Stability and Median Rationalizability for Aggregate Matchings

Abstract

We develop the theory of stability for aggregate matchings used in empirical studies and establish fundamental properties of stable matchings including the result that the set of stable matchings is a non-empty, complete, and distributive lattice. Aggregate matchings are relevant as matching data in revealed preference theory. We present a result on rationalizing a matching data as the median stable matching.

1. Introduction

Following the seminal work of [1], an extensive literature has developed regarding matching markets with non-transferable utility. This literature assumes that there are agent-specific preferences, and studies the existence of stable matchings in which each agent prefers her assigned partner to the outside option of being unmatched, and there are no pairs of agents that would like to match with each other rather than keeping their assigned partners.

In this paper, we develop the theory of stability for aggregate matchings, which we interpret as a dataset of matchings, and consider its testable implications. In an aggregate matching, agents are classified into types according to their observable characteristics: for example age, employment status, gender, or race. The aggregate matching is then a table, a matrix, that records how many women of each type are matched with men of each type. Our notion of matching data is more general than just deterministic matchings and includes randomized matchings of agents to objects (a probabilistic assignment of children to schools, for example, as in [2,3]) as well. Therefore, our results are also applicable to randomized matching environments.

We assume that preferences are at the type-specific level, rather than the individual-level. In our setting, accommodating individual-level preferences would entail losing all empirical bite. Thus we adopt instead the assumption that the definition of types already carries all the information needed to determine agents’ preferences. This differs from the approach in empirical applications of matching theory, which assume transferable utilities (see, e.g., [4] or [5]), but allow for heterogeneous preferences at the individual-level. When our results are taken to data, one needs to introduce an appropriate error structure that allows for individual-level randomness but without individual-level preference heterogeneity. One way of doing this is via measurement error, as in the classical revealed preference theory for consumption [6,7]. One illustration can be found in [8].

We establish fundamental results on the set of stable matchings in Theorem 1. First of all, there exists a stable matching. Then we introduce two partial orders on matchings using preferences of the two sides of the market and show that for every set of stable matchings there exists the greatest lower bound and the least upper bound with respect to the two partial orders in the set of stable matchings, i.e., the set of stable matchings endowed with either partial order is a complete lattice. Furthermore, we show an additional structural result that this lattice is distributive. (The definition of distributive lattices is given in Section 2). Using the lattice structure on the set of stable matchings, we show that there exists a median stable matching, which assigns each agent the median outcome among all stable matching outcomes (Corollary 1). Theorem 1 also provides other structural results including “polarity of interests”, saying that a stable matching is more preferred for men if and only if it is less preferred for women. Furthermore, the outcomes in two stable matchings are always comparable using the partial orders for an agent. The “rural hospitals theorem”, which states that for each type of agent the number of partners in any two stable matchings are the same, also holds.

Once the stability of matchings is understood, we turn to the main question in the paper: when we interpret an aggregate matching as a table of matching data, and when we do not have access to the agents’ preferences, when can we say that the data is consistent with the theory? This is a revealed preference question that has been addressed by [9] (see also [10,11]). Here we focus on a particular selection of stable matchings—the median stable matching, which is defined for markets with deterministic outcomes. We provide a sufficient condition under which matching is rationalizable as the median stable matching (Theorem 3). We show that the condition is not necessary in Appendix A.

Theorem 1 on the structure of stable matchings generalizes well-known results in the literature to allow for randomization, aggregate matchings, and continuum of agents (see [12] for a review). Proposition 1 establishes the existence of generalized median stable matchings in our environment. For earlier existence results of median stable matchings see [13,14,15,16,17,18]. Revealed preference studies include [9,11,19,20]. The works of [9,20] take aggregate matchings as the source of data, but none of these papers considers the question of median-rationalizability.

