A New Proof of the Lattice Structure of Many-to-Many Pairwise-Stable Matchings

Abstract

This paper studies, under substitutable and cardinal monotone preferences, the lattice structure of the set $S(P)$ of many-to-many pairwise-stable matchings. It proves that the selection matchings are increasing functions on $S(P)$. This fact, along with a few auxiliary results, is then used to prove that $S(P)$ is a distributive lattice. The contribution of this paper is not new, but the alternative proof is interesting as it avoids the use of abstract lattice theory.

1 Introduction

In many-to-many two-sided matching markets, definition of stability is a subtle issue; pairwise-stability, setwise-stability, and corewise-stability are common concepts of stability,¹ standard definitions of which imply that “any setwise-stable matching is pairwise-stable and must be in the core.”²

Under strict, substitutable, and cardinal monotone preferences, Konish and Ünver [9] observed, through an example [9, p. 65, Example 2], that the core may be empty, which answers the open question from Sotomayor [14, p. 58, line 10] “whether or not many-to-many core matchings always exist is apparently still an open question”; Blair [2] showed, also through an example, that the set of pairwise-stable matchings and the core might be disjoint [2, p. 622, Example 2.6]. Both examples illustrate, though from different perspectives, that the set of setwise-stable matchings might be empty, which was also proved by Sotomayor [14, p. 60, Example 1].

Roth [11, p. 55, Theorem 2] and Blair [2, p. 623, Theorem 3.2] both proved the existence of many-to-many pairwise-stable matchings. Therefore, currently, only the set of pairwise-stable matchings $S(P)$ may be our study object.

Given two different many-to-many matchings $\mu _1$ and $\mu _2$, let each firm select its most preferred subset of employees from those that assigned to it at $\mu _1$ and $\mu _2$. The selections result in a many-to-many matching, which we denote $\lambda (\mu _1, \mu _2)$ and call selection matching made by firms. We may define, similarly, the selection matching made by workers, $\nu (\mu _1, \mu _2)$.

Alkan [1] proved, under strict, substitutable, and cardinal monotone revealed preferences, that the selection matchings of pairwise-stable matchings are themselves pairwise-stable,³ and that the set of pairwise-stable matchings is a lattice [1, p. 743, Proposition 8].

The current paper proves that the selection matchings are “increasing” functions on the set $S(P)$. This fact, along with a few auxiliary results, is then used to prove that $S(P)$ is a distributive lattice. Although the contribution of this paper is not new, the alternative proof is interesting as it avoids the use of abstract lattice theory.

Hatfield and Milgrom [5] called “law of aggregate demand” to “cardinal monotonicity,” under which they studied many-to-one core [5, p. 917, left column, line 33; right column, line 3] matchings with contracts.⁴ But in many-to-many matching markets, the core and the set of pairwise-stable matchings might be disjoint; furthermore, the core might be empty, as we discussed in the second paragraph. So their results cannot be generalized to the current setting.

Hatfield and Kominers [6, 7] studied properties of setwise-stable matchings [6, p. 183, the last paragraph; p. 184, definition 4; 7, p.7, definition 2] in a one-sided matching game and a many-to-many matching game with contracts, respectively, which they claimed subsume many-to-many matching [6, p. 176, line 5; 7, p. 1, lines 1 to 2 of the Abstract].⁵ But in many-to-many matching markets, the set of setwise-stable matchings is a moot point. So the current setting must be different substantially from theirs.

Blair [2] showed that the set of many-to-many pairwise-stable matchings is a lattice under an “appropriate” partial order.⁶ However, Blair’s lattice was not distributive [2, p. 627, Example 5.2], and the selection matchings might not be pairwise-stable [2, p. 624, line 43].

One distinction between the current paper and Alkan [1] lies in the conditions on preferences necessary for the main results: the current paper works with the complete preference orderings, whereas Alkan [1] worked with the incomplete revealed preference orderings of agents. Another, crucial, distinction is that Alkan proved the distributivity by a result in abstract lattice theory [1, p. 744, lines 25–30], which is an outside fact of his setting [1, p. 739, lines 22–23], whereas the current paper does it without outside fact.

