# Stability and Median Rationalizability for Aggregate Matchings

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary Definitions

#### 2.1. Lattice Theory

**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**.

**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

**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.

**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

- 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.

**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 ${x}_{{m}_{0},w}$.

^{th}row of a matching X is denoted by ${X}_{{m}_{i},\xb7}$, and the j

^{th}column by ${X}_{\xb7,{w}_{j}}$. When it is not ambiguous, we write ${X}_{{m}_{i}}$, or ${X}_{i}$, for ${X}_{{m}_{i},\xb7}$, and ${X}_{{w}_{j}}$, or ${X}_{j}$, for ${X}_{\xb7,{w}_{j}}$. Similarly, we use ${x}_{i,j}$ for ${x}_{{m}_{i},{w}_{j}}$.

**Definition**

**1.**

**individually rational**if ${x}_{m,w}>0$ implies that $w{P}_{m}{w}_{0}$ and $m{P}_{w}{m}_{0}$.

**blocking pair forX**if 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 ${x}_{{m}^{\prime},w}>0$.

**stable**if it is individually rational and there are no blocking pairs for X.

**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).

**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

- $X{\le}_{M}Y$ 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.**

- 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, ${\sum}_{m\in M}{x}_{m,w}={\sum}_{m\in M}{y}_{m,w}$.

**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.

**Proposition**

**1.**

**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.**

## 5. Testable Implications

**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.**

**rationalizable**if 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 is

**M-optimal (W-optimal) rationalizable**if there is a preference profile P such that X is the M-optimal (W-optimal) stable matching in $\langle M,W,P,K\rangle $.

**Definition**

**3.**

**median-rationalizable**if there is a preference profile P such that X is the median stable matching in $\langle M,W,P,K\rangle $.

**Theorem**

**2**

**.**

- 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.

**balanced**if

**Theorem**

**3.**

- $M=\{{m}_{1},{m}_{2}\}$,
- $W=\{{w}_{1},{w}_{2}\}$,
- ${k}_{{m}_{1}}=6$, ${k}_{{m}_{2}}=5$, ${k}_{{w}_{1}}=4$, ${k}_{{w}_{2}}=7$,
- ${w}_{2}{P}_{m1}{w}_{1}$, ${w}_{1}{P}_{m2}{w}_{2}$, ${m}_{1}{P}_{w1}{m}_{2}$, and ${m}_{2}{P}_{w2}{m}_{1}$,

**Corollary**

**2.**

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Examples

#### Appendix A.1. Matchings Rationalizable, but Not Median-Rationalizable

#### Appendix A.2. Median-Rationalizability Condition Is Not Necessary

## Appendix B. Proof of Theorem 1

**Lemma**

**A1.**

**Proof.**

**Claim**

**1.**

**Proof.**

**Proposition**

**A1.**

**Proof.**

**Proposition**

**A2.**

**Proof.**

**Proposition**

**A3.**

**Proof.**

- ${X}^{\left(n\right)}$ and ${Y}^{\left(n\right)}$ have rational entries,
- ${x}_{ij}^{\left(n\right)}=0\iff {x}_{ij}=0$ and ${y}_{ij}^{\left(n\right)}=0\iff {y}_{ij}=0$ for all $i,j$,
- the sum of entries in row i and column j is the same for ${X}^{\left(n\right)}$ and ${Y}^{\left(n\right)}$, and
- ${x}_{ij}^{\left(n\right)}\to {x}_{ij}$ and ${y}_{ij}^{\left(n\right)}\to {y}_{ij}$ as $n\to \infty $ for all $i,j$.

**Proposition**

**A4.**

**Proof.**

**Proposition**

**A5.**

**Proof.**

- ${X}^{\left(n\right)}$ and ${Y}^{\left(n\right)}$ have rational entries,
- ${x}_{ij}^{\left(n\right)}=0\iff {x}_{ij}=0$ and ${y}_{ij}^{\left(n\right)}=0\iff {y}_{ij}=0$ for all $i,j$,
- the sum of entries in row i and column j is the same for ${X}^{\left(n\right)}$ and ${Y}^{\left(n\right)}$, and
- ${x}_{ij}^{\left(n\right)}\to {x}_{ij}$ and ${y}_{ij}^{\left(n\right)}\to {y}_{ij}$ as $n\to \infty $ for all $i,j$.

## Appendix C. Proof of Theorem 3

- ${e}_{i,j}=0$ if $(i,j)$ is not in the cycle c;
- for all i and j, ${\sum}_{h}{e}_{i,h}=0$ and ${\sum}_{l}{e}_{l,j}=0$; and
- $\left|{e}_{i,j}\right|\le \Theta $.

- $({x}_{{i}_{n},{j}_{n}}-{y}_{{i}_{n},{j}_{n}})({x}_{{i}_{n+1},{j}_{n+1}}-{y}_{{i}_{n+1},{j}_{n+1}})<0$
- ${i}_{n}\ne {i}_{n+1}\iff {j}_{n}={j}_{n+1}$.

**Claim**

**2.**

**Proof.**

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Echenique, F.; Lee, S.; Shum, M.; Yenmez, M.B. Stability and Median Rationalizability for Aggregate Matchings. *Games* **2021**, *12*, 33.
https://doi.org/10.3390/g12020033

**AMA Style**

Echenique F, Lee S, Shum M, Yenmez MB. Stability and Median Rationalizability for Aggregate Matchings. *Games*. 2021; 12(2):33.
https://doi.org/10.3390/g12020033

**Chicago/Turabian Style**

Echenique, Federico, SangMok Lee, Matthew Shum, and M. Bumin Yenmez. 2021. "Stability and Median Rationalizability for Aggregate Matchings" *Games* 12, no. 2: 33.
https://doi.org/10.3390/g12020033