EQUIVALENCES BETWEEN TWO MATCHING MODELS: STABILITY

. We study the equivalences between two matching models, where the agents in one side of the market, the workers, have responsive preferences on the set of agents of the other side, the ﬁrms. We modify the ﬁrms’ preferences on subsets of workers and deﬁne a function between the set of many- to-many matchings and the set of related many-to-one matchings. We prove that this function restricted to the set of stable matchings is bijective and that preserves the stability of the corresponding matchings in both models. Using this function, we prove that for the many-to-many problem with substitutable preferences for the ﬁrms and responsive preferences for the workers, the set of stable matchings is non-empty and has a lattice structure.


1.
Introduction. Many-to-many matching models have been useful for studying assignment problems with the distinctive feature that agents can be divided into two disjoint subsets: the set of firms and the set of workers. 1 The nature of the assignment problem consists of matching each agent with a subset of agents from the other side of the market. Thus, each firm may hire a subset of workers while each worker may work for a number of different firms.
Stability has been considered the main property to be satisfied by any matching. A matching is called stable if all agents have acceptable partners and there is no unmatched worker-firm pair who both would prefer to be matched to each other rather than staying with their current partners. Unfortunately, the set of stable matchings may be empty. Substitutability 2 is the weakest condition that has so far been imposed on agents' preferences under which the existence of stable matchings is guaranteed. An agent has substitutable preferences if he wants to continue being matched to an agent on the other side of the market even if other agents become unavailable. 3 The college admissions problem is the name given by Gale and Shapley [4] to a many-to-one matching model. Colleges have responsive preferences over students and students have preference over colleges; each college c has a maximum number of positions to be filled (its quota q c ), it ranks individual students and orders subsets of students in a responsive manner (Roth [12]); namely, to add "good" students to a set leads to a better set, whereas to add "bad" students to a set leads to a worst set. In addition, for any two subsets that differ in only one student, the college prefers the subset containing the most preferred student. In this model the set of stable matchings satisfies the following additional properties: (i) there is a polarization of interests between the two sides of the market along the set of stable matchings, (ii) the set of unmatched agents is the same under every stable matching, (iii) the number of workers assigned to a firm through stable matchings is the same, and (iv) if a firm does not complete its quota under some stable matching then it is matched to the same set of workers at any stable matching. 4 The case in which all quotas are equal to one is called the marriage problem, 5 and is symmetric between the two sides of the market. It was initially thought that the essential features of the college admissions problem could be captured by treating it as a marriage problem in which each of the q c positions available at a college c would be treated as q c different colleges, denote by c 1 , c 2 , . . . , c qc . There is a natural injective correspondence between matchings in the original college admissions problem and matchings in the related marriage problem in this way: a matching µ of the college admissions problem, which matches a college c with the set of students µ(c), corresponds to the matching µ in the related marriage market in which the students in µ(c) are matched, in the order that they occur in its preferences, with the ordered positions of c that appear in the related marriage market; that is, if s is c's most preferred student in µ(c), then µ (c 1 ) = s, and so forth. This correspondence preserves the stability of the matchings. This construction is due to Gale and Sotomayor [5]. Most of the subsequent theoretical literature concerned with these problems focused on the marriage problem, with the assumption that results established for the marriage problem would carry over to the college admissions problem through this kind of transformation. However, Roth [12] observed that certain results, as those of optimality and incentives, cannot be extended from the case of the marriage problem.
Knuth [8] established that the set of stable matchings for the marriage model has a lattice structure and attributed this result to Conway. Roth [11] showed that the least upper bound and the greatest lower bound used by Knuth [8] did not work in a more general many-to-many matching model. Blair [2] proposed a natural extension of the partial ordering used in Knuth [8]. However, its binary operations were unnatural and complicated since they were obtained as the outcomes of nontrivial 3 Kelso and Crawford [7] were the first to use substitutability to show the existence of stable matchings in a many-to-one model with money. Roth [10] shows that, if all agents have substitutable preferences, the set of many-to-many stable matchings is non-empty. 4 Property (i) is a consequence of the decomposition lemma proved by Gale and Sotomayor [5]. Properties (ii) and (iii) were proved independently by Gale and Sotomayor [5] and Roth [10]. Property (iv) was proved by Roth [13]. 5 It is the name given to the one-to-one matching model. See Roth and Sotomayor [15] for a precise and formal definition of such model. sequences of matchings. Roth and Sotomayor [15] extended the result of the marriage problem to the college admission problem and this work was further extended by Martínez et al. [9] by proposing, for a many-to-one model with substitutable and separable with quota preferences, two very natural binary operations that endow to the set of stable matchings with a lattice structure.
