd )H . Notice that this implies zH ∈ / (Xd0 ∪ Zd0 ) \ {z} H . Then z ∈ Cd (Xd0 ∪ Zd ) and z ∈ / Cd (Xd0 ∪ Zd0 ) contradict strongly bilateral substitutes. Thus the only relevant part of the deviation for d is Zd00 ⊂ RH (OH (T )). But then Xd0 ∪ (Zd0 ∪ Zd00 ) ⊂ OH (T ) and Zd0 ⊂ Xd0 and Zd00 ⊂ RH (OH (T )) imply with IRC that Ch (X 0 ∪ (Z 0 ∪ Z 00 )) = Ch (X 0 ∪ Z 0 ) = Ch (X 0 ) = Xh0 , for some h ∈ ZH , contradicting Zi = Ci (X 0 ∪ Z) = Ci (Z) for all i ∈ ZD ∪ ZH . So Z is not an individually rational deviation for the hospitals in ZH . Therefore, no such Z exists. Since X 0 is individually rational and has not blocking set of contracts, it is a stable allocation.

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The following example shows that we cannot further weaken the condition to bilaterally substitutable preferences for the agents on both sides. Example 1. This example shows that no stable allocation exists when contracts satisfy bilateral substitutes for both sides. Let D = {d1 , d2 } and H = {h1 , h2 }. Let X = {w, w0 , x, x0 , y, y 0 , z, z 0 }. 0 0 0 Moreover, wD = wD = xD = x0D = d1 , yD = yD = zD = zD = d2 , wH = 0 0 0 0 wH = yH = yH = h1 , and xH = xH = zH = zH = h2 . The preferences are the following. Pd 1 Pd 2 Ph1 Ph2

: {w0 , x} d1 {w} d1 {x0 } d1 {w0 } d1 {x}. : {y 0 , z} d2 {z 0 } d2 {y} d2 {z} d2 {y 0 }. : {w, y} h1 {y 0 } h1 {w0 } h1 {y} h1 {w}. : {x0 , z 0 } h2 {x} h2 {z} h2 {x0 } h2 {z 0 }.

The preferences of all agents satisfy bilateral substitutes but they do not satisfy unilateral substitutes. For any individual rational allocation, there is at most one agent who holds two contracts. In addition, for an individual rational allocation, if there is one agent holding two contracts, there must be one agent who does not hold any contract. Let X 0 C X 00 to denote the the relationship that X 0 is blocked by X 00 . Then, we have the following: ∅ C {x} C {w0 } C {x0 } C {w} C {w0 , x} C {y 0 } C {z} C {y} C {z 0 } C {y 0 , z} C {x} C {x0 , z 0 } C {w} C {y} C {w0 } C {w, y}. Moreover, {x, y} C {w0 }; {x, y 0 } C {w, y}; {x0 , y} C {w0 , x}; {x0 , y 0 } C {w, y}; {w, z} C {y 0 , z}; {w, z 0 } C {x0 , z 0 }; {w0 , z} C {w0 , x}; {w0 , z 0 } C {x0 , z 0 }. Therefore, a stable allocation does not exist.

3.3

Applications

In this section, we discuss several possible applications. The following example is an application to matching with couples. Example 2 (Based on Hatfield and Kojima (2010), Example 2). Let D = {c, d}, H = {h, h0 }, where c is a couple and d is a single doctor. Let X = 0 0 {x, x0 , y, y 0 , z, z 0 }. Suppose xD = x0D = zD = zD = c, yD = yD = d, xH =

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0 x0H = yH = h, and zH = zH = h0 . Preferences are

Pc Pd Ph Ph0

:{x0 , z} c {x} c {x0 } c {z}. :{y 0 } d {y}. :{x0 }. :{y 0 , z} h0 {y, z 0 } h0 {y, z} h0 {y 0 , z 0 } h0 {y 0 } h0 {y} h0 {z 0 } h0 {z}.

The couple may be interpreted as choosing from {x, x0 } for the husband and from {z, z 0 } for the wife. If both members of the couple cannot work under {x0 , z}, then they prefer that the husband works, and that he works under {x}, which Hatfield and Kojima (2010) motivate as a position with “high wage, long hours” being the best option when the wife is not employed at a hospital. Contracts are strongly bilateral substitutes for doctors and unilateral substitutes for hospitals.10 The allocation {x0 , y 0 } is stable. The second example is an application to airplane seat upgrades. The setting of airplane seat upgrades is based on Section 5.1 in Kominers and S¨onmez (2013). The difference is that we allow more than one flight. This is common for international travel. Example 3 (Based on Section 5.1 in Kominers and S¨onmez (2013)). Two passengers travel from place A to place B. There is no direct flight from A to B, and the only flight option is having two connecting flights. As in Kominers and S¨onmez (2013), there are two methods for upgrading to business class: either by mileage or by cash. Let I = {i1 , i2 } be the set of passengers and F = {f1 , f2 } be the set of connecting flights. The set of contracts is X = {wm , w$ , xm , y $ , z m }, where superscript m stands for upgrade by mileage and $ for upgrade by cash. Moreover11 , (wm )I = (w$ )I = (y $ )I = i1 , (xm )I = (z m )I = i2 , (wm )F = (w$ )F = (xm )F = f1 , and (y $ )F = (z m )F = f2 . 10

The only incidence for agent c to reject a contract when he faces a smaller set and accepts it facing a bigger set is either when x0 ∈ / Cc ({x, x0 }) and x0 ∈ Cc (x, x0 , z) or when 0 z ∈ / Cc ({x, z}) and z ∈ Cc ({x, x , z}). In both incidences, x0D = xD , so the preference satisfies strongly bilateral substitutes. 11 With a similar spirit as in the setting for doctors and hospitals, let subscript I and F be the indicator of the passenger and flight.

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The preferences are the following. Pi1 ={wm , y $ } i1 {wm } i1 {w$ } i1 {y $ }. Pi2 ={xm , z m } i2 {xm } i2 {z m }. Pf1 ={w$ , xm } f1 {wm , xm } f1 {w$ } f1 {xm } f1 {wm }. Pf2 ={y $ , z m } f2 {y $ } f2 {z m }. Passenger i1 only has enough mileage for flight f1 . He is willing to pay cash to upgrade flight f1 or flight f2 , but he does not want to upgrade both flights by cash. Passenger i2 has enough mileage for both flights, but he does not want to upgrade any flight by cash. Pi1 satisfies strongly bilateral substitutes.12 Pi2 and Pf2 satisfy substitutes. Pf1 satisfies unilateral substitutes. The set of contracts {wm , xm , y $ , z m } is a set of stable allocation.

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Conclusion

In this paper, we study conditions that are weaker than substitutable preferences for the existence of a stable allocation in unitary many-to-many matching with contract settings. We find that when the preferences of the agents on one side are strongly bilaterally substitutable and the preferences of the agents on the other side are unilaterally substitutable, a stable allocation exists. We further show that when the preferences of the agents on both sides are bilaterally substitutable, a stable allocation may not exist. Applications of our findings include matchings between couples and hospitals and airplane seat upgrade. Since the first step of any application of matching theory to real-world practice is proving the existence of a stable allocation, our existence result broadens the scope that the theory of many-to-many matching with contracts can apply to.

12

The only incidence that makes a contract rejected in a smaller set but accepted in a bigger set is the following: y $ ∈ / Ci1 ({w$ , y $ }), y $ ∈ Ci1 ({wm , w$ , y $ }). Since (wm )F = $ (w )F , Pi1 satisfies strongly bilateral substitutes.

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