Stable Matchings in the Marriage Model with Indifferences

Abstract

For the marriage model with indifferences, we define an equivalence relation over the stable matching set. We identify a sufficient condition, the closing property, under which we can extend results of the classical model (without indifferences) to the equivalence classes of the stable matching set. This condition allows us to extend the lattice structure over classes of equivalences and the rural hospital theorem.

Appendix

1.1 Strongly Stable Matchings and the Closing Property

Proof of Lemma 1

Let $\mu _{1},$ $\mu _{2},$ $\mu _{3}\in \mathcal {M}$ for all $i\in A$.

1. Let us assume that $\mu _{1}I_{i}\mu _{2}$ and $\mu _{2}P_{i}\mu _{3}$. From the definition of $I_{i}$ and $P_{i}$, we have that

$$\begin{aligned}&\mu _{1}R_{i}\mu _{2}\text { and }\mu _{2}R_{i}\mu _{1} , \end{aligned}$$

(A1)

$$\begin{aligned}&\mu _{2}R_{i}\mu _{3}\text { and }\overline{\mu _{3}R_{i}\mu _{2}}. \end{aligned}$$

(A2)

The transitivity of $R_{i}$ and conditions (A1), (A2) imply that $\mu _{1}R_{i}\mu _{3}.$ Suppose that $\mu _{3}R_{i}\mu _{1}.$ Condition ( A1) and transitivity of $R_{i}$ imply that $\mu _{3}R_{i}\mu _{2}.$ This contradicts (A2). Hence, $\overline{\mu _{3}R_{i}\mu _{1}}$ and $\mu _{1}P_{i}\mu _{3}.$

2. Similarly, we can prove that if $\mu _{1}P_{i}\mu _{2}$ and $\mu _{2}I_{i}\mu _{3},$ then $\mu _{1}P_{i}\mu _{3}.$

3. We assume that $\mu _{1}P_{i}\mu _{2}$ and $\mu _{i}^{\prime }\in \left[ \mu _{i}\right]$ for $i=1,2.$ Then,

$$\begin{aligned} \mu _{1}^{\prime }I_{i}\mu _{1}\text { and }\mu _{1}P_{i}\mu _{2}. \end{aligned}$$

(A3)

Condition (A3) and the fact that $\mu _{1}P_{i}\mu _{2}$ imply that

$$\begin{aligned} \mu _{1}^{\prime }P_{i}\mu _{2}\text {.} \end{aligned}$$

We have that $\mu _{1}P_{i}\mu _{2}$ and $\mu _{2}I_{i}\mu _{2}$. By item 2, we have

$$\begin{aligned} \mu _{1}^{\prime }P_{i}\mu _{2}^{\prime }. \end{aligned}$$

4. It follows from 1, 2 and 3.

Proof of Proposition 1

Let $\left[ \mu _{1}\right] ,$ $\left[ \mu _{2}\right] ,$ $\left[ \mu _{3}\right] \in S(R)/_{\sim }$ for all $i\in A$, and let $\mu _{1}^{\prime }\in \left[ \mu _{1}\right] ,\mu _{2}^{\prime }\in \left[ \mu _{2}\right]$ and $\mu _{3}^{\prime }\in \left[ \mu _{3}\right] .$

• 1. Reflexivity. Since $R_{M}$ is a reflexive relation over $\mathcal {M}$, we have that

  $$\begin{aligned} \mu _{1}^{\prime }R_{M}\mu _{1}^{\prime }, \end{aligned}$$

  so

  $$\begin{aligned} \left[ \mu _{1}\right] R_{M}\left[ \mu _{1}\right] . \end{aligned}$$

• 2. Transitivity. $R_{M}$ is a transitive relation over $\mathcal {M}.$ If $\mu _{1}R_{i}\mu _{2}$ and $\mu _{2}R_{i}\mu _{3}$, then $\mu _{1}R_{i}\mu _{3},$ i.e.