Echenique and Oviedo [3] acknowledged and analyzed the failure of distributivity in their lattice [3, p. 254, lines 28–30].

The paper is organized as follows. In Sect. 2, we present the preliminary notations and definitions. In Sect. 3, we study the properties of the selection matchings and in Sect. 4 we prove the distributivity.

2 Preliminaries

There are two disjoint sets of agents: the set of $n$ firms $F=\{{{f_1},\cdots , {f_n}} \}$, and the set of $m$ workers $W= \{{w_1},\cdots ,{w_m}\}$. Firms can hire groups of workers, and workers can be employed by sets of firms.

Each firm $f\in F$ has a complete, transitive, and strict preference list $P_f$ over all the subsets of $W$; each worker $w \in W$ has a complete, transitive, and strict preference list $P_w$ over all the subsets of $F$. Let $P=\{ P_{f_1}, \cdots , P_{f_n}, P_{w_1}, \cdots , P_{w_m}\}$ denote a Preference profile.

Given a preference profile $P$, for each firm $f\in F$ and any two sets of workers $S, S' \subseteq W$, we write $S P_f S'$ to mean $f$ prefers $S$ to $S'$, and $S R_f S'$ to mean $f$ prefers $S$ at least as well as $S'$; analogously, for each worker $w \in W$ and any two sets of firms $T, T' \subseteq F$, we write $T P_w T'$ and $T R_w T'$. The alternatives, preferred by agent $k$ to the empty set $\emptyset$, are called acceptable to $k$, otherwise are unacceptable to $k$.

A matching specifies which workers are employed by each firm, which firms employ each worker, and which firms and workers are unmatched. Formally, we give the following definition of a matching.

Definition 2.1

A matching $\mu$ is a function from the set $F \cup W$ into the set of all subsets of $F \cup W$ such that for every worker $w \in W$, and for every firm $f \in F$:

• (1) $\mu (w) \subseteq F$;

• (2) $\mu (f) \subseteq W$;

• (3) $w \in \mu (f) \Leftrightarrow f \in \mu (w)$.

We say an agent $k \in F \cup W$ is matched if $\mu (k) \ne \emptyset$, otherwise he is unmatched.

Given a preference profile $P$ and a set $S$ of workers, each firm $f$ can determine which subset of $S$ it would most prefer to hire; we call this $f's$ choice from $S$, and denote it by $C_f(S)$, viz., $C_f(S) \subseteq S$ and $C_f(S) R_f S'$ for all $S' \subseteq S$. Similarly, we define each worker $w's$ choice, $C_w(T)$, from a set $T$ of firms.

We say a matching $\mu$ is individually rational if ${C_k}(\mu (k)) = \mu (k)$ for all agents $k$, $\mu$ is blocked by an agent $k$ if ${C_k}(\mu (k)) \ne \mu (k)$ and by a worker–firm pair $(w,f)$ if $w \notin \mu (f)$, but $w \in C_f(\mu (f) \cup \{w\})$ and $f \in C_w(\mu (w) \cup \{f\})$.

Definition 2.2

A matching is pairwise-stable if it is not blocked by any agent or any worker–firm pair.

It is well known that pairwise-stable matching always exists when agents’ preferences satisfy substitutability, which was introduced by Kelso and Crawford [8], and which we state formally as below.

Definition 2.3

An agent $k's$ preference list $P_k$ satisfies substitutability if, for any subset $S$ of the opposite set that contains agent $i$, $i \in C_k(S)$ then $i \in C_k(S' \cup \{i\})$ for all $S' \subseteq S$.

Remark 2.4

If agent $k's$ preference list $P_k$ is strict and substitutable, then $C_k(S_1 \cup S_2 ) = C_k( C_k(S_1) \cup S_2)$, where ${S_1}$ and ${S_2}$ are subsets of the opposite set.⁷

Alkan [1] introduced cardinal monotonicity, under which he proved the pairwise-stability of the selection matchings, and which we state as follows.

