<p>Yes, this is called a &quot;many to one&quot; matching. Many to many matchings also exist.</p>
<p>Note that not all many to one matchings are a two-sided matching market. E.g., if I buy several oranges, that is not a two-sided matching, I am the only one with any input w.r.t. the matches.
Similarly, entries in different tables of a database can be matched one-to-one, one-to-many or many-to-many. However, there is no market here at all, just a mapping.</p>
<hr />
<p><strong>Examples</strong></p>
<p>Some US municipalities use a many to one matching algorithm to match students to high schools. Since the preferences (rankings) submitted by students as well as schools are considered, this is indeed a two-sided market.</p>
<p><a href="https://www.nrmp.org/intro-to-the-match/how-matching-algorithm-works/" rel="nofollow noreferrer">The US National Medical Resident Program also does this.</a></p>
<p>Recommended reading:</p>
<p><a href="https://economics.mit.edu/files/3024" rel="nofollow noreferrer">The New York City High School Match</a></p>
<blockquote>
<p>Initial discussions focused on whether the
medical match was a good model for New York
City schools, or whether another kind of clearinghouse might be more appropriate. The medical match applied to schools would be a twosided model in which both schools and students
have preferences, with the object of implementing a stable assignment, that is, an efficient
assignment such that no school and student not
matched to one another would both prefer to be.
Thus, the question was, are the students the only
real players in the system, with choices by
schools merely a device for allocating scarce
spaces? If this were the case, there might be
appropriate one-sided clearinghouse models in
which only student preferences determine efficient allocations (cf. Boston Public Schools;
Abdulkadiroglu et al., 2005).</p>
<p>Two things convinced us that New York City
schools are a two-sided market. The first was
that schools withheld capacity to match with
students they preferred. Stable assignments
would eliminate the main incentives for this.
Second, discussions indicated that principals of
different EdOpt schools had different preferences even for students with reading scores in
the lowest category, with some schools preferring higher scores and others preferring students
who had good attendance. If schools have different comparative advantages, allowing scope
for their preferences seemed sensible. Also, the
fact that school administrators gamed the system indicated they were strategic players.</p>
</blockquote>

CC BY-SA 4.0
2022-03-09T09:05:40.653Z
A follow-up: Does the "stable matching problem" define matching markets? My model doesn't have a problem w.r.t stable matching i.e. agents can un-match and re-match at any time. Does that mean I don't really have to worry about matching markets?
Joebevo
CC BY-SA 4.0
2022-03-09T09:39:50.470Z
No, not at all; but please ask new questions as new questions.
Giskard
CC BY-SA 4.0
2022-03-08T14:50:07.330Z
2022-03-08T15:01:07.123Z
2022-03-08T15:01:07.123Z
Giskard
<p>I'm working on a problem where there are two sets of agents, S's and T's. Let's say there are 3 of each, so we have: S1, S2, S3, and T1, T2, T3.</p>
<ul>
<li>Agents S1 through S3 have a valuation for each of T1 through T3.</li>
<li>Agents T1 through T3 have a valuation for each of S1 through S3.</li>
<li>Each of the agents T1 through T3 can each be matched to <em><strong>one or more</strong></em> of
the agents S1 through S3, and will do so according to decreasing valuation of their counterparts. (i.e. they will choose the S which they value highest, followed by the S with the next highest valuation, and so on).</li>
<li>However, agents S1 through S3 can only be matched to one of agents T1 through T3.</li>
</ul>
<p>Does this constitute a two-sided matching market? Usually for two sided matching, I've only seen each one on either side being matched to <em><strong>only one</strong></em> of the other side.</p>

<matching><networks>
Can one have a two-sided matching market where one side is not restricted to being matched to just one of the other side?
Joebevo
# Can one have a two-sided matching market where one side is not restricted to being matched to just one of the other side?

(Asked by: username_0 on 2022-03-08)

I'm working on a problem where there are two sets of agents, S's and T's. Let's say there are 3 of each, so we have: S1, S2, S3, and T1, T2, T3.

* Agents S1 through S3 have a valuation for each of T1 through T3.
* Agents T1 through T3 have a valuation for each of S1 through S3.
* Each of the agents T1 through T3 can each be matched to ***one or more*** of
  the agents S1 through S3, and will do so according to decreasing valuation of their counterparts. (i.e. they will choose the S which they value highest, followed by the S with the next highest valuation, and so on).
* However, agents S1 through S3 can only be matched to one of agents T1 through T3.

Does this constitute a two-sided matching market? Usually for two sided matching, I've only seen each one on either side being matched to ***only one*** of the other side.

**Tags:** matching, networks

---

## 1 answers

### Answer by username_1 on 2022-03-08. Score: 2

Yes, this is called a "many to one" matching. Many to many matchings also exist.

Note that not all many to one matchings are a two-sided matching market. E.g., if I buy several oranges, that is not a two-sided matching, I am the only one with any input w.r.t. the matches.
Similarly, entries in different tables of a database can be matched one-to-one, one-to-many or many-to-many. However, there is no market here at all, just a mapping.

---

**Examples**

Some US municipalities use a many to one matching algorithm to match students to high schools. Since the preferences (rankings) submitted by students as well as schools are considered, this is indeed a two-sided market.

[The US National Medical Resident Program also does this.](https://www.nrmp.org/intro-to-the-match/how-matching-algorithm-works/)

Recommended reading:

[The New York City High School Match](https://economics.mit.edu/files/3024)

> Initial discussions focused on whether the
> medical match was a good model for New York
> City schools, or whether another kind of clearinghouse might be more appropriate. The medical match applied to schools would be a twosided model in which both schools and students
> have preferences, with the object of implementing a stable assignment, that is, an efficient
> assignment such that no school and student not
> matched to one another would both prefer to be.
> Thus, the question was, are the students the only
> real players in the system, with choices by
> schools merely a device for allocating scarce
> spaces? If this were the case, there might be
> appropriate one-sided clearinghouse models in
> which only student preferences determine efficient allocations (cf. Boston Public Schools;
> Abdulkadiroglu et al., 2005).
>
> Two things convinced us that New York City
> schools are a two-sided market. The first was
> that schools withheld capacity to match with
> students they preferred. Stable assignments
> would eliminate the main incentives for this.
> Second, discussions indicated that principals of
> different EdOpt schools had different preferences even for students with reading scores in
> the lowest category, with some schools preferring higher scores and others preferring students
> who had good attendance. If schools have different comparative advantages, allowing scope
> for their preferences seemed sensible. Also, the
> fact that school administrators gamed the system indicated they were strategic players.

> **Comments:**
> **username_0**: A follow-up: Does the "stable matching problem" define matching markets? My model doesn't have a problem w.r.t stable matching i.e. agents can un-match and re-match at any time. Does that mean I don't really have to worry about matching markets?  
> **username_1**: No, not at all; but please ask new questions as new questions.  

---