  $$\begin{aligned} \left[ \mu _{1}\right] R_{M}\left[ \mu _{2}\right] ,\left[ \mu _{2}\right] R_{M}\left[ \mu _{3}\right] \text { and }\left[ \mu _{1}\right] R_{M}\left[ \mu _{3}\right] . \end{aligned}$$

• 3. Antisymmetric. Assume that

  $$\begin{aligned} \left[ \mu _{1}\right] R_{M}\left[ \mu _{2}\right] \text { and }\left[ \mu _{2}\right] R_{M}\left[ \mu _{1}\right] , \end{aligned}$$

  i.e.

  $$\begin{aligned} \mu _{1}R_{M}\mu _{2}\text { and }\mu _{2}R_{M}\mu _{1}. \end{aligned}$$

  Hence, $\mu _{1}I_{M}\mu _{2}$, i.e. $\left[ \mu _{1}\right] =\left[ \mu _{2} \right]$.

The following Manlove’s result will be useful in proof of Proposition 4.

Lemma A1

(Manlove) Let (M, W, R) be a marriage model with indifferences, and let $\mu$ and $\mu ^{\prime }$ be two strongly stable matchings. Suppose that for any men $m_{1}\in M,$ $\mu (m_{1})=w_{1}$ and $\mu ^{\prime }(m_{1})=w_{2},$ where $w_{1}\ne w_{2}$ and $w_{2}R_{m_{1}}w_{1}$. Then there are sequences of agents involving, $m_{1}, w_{1}$ and $w_{2}$ as follows: for some $r>1,$ there are r men $m_{1},\cdots ,m_{r}$ and r women $w_{1},\cdots ,w_{r}$ which satisfies

$$\begin{aligned} \mu (m_{i})=w_{i}\quad (1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r)\text { and }\mu ^{\prime }(m_{i})=w_{i+1}\quad (1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r). \end{aligned}$$

(1) If $w_{2}I_{m_{1}}w_{1},$ then

$$\begin{aligned} w_{i+1}I_{m_{i}}w_{i}\quad (1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r)\text { and } m_{i}I_{w_{i}}m_{i-1}\quad (1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r), \end{aligned}$$

(2) If $w_{2}P_{m_{1}}w_{1},$ then

$$\begin{aligned} w_{i+1}P_{m_{i}}w_{i}\quad (1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r)\text { and } m_{i}P_{w_{i}}m_{i-1}\quad (1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r), \end{aligned}$$

where $m_{0}=m_{r}, m_{r+1}=m_{1}$ and $w_{r+1}=w_{1}.$

We could formulate a symmetric lemma by exchanging the role of women and men.

Proof of Proposition 4

Let $\mu ,\mu ^{\prime }\in \mathrm{SS}(R)$, if $J_{M}(\mu ,\mu ^{\prime })=J_{W}(\mu ,\mu ^{\prime })=\varnothing ,$ then the result follows. We assume that $J_{M}(\mu ,\mu ^{\prime })\ne \varnothing .$ We will show that

$$\begin{aligned} J_{M}(\mu ,\mu ^{\prime })\subseteq \mu (J_{W}(\mu ,\mu ^{\prime })). \end{aligned}$$

Let $m_{1}\in J_{M}(\mu ,\mu ^{\prime }).$ This means that

$$\begin{aligned} \mu (m_{1})I_{m_{1}}\mu ^{\prime }(m_{1}). \end{aligned}$$

Since $m_{1}\in J_{M}(\mu ,\mu ^{\prime }),$ we have that $\mu (m_{1})\ne \mu ^{\prime }(m_{1}).$ Then there exist $w_{1}$ and $w_{2}$ such that $\mu (m_{1})=w_{1}$ and $\mu ^{\prime }(m_{1})=w_{2}.$ Since $m_{1}\in J_{M}(\mu ,\mu ^{\prime })$, we have that

$$\begin{aligned} w_{1}=\mu (m_{1})I_{m_{1}}\mu ^{\prime }(m_{1})=w_{2}. \end{aligned}$$