2. Preliminary Definitions

2.1. Lattice Theory

If S is a set and ≤ is a partial order on S, we say that the pair $(S,\le )$ is a partially ordered set. We say that $x,y\in S$ are comparable if $x\le y$ or $y\le x$. A partially ordered set $(S,\le )$ is a lattice if, for every $x,y\in S$, the least upper bound and the greatest lower bound of $x,y$ with respect to the partial order ≤ exist in S. We denote the least upper bound of $x,y$ by $x\vee y$; and the greatest lower bound of $x,y$ by $x\wedge y$. Similarly, if for every $S^{\prime }\subset S$, the least upper bound and the greatest lower bound of $S^{\prime }$ exist in S, the lattice $(S,\le )$ is called complete.

We say that a lattice $(S,\le )$ is distributive if the following holds for all $x,y,z\in S$:

• $x\vee (y\wedge z)=(x\vee y)\wedge (x\vee z)$, and
• $x\wedge (y\vee z)=(x\wedge y)\vee (x\wedge z)$.

2.2. Graph Theory

An (undirected) graph is a pair $G=(V,L)$, where V is a set and L is a subset of $V\times V$. A path in G is a sequence $p=\langle v_0,\dots ,v_N\rangle$ such that for $n\in \{0,\dots ,N-1\}$, $(v_n,v_{n+1})\in L$. We write $v\in p$ to denote that v is a vertex in p. A path $\langle v_0,\dots ,v_N\rangle$ connects the vertices $v_0$ and $v_N$. A path $\langle v_0,\dots ,v_N\rangle$ is minimal if there is no proper subsequence of $\langle v_0,\dots ,v_N\rangle$ which is also a path connecting the vertices $v_0$ and $v_N$. The length of path $\langle v_0,\dots ,v_N\rangle$ is N.

A cycle in G is a path $c=\langle v_0,\dots ,v_N\rangle$ with $v_0=v_N$. A cycle is minimal if for any two vertices $v_n$ and $v_{n^{\prime }}$ in c, the paths in c from $v_n$ to $v_{n^{\prime }}$ and from $v_{n^{\prime }}$ to $v_n$ are distinct and minimal. We call v and w adjacent in c if there is n such that $v_n=v$ and $v_{n+1}=w$ or $v_n=w$ and $v_{n+1}=v$. If c and $c^{\prime }$ are two cycles, and there is a path from a vertex of c to a vertex of $c^{\prime }$, then we say that c and $c^{\prime }$ are connected.

3. Model

The primitives of the model are represented by a four-tuple $\langle M,W,P,K\rangle$, where

• M and W are finite and disjoints sets of, respectively types of men, and types of women.
• P is a preference profile: a list of preferences $P_m$ for every type-m man and $P_w$ for every type-w woman. Each $P_m$ is a linear order over $W\cup w_0$, and each $P_w$ is a linear order over $M\cup m_0$. Here, $w_0$ and $m_0$ represent the outside option of being unmatched. The weak order associated with $P_m$ is denoted by $R_m$ and the weak order associated with $P_w$ is denoted by $R_m$.
• K is a list of non-negative real numbers $K_m$ for each $m\in M$ and $K_w$ for each $w\in W$. There are $K_m$ type-m men and $K_w$ type-w women.

We enumerate M as $m_1,\dots ,m_{\left|M\right|}$ and W as $w_1,\dots ,w_{\left|W\right|}$. A matching is a $\left|M\right|\times \left|W\right|$ matrix $X=\left(x_{m_i,w_j}\right)$ such that $x_{m_i,w_j}\in {\Re }_{+}$, ${\sum }_j{x}_{m_i,w_j}\le K_{m_i}$ for all i, and ${\sum }_i{x}_{m_i,w_j}\le K_{w_j}$ for all j. (${\Re }_{+}$ denotes the set of non-negative real numbers.) We denote the mass of single m-agents in X by $x_{m,w_0}$, and denote the mass of single w-agents in X by $x_{m_0,w}$.