The college admissions problem with substitutable preferences is the name given by Roth and Sotomayor [15] to the most general many-to-one model with ordinal preferences. Firms are restricted to have substitutable preferences over subsets of workers, while workers may have all possible preferences over the set of firms. Under this hypothesis Roth and Sotomayor [15] showed that the deferred-acceptance algorithms produce either the firm-optimal stable matching or the worker-optimal stable matching, depending on whether the firms or the workers make the offers. The firm (worker)-optimal stable matching is unanimously considered by all firms (respectively, workers) to be the best among all stable matchings.
It is natural to think that there exists a natural bijective function between the set of stable matchings in the original many-to-many model with responsive preferences for the workers and the set of stable matchings in the related many-to-one model. However, we show that the natural extension presented by Gale and Sotomayor [5] does not preserve the equivalence between the set of stable matchings in the manyto-many matching model and the corresponding set of stable matchings of the related many-to-one model. 6 We study a many-to-many matching model with responsive preferences for the workers, by investigating how to define agents' preferences in its related many-to-one model to preserve the stability between matchings in the original many-to-many model and matchings in the related many-to-one model. For this reason, over the subsets of copies of workers in the related many-to-one model, we modify the firms' preferences and define a function between the set of many-tomany matchings and the set of related many-to-one matchings. We show that this function, restricted to the set of stable matchings, is bijective. Hence, it preserves the stability between both sets of stable matchings, obtaining our main theorem.
Moreover, we prove that the modified firms' preferences, defined over the subsets of copies of students in the related many-to-one market, inherit the restriction of substitutability when the firms have substitutable preferences in the original manyto-many model. Since the function defined preserves the stability between both sets of stable matchings, we give an alternative proof that the set of stable matchings is non-empty for the many-to-many model with substitutable preferences for the firms, and responsive preferences for the workers. Also, we prove that this function preserves the Blair partial order of the agents on each side of the market over the set of stable matchings, and that this set has a lattice structure for the many-to-many model with substitutable preferences for the firms, and responsive preferences for the workers. From here, it follows that both sets of stable matchings have equivalent lattice structures.
Our paper contributes to this literature by proposing a new tool to prove results in the many-to-many model that follow from results in the many-to-one model. In this paper, we present novel proofs of two already well-known results in the manyto-many model. However, it is important to mention that this new technique of demonstration introduced in this paper, could be used to prove results not known yet in the many-to-many model. The paper is organized as follows. In Section 2, we present the preliminary notations and definitions. In Section 3, we introduce a many-to-many matching model and its related many-to-one matching model, and we present the principal result of this paper. In Section 4, we prove that the set of stable matchings in the many-to-many model has a lattice structure. Finally, in Section 5, we conclude with some final remarks.

2.
Preliminaries. There are two finite and disjoint sets of agents, the set of n firms F = {f 1 , ..., f n } and the set of m workers W = {w 1 , ..., w m }. Each worker w j ∈ W with j = 1, . . . , m has a maximum number of positions to be filled: its quota, 7 denoted by s j . Observe that, the quota of worker w imposes only a restriction on the maximal number of firms to which w can be assigned. Let s = (s 1 , . . . , s m ) be the list of quotas, one for each worker w j ∈ W. To simplify the notation, sometimes, we denote a generic firm by f (instead of f i ) and a generic worker by w (instead of w j ), and its quota by s w . Each firm f ∈ F has an antisymmetric, transitive and complete preference relation f over the set of all subsets of W , and each worker w ∈ W has an antisymmetric, transitive and complete preference relation w over the set of all subsets of F . Given A, B ⊆ W, we write A f B to indicate that firm f likes A at least as well as B. Given the preference relation f , we say that A f B when A f B and A = B. Analogously, for each worker w ∈ W and any two sets of firms C, D ⊆ F, we write C w D and C w D. Preferences profiles are (n + m)-tuples of preference relations and they are represented by = ( f1 , . . . , fn , w1 , . . . , wm ) = (( f ) f ∈F , ( w ) w∈W ). We denote by a ∈ F ∪ W a generic agent of either set. Given a preference relation of an agent a , the subsets of partners preferred to the empty set by a are called acceptable.