By Lemma A1 (Manlove), we have that there exist successions of agents $m_{1},\cdots ,m_{r}$ and $w_{1},\cdots ,w_{r}$ such that

$$\begin{aligned} \mu (m_{i})=w_{i}\quad (1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r)\text { and }\mu ^{\prime }(m_{i})=w_{i+1}\quad (1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r). \end{aligned}$$

Also

$$\begin{aligned} w_{i+1}I_{m_{i}}w_{i}(1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r)\text { and }m_{i}I_{w_{i}}m_{i-1}(1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r). \end{aligned}$$

If $i=1$, then

$$\begin{aligned} m_{1}=\mu (w_{1})I_{w_{1}}\mu ^{\prime }(w_{1}), \end{aligned}$$

i.e. $w_{1}\in J_{W}(\mu ,\mu ^{\prime }).$ Hence, $\mu (w_{1})=m_{1}\in \mu (J_{W}(\mu ,\mu ^{\prime })).$

Now, we will show that

$$\begin{aligned} \mu (J_{W}(\mu ,\mu ^{\prime }))\subseteq J_{M}(\mu ,\mu ^{\prime }). \end{aligned}$$

Let $\mu (w_{1})\in \mu (J_{W}(\mu ,\mu ^{\prime })).$ Since $w_{1}\in J_{W}(\mu ,\mu ^{\prime })$, we have that

$$\begin{aligned} \mu (w_{1})I_{w_{1}}\mu ^{\prime }(w_{1}). \end{aligned}$$

Notice that $\mu (m_{1})\ne \mu ^{\prime }(m_{1}).$ Then there exist $m_{1}$ and $m_{2}$ such that $\mu (w_{1})=m_{1}$ and $\mu ^{\prime }(w_{1})=m_{2}.$ By Lemma A1 (Manlove), we have that there exist successions of agents $m_{1},\cdots ,m_{r}$ and $w_{1},\cdots ,w_{r}$ such that

$$\begin{aligned} \mu (w_{i})=m_{i}\quad (1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r)\text { and }\mu ^{\prime }(w_{i})=m_{i+1}\quad (1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r). \end{aligned}$$

Also

$$\begin{aligned} m_{i+1}I_{m_{i}}m_{i}\quad (1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r)\text { and } w_{i}I_{w_{i}}w_{i-1}(1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r). \end{aligned}$$

If $i=1$, then

$$\begin{aligned} w_{1}=\mu (m_{1})I_{m_{1}}\mu ^{\prime }(m_{1}), \end{aligned}$$

i.e. $m_{1}\in J_{M}(\mu ,\mu ^{\prime }).$ Hence, $m_{1}=\mu (w_{1})\in J_{M}(\mu ,\mu ^{\prime }).$

Proof of Proposition 5

Let $\mu ,\mu ^{\prime }\in \mathrm{SSS}(R)$, if $J_{M}(\mu ,\mu ^{\prime })=J_{W}(\mu ,\mu ^{\prime })=\varnothing ,$ then the result follows. We assume that $J_{M}(\mu ,\mu ^{\prime })\ne \varnothing .$

Let $m_{1}\in J_{M}(\mu ,\mu ^{\prime }).$ This means that

$$\begin{aligned} \mu (m_{1})I_{m_{1}}\mu ^{\prime }(m_{1}). \end{aligned}$$

(A4)

Since $m_{1}\in J_{M}(\mu ,\mu ^{\prime }),$ we have that $\mu (m_{1})\ne \mu ^{\prime }(m_{1}).$ Then there exist $w_{1}$ and $w_{2}$ such that $\mu (m_{1})=w_{1}$ and $\mu ^{\prime }(m_{1})=w_{2}.$ Since $m_{1}\in J_{M}(\mu ,\mu ^{\prime })$, we have that

$$\begin{aligned} w_{1}=\mu (m_{1})I_{m_{1}}\mu ^{\prime }(m_{1})=w_{2}. \end{aligned}$$