The i^th row of a matching X is denoted by $X_{m_i,·}$, and the j^th column by $X_{·,w_j}$. When it is not ambiguous, we write $X_{m_i}$, or $X_i$, for $X_{m_i,·}$, and $X_{w_j}$, or $X_j$, for $X_{·,w_j}$. Similarly, we use $x_{i,j}$ for $x_{m_i,w_j}$.

Our solution concept is stability:

Definition 1.

A matching X isindividually rationalif $x_{m,w}>0$ implies that $w{P}_m{w}_0$ and $m{P}_w{m}_0$.

A pair $(m,w)$ is ablocking pair forXif there exist $m^{\prime }\in M\cup \left\{m_0\right\}$ and $w^{\prime }\in W\cup \left\{w_0\right\}$ such that $m{P}_w{m}^{\prime }$, $w{P}_m{w}^{\prime }$, $x_{m,w^{\prime }}>0$, and $x_{m^{\prime },w}>0$.

A matching X isstableif it is individually rational and there are no blocking pairs for X.

We denote by $S(M,W,P,K)$ the set of all stable matchings in $\langle M,W,P,K\rangle$.

Individual rationality states that if $x_{m,w}>0$, then for type-m men type-w women are more preferred than being unmatched and for type-w women type-m men are more preferred than being unmatched. A pair $(m,w)$ is a blocking pair if type-m men have a positive measure of partners who are less preferred than type-w women and type-w women have a positive measure of partners who are less preferred than type-m men.

Two special cases of our model are worth emphasizing: The model of random matching is obtained when $K_m=1$ for all m, and each $K_w$ is a natural number. The interpretation of random matching is that men are “students” and women are “schools”. Students are assigned to schools at random, and each school w has $K_w$ seats available for students. In real-life school choice, the randomization often results from indifferences in schools’ preferences over students [3]: Matching theory often requires strict preferences, so a random “priority order” is produced for the schools in order to break indifferences. Random matchings arise in many other situations as well, because random assignment is often a basic consequence of fairness considerations (if two children want the same toy, fairness demands that they flip a coin to determine who gets it).

The second model is that of integral complete matching, where all $K_m$ and $K_w$ are natural numbers, and all entries of a matching X are natural numbers. The interpretation is that there are $K_m$ type-m men and $K_w$ type-w women, and that a matching X exhibits in $x_{m,w}$ how many type m men matched to type-w women. This special case captures actual observations in marriage models (see, for example, [8]). We observe that men and women are partitioned into types according to their observable characteristics (age, income, education, etc.); and we are given a table showing how many type-m men married type-w women. These observations are essentially “flow” observations (marriages in a given year), so the integral complete matchings do not have any single agents.

4. Structure of Stable Matchings

In this section, we study the structure of stable matchings. For each $m\in M$, define

$${\chi }_m=x\in {\Re }_{+}^{\left|W\right|}:\sum _{1\le j\le \left|W\right|}{x}_j\le K_m,$$

and a partial order ${\le }_m$ on ${\chi }_m$ by

$$y{\le }_{m}x\mathrm{iff}\forall w\in W\cup \left\{w_0\right\},\sum _{0\le j\le \left|W\right|w_j{R}_{m}w}{y}_j\le \sum _{0\le j\le \left|W\right|w_j{R}_{m}w}{x}_j,$$

where $x_0=K_m-{\sum }_{1\le j\le \left|W\right|}{x}_j$, and $y_0=K_w-{\sum }_{1\le j\le \left|W\right|}{y}_j$.

Note that ${\le }_m$ is defined by analogy with the first-order stochastic dominance order on probability distributions. In the case where $K_m=1$, vectors x and y in ${\chi }_m$ represent probability distributions over the types of women that m may match to. In that case $y{\le }_{m}x$ if and only if the lottery induced by y over $W\cup \left\{w_0\right\}$ is worse than the lottery induced by x, for any von Neumann-Morgenstern utility function that is compatible with $R_m$.