Definition 2.5

An agent $k's$ preference list $P_k$ satisfies cardinal monotonicity if, for all subsets $S$ of the opposite set and all $S' \subseteq S,\, \left| C_{k}(S')\right| \leqslant \left| C_{k}(S)\right|$.⁸

3 Monotonicity

Given a preference profile $P$, let $M(P)$ denote the set of many-to-many matchings, and let $S(P)$ denote the set of many-to-many pairwise-stable matchings. For any two matchings $\mu _1, \mu _2 \in M(P)$, let each firm select its most preferred subset of workers from those that assigned to it at $\mu _1$ and $\mu _2$, then the selections produce a many-to-many matching.

Formally, the selection matching made by firms of $\mu _1$ and $\mu _2,\, \lambda (\mu _1, \mu _2)$, is defined by $\lambda (\mu _1, \mu _2)(f) = {C_f}( {{\mu _1}(f) \cup {\mu _2}(f)} )$ for all firms $f \in F$, and $\lambda (\mu _1, \mu _2)(w) = \{f| w \in C_f(\mu _1(f) \cup \mu _2(f) ) \}$ for all workers $w \in W$. Similarly, the selection matching made by workers of $\mu _1$ and $\mu _2,\, \nu (\mu _1, \mu _2)$, is defined by $\nu (\mu _1, \mu _2)(w) = {C_w}( \mu _1(w) \cup \mu _2(w) )$ for all workers $w \in W$, and $\nu (\mu _1, \mu _2)(f) = \{w| f \in C_w(\mu _1(w) \cup \mu _2(w) ) \}$ for all firms $f \in F$. So $\lambda$ and $\nu$ are binary operations on the set $M(P)$.

Let $\mu _1, \mu _2 \in M(P)$, define $\mu _1 \succeq _F^{B} \mu _2$ iff $C_f( \mu _1(f) \cup \mu _2(f) )=\mu _1(f)$ for all firms $f \in F$, and $\mu _1 \succeq _W^{B} \mu _2$ iff $C_w( \mu _1(w) \cup \mu _2(w) )=\mu _1(w)$ for all workers $w \in W$. Then for every matching $\mu \in M(P)$, $\mu \succeq _F^{B} \mu$ and $\mu \succeq _W^{B} \mu$. So we have two partial orders, $\succeq _F^{B}$ and $\succeq _W^{B}$, on the set $M(P)$, which include the stable ones.

Fix a matching $\mu \in M(P)$, let $\lambda _{\mu }(\mu ')=\lambda (\mu , \mu ')$ and $\nu _{\mu }(\mu ')=\nu (\mu , \mu ')$ for all $\mu ' \in M(P)$, then $\lambda _{\mu }$ and $\nu _{\mu }$ are functions on $M(P)$. We investigate, on the partial sets $(M(P), \succeq _F^{B})$ and $(M(P), \succeq _W^{B})$, the monotonicity properties of $\lambda _{\mu }$ and $\nu _{\mu }$, which are sufficient for the distributivity proved in the next section, and make our proof substantially different than that of Alkan [1].

Proposition 3.1

When agents have substitutable preferences, let $\mu , \mu _1,\, \mu _2 \in M(P)$, if $\mu _1 \succeq _F^{B} \mu _2$ then $\lambda _{\mu }(\mu _1) \succeq _F^{B} \lambda _{\mu }(\mu _2)$.