To express preference relations in a concise manner, and since only acceptable sets of partners will matter, we will represent preference relations as lists of acceptable partners. For instance, The assignment problem consists of matching workers with firms keeping the bilateral nature of their relationship and allowing for the possibility that both, firms and workers, may remain unmatched. Formally, Definition 2.1. A matching µ is a mapping from the set F ∪ W into the set of all subsets of F ∪ W such that, for all w ∈ W and f ∈ F : We say that an agent a is single in a matching µ if µ (a) = ∅. Otherwise, the agent is matched. A matching is said to be one-to-one (known as the marriage problem) if firms can hire at most one worker, and workers can work for at most one firm. A matching is said to be many-to-one if workers can work for at most one firm but firms may hire many workers.
Suppose each worker w gives its ranking of individual firms and orders subsets of firms in a responsive manner; namely, to add "good" firms to a set leads to a better set, whereas to add "bad" firms to a set leads to a worst set. In addition, for any two subsets that differ in only one firm, a worker prefers the subset containing the most preferred firm. Given an ordered list of quotas s = (s 1 , . . . , s m ) of the workers, we state the definition formally, as follow, Definition 2.2. The preference relation w over 2 F is responsive if satisfies the following conditions: Given a set of firms S ⊆ F , each worker w ∈ W can determine which subset of S would most prefer to hire. We will call this the w's choice set from S, and denote it by Ch (S, w ). Formally, Symmetrically, given a set of workers S ⊆ W , let Ch (S, f ) denote firm f 's most preferred subset of S according to its preference relation f . Formally, We assume that firms' preferences for groups of workers are such that the firms regard individual workers more as substitutes for each other than as complements. See Chapter 6 in Roth and Sotomayor [15]'s book for the complete bibliography. Formally, That is, if f has substitutable preferences, then if its preferred set of employees from S includes w, so will its preferred set of employees from any subset of S that still includes w. 9 A preference profile ( f ) f ∈F is substitutable if each f satisfies substitutability. Note that substitutability is a weaker condition than responsiveness.
A matching µ is blocked by agent a if µ (a) = Ch (µ (a) , a ). A matching µ is individually rational if it is not blocked by any individual agent. A matching µ is blocked by a worker-firm pair A matching µ is stable if it is not blocked by any individual agent or any worker-firm pair.
Let M = (F, W, s, ) be a specific many-to-many matching problem such that firms have substitutable preferences and workers have responsive preferences. Remark 1. When µ ∈ M is blocked by a worker-firm pair (w, f ) , and since we assume that ( w ) w∈W is responsive, an equivalent formulation of the condition 3. The connection between the models. Gale and Sotomayor [5] give a formal proof of the equivalence between the college admissions problem and its related marriage problem. In the present section, we study if it is possible to extend in a natural way the methodology developed by Gale and Sotomayor [5] to study the equivalence between the many-to-many matching model and its related many-to-one matching model.

3.1.