Letting ${\chi }_w=x\in {\Re }_{+}^{\left|M\right|}:{\sum }_{1\le i\le \left|M\right|}{x}_i\le K_w$, we define ${\le }_w$ in an analogous way.

We introduce two partial orders on matchings. Suppose X and Y are matchings, then:

• $X{\le }_{M}Y$ if, for all $m\in M$, $X_m{\le }_m{Y}_m$
• $X{\le }_{W}Y$ if, for all $w\in W$, $X_w{\le }_w{Y}_w$.

Theorem 1.

$(S(M,W,P,K),{\le }_M)$ and $(S(M,W,P,K),{\le }_W)$ are nonempty, complete, and distributive lattices; in addition, for $X,Y\in S(M,W,P,K)$

1. 

$X{\le }_{M}Y$ if and only if $Y{\le }_{W}X$;

2. 

for all $a\in M\cup W$, either $X_a{\le }_a{Y}_a$ or $Y_a{\le }_a{X}_a$;

3. 

for all m, ${\sum }_{w\in W}{x}_{m,w}={\sum }_{w\in W}{y}_{m,w}$ and for all w, ${\sum }_{m\in M}{x}_{m,w}={\sum }_{m\in M}{y}_{m,w}$.

Theorem 1 presents a version, adapted to our model, of the standard results on matching theory. The proof is in Appendix B. Statement (1) is a “polarity of interests” result, saying that a stable matching X is better for men if and only if it is worse for women. Statement (2) says that the outcomes in two stable matchings are always comparable for an agent: note that ${\le }_a$ is an incomplete preference relation. Statement (3) is the “rural hospitals theorem,” which says that the same number of agents of a given type are matched in any two stable matchings.

Theorem 1 also implies that there are two stable matchings, $X^M$ and $X^W$, such that for all stable matchings X,

$$\begin{array}{c}{X}^W{\le }_{M}X{\le }_M{X}^M,\mathrm{and}\hfill \\ X^M{\le }_{W}X{\le }_W{X}^W.\hfill \end{array}$$

We refer to $X^M$ as the man-optimal (M-optimal) stable matching, and to $X^W$ as the woman-optimal (W-optimal) stable matching. We also call $X^M$ and $X^W$ extremal stable matchings. A matching X is the unique stable matching if $S(M,W,P,K)=X$; in this case X coincides with the man-optimal and the woman-optimal stable matchings.

Perhaps extremal stable matchings are unlikely to arise because they favor one side of the market over the other. One may instead be interested in matchings that present a compromise. The median stable matching provides such a compromise by assigning each agent a partner who is median ranked amongst all of his/her stable matching partners. Indeed in an experimental study on decentralized market, ref. [21] find that the median stable matching tends to emerge among stable matchings. As such, we may want to test whether an observed matching in field data is rationalizable as a median stable matching.

It is not obvious that median stable matchings exist. We shall first show that median stable matchings exist by only considering integral matchings that have integer entries, and then present a sufficient condition for rationalizability as a median stable matching.

We consider a market $\langle M,W,P,K\rangle$, where all numbers $K_m$, $K_w$, and entries of matchings X are non-negative integers. Consequently, the number of stable matchings is finite, say n. Let $S(M,W,P,K)=\{X^1,\dots ,X^n\}$ be the set of stable matchings. For each agent a we consider all stable outcomes and rank them according to ${\ge }_a$. All the outcomes are comparable by (2) of Theorem 1. More formally, let $\{X_a^{\left(1\right)},\dots ,X_a^{\left(n\right)}\}=\{X_a^1,\dots ,X_a^n\}$ and $X_a^{\left(1\right)}{\ge }_a\dots {\ge }_a{X}_a^{\left(n\right)}$. Using these ranked outcomes and (1) of Theorem 1, we construct the matrices $Y^{\left(l\right)}$ for all $l=1,\dots ,n$ such that $Y_m^{\left(l\right)}=X_m^{\left(l\right)}$ and $Y_w^{\left(l\right)}=X_w^{(n+1-l)}$. Matrix $Y^{\left(l\right)}$ assigns each type of men the $l^{th}$ best outcome, and each type of women the $l^{th}$ worst outcome among stable matching partners.