Proof

Suppose $\mu _1 \succeq _F^{B} \mu _2$, then $C_f( \mu _1(f) \cup \mu _2(f) )=\mu _1(f)$ for all firms $f \in F$ by the definition of $\succeq _F^{B}$. By the definition of the selection matching made by firms, for all $f \in F$,

$$\begin{aligned}&C_f( \lambda (\mu , \mu _1)(f) \cup \lambda (\mu , \mu _2)(f) )\\&\quad= C_f( C_f( \mu (f) \cup \mu _1(f) ) \cup C_f(\mu (f) \cup \mu _2(f) ) )\\&\quad= C_f( \mu (f) \cup \mu _1(f) \cup \mu _2(f) )\\&\quad= C_f( \mu (f) \cup C_f( \mu _1(f) \cup \mu _2(f) ) )\\&\quad= C_f( \mu (f) \cup \mu _1(f) )\\&\quad= \lambda (\mu , \mu _1)(f), \end{aligned}$$

where the first and the last equations hold by the definition of $\lambda$, the fourth equation holds by the assumption and all the others by substitutability (see Remark 2.4). Thus $\lambda _{\mu }(\mu _1) \succeq _F^{B} \lambda _{\mu }(\mu _2)$. $\square$

Because workers and firms play a symmetric role in our model, we have directly the following proposition.

Proposition 3.2

When agents have substitutable preferences, let $\mu , \mu _1,\,\mu _2 \in M(P)$ , if $\mu _1 \succeq _W^{B} \mu _2$ then $\nu _{\mu }(\mu _1) \succeq _W^{B} \nu _{\mu }(\mu _2)$.

Propositions 3.1 and 3.2 exhibit that $\lambda _{\mu }$ and $\nu _{\mu }$ are “increasing” functions on $(M(P), \succeq _F^{B})$ and $(M(P), \succeq _W^{B})$, respectively. The following lemma comes from Theorem 4.5 of Blair [2].

Lemma 3.3

When agents have substitutable preferences, if $\mu _1, \mu _2 \in S(P)$ then $\mu _1 \succeq _F^{B} \mu _2 \Leftrightarrow \mu _2 \succeq _W^{B} \mu _1$.

Lemma 3.3 sets a bridge between the two partial orders $\succeq _F^{B}$ and $\succeq _W^{B}$, which hence enlarges the domain of both functions, as we will prove below.

The following lemma comes from Proposition 8 of Alkan [1, p. 743].

Lemma 3.4

When agents have substitutable and cardinal monotone preferences, the set $S(P)$ is closed under $\lambda$ and $\nu$.

Examples in Li [10, p. 390, Example 3.1] and Alkan [1, p. 745] showed that cardinal monotonicity is a necessary condition for the pairwise-stability of the selection matchings.

Proposition 3.5

When agents have substitutable and cardinal monotone preferences, let $\mu , \mu _1, \mu _2 \in S(P)$ , if $\mu _1 \succeq _F^{B} \mu _2$ then $\nu _{\mu }(\mu _1) \succeq _F^{B} \nu _{\mu }(\mu _2)$.

Proof

Suppose $\mu _1 \succeq _F^{B} \mu _2$, then Lemma 3.3 implies $\mu _2 \succeq _W^{B} \mu _1$, so $\nu _{\mu }(\mu _2) \succeq _W^{B} \nu _{\mu }(\mu _1)$ by Proposition 3.2. Since $\mu$, $\mu _1$, and $\mu _2$ are pairwise-stable, Lemma 3.4 gives both $\nu _{\mu }(\mu _2)$ and $\nu _{\mu }(\mu _1)$ are pairwise-stable matchings, hence Lemma 3.3 implies $\nu _{\mu }(\mu _1) \succeq _F^{B} \nu _{\mu }(\mu _2)$. $\square$

The following proposition holds immediately by symmetry.

Proposition 3.6

When agents have substitutable and cardinal monotone preferences, let $\mu , \mu _1, \mu _2 \in S(P)$ , if $\mu _1 \succeq _W^{B} \mu _2$ then $\lambda _{\mu }(\mu _1) \succeq _W^{B} \lambda _{\mu }(\mu _2)$.

4 Distributivity

We will prove the distributivity in this section. The monotonicity properties proved in the previous section along with the properties proved below are the bases of our proof and make the proof substantially different than that of Alkan [1].

Proposition 4.1

When agents have substitutable preferences, let $\mu _1, \mu _2 \in S(P)$ , then the following equations hold for all firms $f \in F$:

$$\begin{aligned} C_f(\nu (\mu _1, \mu _2)(f) \cup \mu _i(f))=\mu _i(f),\quad i=1, 2. \end{aligned}$$

Proof

We prove the equation for $i=1$.