(A5)

Since $\mu ,\mu ^{\prime }\in \mathrm{SSS}(R)\subseteq \mathrm{SS}(R)$, we can apply Lemma A1 (Manlove). Then there exist successions of agents $m_{1},\cdots ,m_{r}$ and $w_{1},\cdots ,w_{r}$ such that

$$\begin{aligned} \mu (m_{i})=w_{i}\quad (1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r)\text { and }\mu ^{\prime }(m_{i})=w_{i+1}\quad (1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r). \end{aligned}$$

Also

$$\begin{aligned} w_{i+1}I_{m_{i}}w_{i}(1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r)\text { and }m_{i}I_{w_{i}}m_{i-1}(1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}r). \end{aligned}$$

If $i=1$, then

$$\begin{aligned} m_{1}=\mu (w_{1})I_{w_{1}}\mu ^{\prime }(w_{1}). \end{aligned}$$

(A6)

Conditions A5 and A6 imply that the pair $(m_{1},w_{1})$ superblocks $\mu ^{\prime }.$ This contradicts that $\mu ^{\prime }$ is a superstable matching. This contradiction came from assuming that $J_{M}(\mu ,\mu ^{\prime })\ne \varnothing .$ So the closing property holds.

The next example shows that the closing property is not a necessary condition to the lattice structure of the stable matching set.

Example 6

Let $M=\left\{ m_{1},m_{2},m_{3}\right\}$ be the set of men and $W=\left\{ w_{1},w_{2},w_{3}\right\}$ be the set of women. Consider the preference profile R : 

$$\begin{aligned} \begin{array}{ll} R_{m_{1}}:\left[ w_{1},w_{2}\right] ,m_{1}, &{} R_{w_{1}}:m_{3},m_{2},m_{1},w_{1}, \\ R_{m_{2}}:w_{2},w_{1},w_{3},m_{2}, &{} R_{w_{2}}:m_{1},m_{2},w_{2}, \\ R_{m_{3}}:w_{3},w_{1},m_{3}, &{} R_{w_{3}}:m_{2},m_{3},w_{3}. \end{array} \end{aligned}$$

The set of stable matchings consists of the following matchings:

$$\begin{aligned} \mu _{1}=\left( \begin{array}{ccc} m_{1} &{} m_{2} &{} m_{3} \\ w_{1} &{} w_{2} &{} w_{3} \end{array} \right) ,\quad \mu _{2}=\left( \begin{array}{ccc} m_{1} &{} m_{2} &{} m_{3} \\ w_{2} &{} w_{1} &{} w_{3} \end{array} \right) ,\quad \mu _{3}=\left( \begin{array}{ccc} m_{1} &{} m_{2} &{} m_{3} \\ w_{2} &{} w_{3} &{} w_{1} \end{array} \right) . \end{aligned}$$

Observe that $\mu _{1}R_{m_{i}}\mu _{2}R_{m_{i}}\mu _{3}$, for $i=1,2,3.$ Also $\mu _{3}R_{w_{i}}\mu _{2}R_{w_{i}}\mu _{1}$ for$i=1,2,3.$ In this example, we have a lattice structure over stable matching set. Nevertheless, the stable matching set does not satisfy the closing property. We have

$$\begin{aligned} J_{M}(\mu _{1},\mu _{2})=\left\{ m_{1}\right\} \text { and }J_{W}(\mu _{1},\mu _{2})=\varnothing . \end{aligned}$$

So

$$\begin{aligned} \mu _{1}(J_{M}(\mu _{1},\mu _{2}))=\mu _{1}(m_{1})=w_{1}\ne J_{W}(\mu _{1},\mu _{2}). \end{aligned}$$