Proposition 1.

$Y^{\left(l\right)}$ is a stable matching.

The proof of Proposition 1 follows from the lattice structure of the set of stable matchings. It is essentially the same as the proofs of Theorem $3.2$ in [22] and Theorem 1 of [17].

If n is odd we term $Y^{\left(\frac{n+1}{2}\right)}$ the median stable matching. If n is even we refer to $Y^{\left(\frac{n}{2}\right)}$ (or $Y^{(\frac{n}{2}+1)}$) as the median stable matching (of course the choice is arbitrary).

Corollary 1.

The median stable matching exists.

5. Testable Implications

Suppose that we are given a data of a matching X, and the corresponding M, W and K. Similar to [9], we assume that there are no “singles,” or unmatched agents, and so ${\sum }_m{K}_m={\sum }_w{K}_w$, and ${\sum }_w{x}_{m,w}=K_m$ for all m, and ${\sum }_m{x}_{m,w}=K_w$ for all w. We call such a matching in which there are no singles complete. We ask when there are preferences for the different types of men and women such that X is a stable matching or an extremal stable matching (which is shown to exist in Section 4).

Definition 2.

A matching X isrationalizableif there exists a preference profile $P=\left({\left(P_m\right)}_{m\in M},{\left(P_w\right)}_{w\in W}\right)$ such that X is a stable matching in $\langle M,W,P,K\rangle$. A matching X isM-optimal (W-optimal) rationalizableif there is a preference profile P such that X is the M-optimal (W-optimal) stable matching in $\langle M,W,P,K\rangle$.

We also want to know when a matching can be rationalized as the median stable matching.

Definition 3.

An integral matching X ismedian-rationalizableif there is a preference profile P such that X is the median stable matching in $\langle M,W,P,K\rangle$.

To answer the questions, we define a graph $(V,L)$, where V consists of all the positive entries in X, and there is an edge $(v,v^{\prime })\in L$ if and only if v and $v^{\prime }$ are either in the same row or in the same column in X. More formally, for each matching X we associate a graph $(V,L)$ defined as follows. The set of vertices V is $\{(m,w):m\in M,w\in W\mathrm{such}\mathrm{that}{x}_{m,w}>0\}$, and an edge $((m,w),(m^{\prime },w^{\prime }))\in L$ is formed for every pair of vertices $(m,w)$ and $(m^{\prime },w^{\prime })$ with $m=m^{\prime }$ or $w=w^{\prime }$.

A previous study finds that the rationalizability of a matching depends on minimal cycles (defined in Section 2.2) of its associated graph: (The result in [9] is more general than what is stated here.)

Theorem 2

([9]).

1. 

A complete matching X is rationalizable if and only if its associated graph does not contain two connected, distinct, minimal cycles.

2. 

A complete matching X is M-optimal (W-optimal) rationalizable if and only if its associated graph has no minimal cycle.

For example, let matching X be the matrix on the left, with three types of men and women. Its associated graph is depicted on the right:

The following two minimal cycles in this graph

are connected; thus the matching is not rationalizable.

The rationalizability by stable matchings and extremal stable matchings only depends on whether each number $x_{m,w}$ is positive or 0. However, we find that median rationalizability depends on the value of $x_{m,w}$ as well as whether $x_{m,w}$ is positive or zero.

If $\langle v_0,\dots ,v_N\rangle$ is a minimal cycle, then N is an even number. We say that a minimal cycle c is balanced if

$$min\{v_0,v_2,\dots ,v_{N-2}\}=min\{v_1,v_3,\dots ,v_{N-1}\}.$$

(See the example below for an illustration of this condition.)