Suppose not then there exists a firm $f \in F$ such that $C_f(\nu (\mu _1, \mu _2)(f) \cup \mu _1(f)) \ne \mu _1(f)$. Since $\mu _1$ is pairwise-stable, $C_f(\nu (\mu _1, \mu _2)(f) \cup \mu _1(f)) P_f \mu _1(f) R_f \emptyset$, there exists a worker $w \in W$ such that $w \in C_f(\nu (\mu _1, \mu _2)(f) \cup \mu _1(f))$ and $w \notin \mu _1(f)$; so $w \in \nu (\mu _1, \mu _2)(f)$, and so $w \in \mu _2(f)$ ($f \in \mu _2(w)$ by the definition of a matching) by the definition of $\nu$, and $w \in C_f(\mu _1(f) \cup \{w\} )$ by substitutability. Because $\mu _1$ is pairwise-stable and $w \notin \mu _1(f)$, $f \notin C_w(\mu _1(w) \cup \{f\} )$. Since $f \in \mu _2(w)$ by the above analysis, $f \notin C_w(\mu _1(w) \cup \mu _2(w))$ by substitutability, viz., $f \notin \nu (\mu _1, \mu _2)(w)$, which contradicts the above analysis that $f \in \nu (\mu _1, \mu _2)(w)$. $\square$

Because firms and workers play a symmetric role in our model, we have the following result.

Proposition 4.2

When agents have substitutable preferences, let $\mu _1, \mu _2 \in S(P)$ , then the following equations hold for all workers $w \in W$

$$\begin{aligned} C_w(\lambda (\mu _1, \mu _2)(w) \cup \mu _i(w))=\mu _i(w),\quad i=1, 2. \end{aligned}$$

Lemmas 4.3 and 4.4 below come from Propositions 8 and 7 of Alkan [1], respectively.

Lemma 4.3

When agents have substitutable and cardinal monotone preferences, $(S(P), \lambda , \nu , \succeq _F^{B})$ and $(S(P), \nu , \lambda , \succeq _W^{B})$ are lattice.

Lemma 4.4

When agents have substitutable and cardinal monotone preferences, the property of complementarity holds on $S(P)$ , viz., let $\mu , \mu ' \in S(P)$ , then (i) $\lambda (\mu , \mu ')(k) \cap \nu (\mu , \mu ')(k)=\mu (k) \cap \mu '(k)$ and (ii) $\lambda (\mu , \mu ')(k) \cup \nu (\mu , \mu ')(k)=\mu (k) \cup \mu '(k)$ for all agents $k \in F \cup W$.

The following proposition exhibits that $\lambda$ is distributive over $\nu$ on $S(P)$, and propositions 3.1, 4.1, and the complementarity are the foundation of the proof.

Proposition 4.5

When agents have substitutable and cardinal monotone preferences, let $\mu _1, \mu _2, \mu _3 \in S(P)$ , then

$$\begin{aligned} \lambda ( \mu _1, \nu (\mu _2, \mu _3) )= \nu ( \lambda (\mu _1, \mu _2), \lambda (\mu _1, \mu _3) ). \end{aligned}$$

Proof

Proposition 4.1 in conjunction with the definition of $\succeq _F^{B}$ gives $\mu _2 \succeq _F^{B} \nu (\mu _2, \mu _3)$ and $\mu _3 \succeq _F^{B} \nu (\mu _2, \mu _3)$, then Proposition 3.1 implies $\lambda (\mu _1, \mu _2) \succeq _F^{B} \lambda (\mu _1,\nu (\mu _2, \mu _3))$ and $\lambda (\mu _1, \mu _3) \succeq _F^{B} \lambda (\mu _1, \nu (\mu _2, \mu _3))$. Thus, $\lambda (\mu _1, \nu (\mu _2, \mu _3))$ is a lower bound of both $\lambda (\mu _1, \mu _2)$ and $\lambda (\mu _1, \mu _3)$. Since $\nu$ is the meet operation of partial order $\succeq _F^{B}$ (Lemma 4.3), $\nu ( \lambda (\mu _1, \mu _2), \lambda (\mu _1, \mu _3) ) \succeq _F^{B} \lambda ( \mu _1, \nu (\mu _2, \mu _3) )$. Then by the definition of $\succeq _F^{B}$, for all firms $f \in F$,