The related many-to-one problem. Given the many-to-many matching problem M we can consider its related many-to-one problem, in which each worker w j with quota s j is broken into s j "pieces" of itself. In the related market each worker has a quota of one. In other words, we replace each worker w j by s j positions (copies) of w j , denoted by w 1 j , . . . , w sj j , where the superscript of w t j indicates the t-th copy of w j . 11 Each one of them, in that related many-to-one problem, has preferences over F ∪ ∅ that are identical with those of w j with a quota of one. We denote by W s the set of copies of W ; that is, Observe that, |W s | = m j=1 s j . In addition, in the related market, each firm has preferences over W s ∪ ∅ as follow: the preference relation of each firm f ∈ F is modified replacing w j , where this appears, by the list w 1 j , . . . , w sj j in that order. That is, if a firm f in the original many-to-many problem prefers w to w , then the firm f in the related many-to one problem, will prefer all of the positions of w over all of the positions of w . Since every firm f ∈ F has preferences over the set of positions (copies) of workers, we will suppose that, if w j f ∅, then w t j f w t j if and only if t < t . We denote by I j = {1, . . . , s j } the set of indexes of the copies of every worker w j .
The following example shows that the natural correspondence used by Gale and Sotomayor [5] to prove the equivalence between the college admissions problem and its related marriage problem can not be extended in a natural way to many-to-many matching problems and their related many-to-one matching problems. 2). The firms have the following substitutable preferences, The workers have the following responsive preferences, observe that our many-to-many problem is more general that the many-to-many problem with capacity restrictions in both sides of the market. 11 If some worker w j has quota equal to one, then simply will denote its unique copie as w 1 j .
{f 1 , f 2 } w2 {f 1 } w2 {f 2 } w2 ∅ Consider the related many-to-one problem in which each worker w j is replaced by two positions (copies) of himself, then W s = {w 1 1 , w 2 1 , w 1 2 , w 2 2 }. Now, preferences of each w t j are over F ∪ ∅, and preferences of each f i are over W s ∪ ∅. Each copy has the same preferences over firms that are identical with those of w j with a quota of one, that is, {f 1 } w t j {f 2 } w t j ∅ for all j = 1, 2 and t = 1, 2. To construct firms' preferences is only needed the firms preference on singleton subsets of workers. Thus, each firm f i has the following preferences: Let ν be the following stable matching of the related many-to-one problem, Following Gale and Sotomayor [5], note that since all copies w 1 1 , w 2 1 corresponding to worker w 1 have the same preferences then, in order for the matching ν in the related many-to-one problem, corresponds to a matching µ in the many-to-many problem, it must be that the most preferred firm in µ(w 1 ) is matched to w 1 1 and the second most preferred to w 2 1 . Thus, µ(w 1 ) = f 1 . Similarly, µ(w 2 ) = f 2 . To sum up, However, matching µ is not stable in the many-to-many problem because is blocked, for example,by the worker-firm pair (w 2 , f 1 ) since the quota of worker w 2 is two.
Example 1 shows that the straightforward extended of the procedure proposed by Gale and Sotomayor [5] for the college admission problem and the marriage problem does not preserve the stability of the matchings between the many-tomany matching problem and the corresponding related many-to-one problem. This negative result, led us to modify the agents' preferences in the related many-to-one model and investigate if such equivalence exists. Definition 3.1. Given S ⊆ W s , we will say that S has clones if there exists w j and t, t ∈ I j (t = t ) such that w t j , w t j ⊆ S. If S ⊆ W s has no clones, we call it a simple set.
Definition 3.1 specifies that a subset of copies of workers, S ⊆ W s , has clones if there exists at least two copies of a same worker in S ⊆ W s .Given S ⊆ W s , let S = {w j ∈ W | ∃ t ∈ I j such that w t j ∈ S}. We define for each firm a strict preference relation * f over 2 W s derived from f over 2 W as follows, Definition 3.2. Let S, S ⊆ W s and f over 2 W be given. Denote by * f any complete preference relation 12 over 2 W s that satisfies the following properties: 1. If S is not a simple set, then ∅ * f S. 2. If S and S are simple sets: (a) If w t j ∈ S and w t j / ∈ S are such that t < t , and w j f ∅, then (b) If S = S , then S * f S if and only if S f S . The following lemmata characterize the choice set of firm f ∈ F in the related many-to-one market.