Theorem 3.

An integral complete matching X is median-rationalizable if it is rationalizable and if all minimal cycles of $(V,L)$ are balanced.

The proof is in Appendix C. It is worth observing that, in our proof, we construct preferences such that the resulting number of stable matchings is odd. Therefore, our results do not depend on the choice of $Y^{\left(\frac{n}{2}\right)}$ or $Y^{(\frac{n}{2}+1)}$ as the median stable matching when n is even.

To provide some intuition for this result, consider the following example:

• $M=\{m_1,m_2\}$,
• $W=\{w_1,w_2\}$,
• $k_{m_1}=6$, $k_{m_2}=5$, $k_{w_1}=4$, $k_{w_2}=7$,
• $w_2{P}_{m1}{w}_1$, $w_1{P}_{m2}{w}_2$, $m_1{P}_{w1}{m}_2$, and $m_2{P}_{w2}{m}_1$,

where each man prefers all women to being unmatched and each woman prefers all men to being unmatched. In this market, there are five stable matchings as shown in Figure 1. Note that the preferences of each type of agent is shown by an arrow, for example the arrow in the first row represents the preference ranking of $m_1$ that type-$w_2$ woman is more preferred to type-$w_1$ woman. Therefore, the preferences are “clockwise” cyclical. These matchings differ from each other in that, starting with matching A, matchings B-E are formed by clockwise “rotations”—that is, by dissolving two couples $\{(m_1,w_1),(m_2,w_2)\}$ and repairing them as $\{(m_1,w_2),(m_2,w_1)\}$. Given the clockwise preferences, moving from matching A to E, the matchings become increasingly preferred by the men, and at the same time, less preferred by the women.

Hence, matching C is the median stable matching, as the two stable matchings obtained by rotating this matching counterclockwise (i.e., A and B) are more preferred by the women, while the two obtained by rotating clockwise (i.e., D and E) are more preferred by the men. (In addition, one cannot rotate further clockwise beyond E, nor further counterclockwise beyond A, as one “runs out” of pairs in both cases.) Note also that only in the third matching is the cycle balanced: In the minimal cycle there are four vertices with entries $2,4,3,2$ and when we take the minimum of the entries of the odd-numbered vertices and the minimum of the entries of the even-numbered vertices for any numbering of vertices preserving the order in which they appear in the cycle we get:

$$min\{2,3\}=2=min\{4,2\}.$$

Therefore, the minimal cycle for matching C is balanced. The balancedness condition ensures that one can rotate an equal number of times both in the clockwise direction and in the counterclockwise direction. Furthermore, note that the balancedness condition fails for the other matchings. For example, in matching B, $min\{3,4\}=3\ne 1=min\{3,1\}$, there the minimal cycle in matching B is not balanced.

Note that the stability and rationalizability of a matching only depend on which entries are zero or positive. Thus if an observed matching X is canonicalized into $X^C$ ($x_{m,w}^c=0$ if $x_{m,w}=0$, and $x_{m,w}^c=1$ if $x_{m,w}>0$), then obviously X is rationalizable if and only if $X^C$ is rationalizable. Given the role of canonical matchings in rationalizability, the following corollary is of some interest.

Corollary 2.

A canonicalized matching $X^C$ is either not rationalizable or it is median-rationalizable.

Unfortunately, Theorem 3 provides only a sufficient condition for median-rationalizability. Corollary 2 gives a characterization of necessary conditions, but only for the case of canonical matchings. To sketch the boundaries of this result, we present two examples in Appendix A. Example Appendix A.1 shows that there are indeed matchings that are not rationalizable as median stable matchings, so Corollary 2 on canonical matchings does not extend to all aggregate matchings. Example Appendix A.2 shows that the sufficient condition in Theorem 3 is not necessary.