$${{C_f(\nu (\lambda (\mu _1, \mu _2), \lambda (\mu _1, \mu _3))(f) \cup \ \lambda (\mu _1, \nu (\mu _2, \mu _3))(f))}} ={{ \nu (\lambda (\mu _1, \mu _2), \lambda (\mu _1, \mu _3))(f)}}.$$

(4.1)

Suppose there is a firm $f \in F$ such that

$$\nu ( \lambda (\mu _1, \mu _2), \lambda (\mu _1, \mu _3) )(f) \ne \lambda ( \mu _1,\,\nu (\mu _2, \mu _3) )(f)$$

. Since $\lambda ( \mu _1, \nu (\mu _2, \mu _3))$ is pairwise-stable by Lemma 3.4 and agents have strict preferences, Eq. (4.1) gives

$$\begin{aligned} \nu ( \lambda (\mu _1, \mu _2), \lambda (\mu _1, \mu _3) )(f) P_f \lambda ( \mu _1, \nu (\mu _2, \mu _3) )(f) R_f \emptyset . \end{aligned}$$

So there exists a worker $w \in \nu ( \lambda (\mu _1, \mu _2), \lambda (\mu _1, \mu _3) )(f)$, but $w \notin \lambda ( \mu _1, \nu (\mu _2, \mu _3) )(f)$. Because agents have substitutable preferences, Eq. (4.1) implies

$$\begin{aligned} w \in C_f( \lambda (\mu _1, \nu (\mu _2, \mu _3))(f) \cup \{w\} ). \end{aligned}$$

Since $\lambda ( \mu _1, \nu (\mu _2, \mu _3) )(f) =C_f( \mu _1(f) \cup \nu (\mu _2, \mu _3)(f) ),\, w \notin \mu _1(f) \cup \nu (\mu _2, \mu _3)(f)$ by substitutability. Then

$$\begin{aligned} w \notin \mu _1(f) \ and \ w \notin \nu (\mu _2, \mu _3)(f). \end{aligned}$$

(4.2)

Since $w \in \mu _1(f) \cup \mu _2(f) \cup \mu _3(f)$, $w \in \mu _2(f) \cup \mu _3(f)$. Then $w \in \lambda (\mu _2, \mu _3)(f)$ and $w \notin \mu _2(f) \cap \mu _3(f)$ by complementarity. So $w \notin \mu _1(f) \cup \mu _2(f)$ or $w \notin \mu _1(f) \cup \mu _3(f)$. Since $\lambda (\mu _1, \mu _2)(f) \subseteq \mu _1(f) \cup \mu _2(f)$ and $\lambda (\mu _1, \mu _3)(f) \subseteq \mu _1(f) \cup \mu _3(f)$, $w \notin \lambda (\mu _1, \mu _2)(f)$ or $w \notin \lambda (\mu _1, \mu _3)(f)$. So $w \notin \lambda (\mu _1, \mu _2)(f) \cap \lambda (\mu _1, \mu _3)(f)$ and consequently $w \notin \lambda ( \lambda (\mu _1, \mu _2), \lambda (\mu _1, \mu _3) )(f) \cap \nu ( \lambda (\mu _1, \mu _2),$ $\lambda (\mu _1, \mu _3) )(f)$ by complementarity. Since $w \in \nu ( \lambda (\mu _1, \mu _2), \lambda (\mu _1, \mu _3) )(f)$ by the assumption, again by complementarity $w \notin \lambda ( \lambda (\mu _1, \mu _2), \lambda (\mu _1, \mu _3) )(f)$. Since $\lambda ( \lambda (\mu _1, \mu _2), \lambda (\mu _1, \mu _3) )(f) =C_f( \mu _1(f) \cup \mu _2(f) \cup \mu _3(f) )$ by the definition of $\succeq _F^{B}$ and by substitutability, then $w \notin C_f( \mu _1(f) \cup \mu _2(f) \cup \mu _3(f) )$.