Proof. Suppose that S ⊆ W s is not a simple set. Let which implies, by definition of * f , Ch(S, * f ) f S for all S ⊆ S; that is, Let w t j ∈ Ch(S, * f ). By definition, w j ∈ Ch(S, * f ), thus by the Claim, w j ∈ Ch(S, f ). Hence, w t j ∈ A. Let w t j ∈ A ⊆ S; that is, w j ∈ Ch(S, f ) and t ≤ t for all w t j ∈ S. It follows from the Claim that w j ∈ Ch(S, * f ), and there exists t ∈ I j such that w t j ∈ Ch(S, * f ) and t < t for all w t j ∈ S. Thus, w t j ∈ Ch(S, * f ). Particularly, if S ⊆ W s is a simple set, we obtain that Therefore, denote the many-to-one problem related to M by: where, for each f ∈ F, * f is a preference relation over 2 W s and, for each w t ∈ W, w t is the preference relation over 2 F generated by w over 2 F . Let M s be the set of all matchings in M s .
For simplicity of notation we will use the letter µ to denote a matching in M and will use the letter ν to denote a matching in M s .
A matching ν ∈ M s is individually rational if it is not blocked by any individual agent. In this case, a matching ν ∈ M s is blocked by a position w t ∈ W s , if ∅ w t ν(w t ); that is, if the agent w t prefers remaining alone to be matched to ν(w t ).
Note that a matching ν in M s is blocked by a worker-firm pair f ), and f w t ν(w t ). A matching ν ∈ M s is stable if it is not blocked by any individual agent or any worker-firm pair. Let S(M s ) be the set of stable matchings in M s .
We will define a correspondence between matchings in the original many-to-many model and matchings in the related many-to-one model as follows. We say that matching µ ∈ M corresponds to matching ν ∈ M s if f is the w's most preferred firm in µ(w), then ν(w 1 ) = f, where w 1 is the first copy of worker w. Analogously, if f is the second w's most preferred firm in µ(w), then ν(w 2 ) = f , where w 2 is the second copy of worker w, and so forth. Also, if w does not fill his quota at µ(w), i.e., |µ(w)| = r < s w , ν(w t ) = ∅ for those copies w t of w, with t > r. Formally: Let µ ∈ M and w j ∈ W be such that µ(w j ) = {f i1 , f i2 , . . . , f ir } , r ≤ s j and Define, φ : M → M s , by denoting φ(µ) ≡ φ µ , as follows; Since each worker w j ∈ W has responsive preferences over 2 F , we have that φ µ (w t j ) wj φ µ (w t+1 j ) for all t ∈ I j . The next theorem states that the sets of stable matchings of the many-to-many problem M, and its related many-to-one problem M s , are equivalent. Thus, we obtain a generalization of the result of Gale and Sotomayor [5]. 13 Theorem 3.5. Let φ : M → M s be defined as in (2). Let M = (F, W, s, ) be a many-to-many problem such that workers' preferences are responsive and let M s = (F, W s , s, ( * f ) f ∈F , ( w t ) w t ∈W s ) be its related many-to-one problem. Then, µ ∈ M is stable if and only if φ(µ) ∈ M s is stable.
It is important to highlight that Theorem 3.5 does not require the condition of substitutability for the firms' preferences.
The proof of Theorem 3.5 will use the following lemmata: Lemma 3.6. The function φ is injective.

Remark 2.
Note that when firms can employ only one worker, the natural injective correspondence defined by Gale and Sotomayor [5] is a particular case of our injective application φ : M → M s .
Lemma 3.7. Let µ ∈ M be a stable matching. Then, φ(µ) ∈ M s is a stable matching.
Proof. Suppose that φ(µ) / ∈ S(M s ) and denote ν ≡ φ(µ). We will consider the following two cases: Case 1. The matching ν is not individually rational in M s .
(a) If w t ∈ W s blocks ν, ∅ w t ν(w t ). Let f t = ν(w t ); hence by definition, f t ∈ µ(w). We obtain ∅ w t f t , and so ∅ w f t .
Case 2. The matching ν is individually rational but is blocked by a worker-firm pair.