By the assumption, $w \in \nu ( \lambda (\mu _1, \mu _2), \lambda (\mu _1, \mu _3) )(f) \subseteq \lambda (\mu _1, \mu _2)(f) \cup \lambda (\mu _1, \mu _3)(f)$, and $w \notin \nu (\mu _2, \mu _3)(f)$ and $w \in \mu _2(f) \cup \mu _3(f)$ (see Eq. (4.2)) give $w \in \lambda (\mu _2, \mu _3)(f)$ by complementarity, thus $w \in \lambda (\mu _1, \mu _2)(f) \cap \lambda (\mu _2, \mu _3)(f)$ or $w \in \lambda (\mu _1, \mu _3)(f) \cap \lambda (\mu _2, \mu _3)(f)$. Therefore, $w \in \lambda ( \lambda (\mu _1, \mu _2), \lambda (\mu _2, \mu _3) )(f)$ or $w \in \lambda ( \lambda (\mu _1, \mu _3),$ $\lambda (\mu _2, \mu _3) )(f)$ by complementarity. But $\lambda ( \lambda (\mu _1, \mu _2), \lambda (\mu _2, \mu _3) )(f) = C_f( \mu _1(f) \cup \mu _2(f) \cup \mu _3(f) )$ and $\lambda ( \lambda (\mu _1, \mu _3), \lambda (\mu _2, \mu _3) )(f)=C_f( \mu _1(f) \cup \mu _2(f) \cup \mu _3(f) )$ by substitutability. Therefore, $w \in C_f( \mu _1(f) \cup \mu _2(f) \cup \mu _3(f) )$ which contradicts the above conclusion that $w \notin C_f( \mu _1(f) \cup \mu _2(f) \cup \mu _3(f) )$.

Thus, $\nu (\lambda (\mu _1, \mu _2), \lambda (\mu _1, \mu _3))(f) = \lambda (\mu _1, \nu (\mu _2, \mu _3))(f)$ for all firms $f \in F$, viz., $\lambda (\mu _1, \nu (\mu _2, \mu _3)) = \nu (\lambda (\mu _1, \mu _2), \lambda (\mu _1, \mu _3))$. $\square$

Because workers and firms play a symmetric role, Proposition 3.2, combined with the definition of $\succeq _W^{B}$ and Proposition 4.2, yields the following result.

Proposition 4.6

When agents have substitutable and cardinal monotone preferences, let $\mu _1, \mu _2, \mu _3 \in S(P)$ , then

$$\begin{aligned} \nu ( \mu _1, \lambda (\mu _2, \mu _3) )= \lambda ( \nu (\mu _1, \mu _2), \nu (\mu _1, \mu _3))\end{aligned}.$$

Example 5.2 of Blair [2] shows that, when agents do not have cardinal monotone preferences, the set of pairwise-stable matchings might not be a distributive lattice.

5 Conclusion

Lattice is a fundamental concept of optimization theory; it provides a concrete path to the optimal objectives. In many-to-many two-sided matching markets, the interests of agents on the same side of the market can be simultaneously maximized [4, p. 55, theorem 2]. Alkan [1] studied the lattice structure of many-to-many pairwise-stable matchings; he proved the distributivity by an abstract lattice theory which is outside his setting.

The current paper provides an alternative proof of the distributive lattice of many-to-many pairwise-stable matchings, which improves the study of the lattice structure in two-sided matching markets. The generality of our analysis is not only theoretically interesting but may lead to new insights on some open problems, such as the necessary and sufficient conditions for existence of a setwise-stable matching and a corewise-stable matching. These topics are left for future research.