By assumption, there exists a pair (w t j , f i ) which blocks ν; that is, . From (1) and Lemma 3.3, w j ∈ Ch(ν(f i ) ∪ w t j , fi ) and t < t for all w t j ∈ ν(f i ) ∪ w t j . Claim Proof of Claim. Remember that and from (1) and Lemma 3.3, t < t , which gives ν(w t j ) = f i . Then by (2), ν(w t j ) wj ν(w t j ), thus t < t, contrary to t < t . This proves that there does not exist t = t such that j . This establishes that, . Suppose that ν(w t j ) = f i for some i = i; thus, f i ∈ µ(w j ), then from condition (2) we have f i w t j f i ; that is, f i wj f i ; since wj is responsive. By Remark 1, we obtain: Finally, from (4) and (5), µ is blocked by a worker-firm pair (f i , w j ), which is a contradiction. Therefore, ν ∈ S(M s ).
By Lemma 3.6 the function φ is injective. However, the function φ is not surjective. For example, let F = {f 1 , f 2 }, W = {w} and s w = 2, be such that w is acceptable by f 1 and f 2 , and w has the following responsive preference, Consider the related many-to-one problem in which the worker w is replaced by two copies of himself; then W s = {w 1 , w 2 }. Now, preferences of each w t are over F ∪ ∅, and preferences of each f i are over W s ∪ ∅, for all t = 1, 2 and for all i = 1, 2. Each copy has the same preferences over firms that are identical with those of w with a quota of one, that is, {f 1 } w t {f 2 } w t ∅ for all t = 1, 2. In addition, each firm f i has the following preferences: Let ν be a matching of the related many-to-one problem M s Let µ be a matching of the many-to-many problem M Then, Thus, ν = φ µ . Since µ is the unique matching in M, we conclude that the function φ is not surjective. Next, we will consider the restriction of φ over the set of stable matchings, denoted by φ S . The following lemma states that φ S is a surjective function. Proof. Fix ν ∈ S(M s ) and define for each f i ∈ F and w j ∈ W µ(f i ) = w j | ∃t ∈ I j such that ν(w t j ) = f i and µ(w j ) = ν(w t j ) | t ∈ I j . By definition, φ S (µ) ≡ ν. We show that µ ∈ S(M). Suppose that µ / ∈ S(M). We will distinguish between two cases: Case 1. The matching µ is not individually rational.
Then, since w is responsive, ∅ w t f for all t ∈ I w . As f ∈ µ (w) , there exists t ∈ I w such that f = ν(w t ). Since ∅ w t ν(w t ), worker w t blocks the matching ν, which is a contradiction. Therefore, µ ∈ S(M). ( In the previous section we defined the function φ mapping matchings of the original problem M into matchings of the related problem M s and we proved that both problems have sets of equivalent stable matchings. In this section, using the function φ S , we will establish that the set of stable matchings of M has a lattice structure. 18 Following Blair [2], let µ 1 and µ 2 two individually rational matchings for all firms in the many-to-many problem M. Define the partial order B F , by setting Similarly, we define the partial order B W for all workers in the many-to-many model M.
Let ν 1 and ν 2 two individually rational matchings for all firms in the related many-to-one model M s . Define the partial order * B F , by setting Finally, we define the partial order W s in the related many-to-one model M s , by setting  Corollary 2. Let µ, µ ∈ S(M). If for some w we have that µ(w) B w µ (w), then f w f for all f ∈ µ(w) and f ∈ µ (w)\µ(w).

Final
Remarks. This paper contributes to the literature by proposing a new tool (the bijection φ S ) to prove results in the many-to-many model that follow from results in the many-to-one model. In particular, we gave an alternative proof of the non-emptiness and lattice structure of the set of stable matchings in the manyto-many model with substitutable preferences for firms, and responsive preferences for workers. The proof follows from results that hold in the related many-to-one model. We finish the paper with two final remarks.
First, Lemma 3.10 shows that, if the preference relation f is substitutable in the model M, then the preference relation * f is substitutable in the model M s . Nevertheless, if we assume that the preference relation f is responsive in the to find a sufficient condition for the existence of setwise stable matchings. Thus, it would be interesting to ask, using the bijection φ S , whether one could prove that the set of setwise stable matchings is non-empty. But, we leave the answers of these questions for further research.