<p>Yes, this has been studied. Look up "matching with contracts" and the earlier papers by Crawford &amp; Knoer (1981), and Kelso &amp; Crawford (1982) in Econometrica. </p>

CC BY-SA 3.0
2017-09-13T01:44:29.810Z
Thanks a lot! Crawford & Knoer (1981) and Kelso & Crawford (1982) seem to be more restricted, but "matching with contracts" (which seems to be introduced in Hatfield & Milgrom (2005)) seems to be exactly what I was looking for!
dankness
CC BY-SA 3.0
2017-09-13T06:10:48.437Z
@dankness It is actually possible to embed the Hatfield & Milgrom model in the Kelso and Crawford model. See [this](http://people.hss.caltech.edu/~fede/published/embed.pdf) paper.
Michael Greinecker
CC BY-SA 3.0
2017-09-11T05:56:17.213Z
2017-09-11T05:56:17.213Z
Michael Greinecker
<p>I'm thinking of the following problem: There's a set $M$ of men and a set $W$ of women; furthermore there's a set $S$ of "possible marriages". Each possible marriage is a triple $(m, w, x)$, consisting of a man $m \in M$, a woman $w \in W$ and some arbitrary piece of extra information $x$.</p>

<p>For a man $m$, let $S_m = \{(m', w', x) \in S \mid m = m'\}$ be the set of possible marriages involving him, and for any woman $w$, let $S_w = \{(m', w', x) \in S \mid w = w'\}$ be the set of possible marriages involving her.</p>

<p>Instead of ranking the women he would marry, a man $m$ ranks the possible marriages: There is a total ordering $\leq_m$ defined on $S_m$. Similarly, for any woman $w$, there is a total ordering $\leq_w$ defined on $S_w$.</p>

<p>The problem is to find a stable matching. A stable matching is now a subset $X \subseteq S$ such that
<ul>
<li> every person is matched at most once, that is, for each man $m$, the intersection $X \cap S_m$ has at most one element, and for each woman $w$, the intersection $X \cap S_w$ has at most one element, and </li>
<li> $X$ is <em>stable</em> in the sense that for each possible marriage $s \in S$ between a man $m$ and a woman $w$, either</p>


<li> $m$ is in a marriage that is at least as good for him as $s$, that is, there is some $s' \in X \cap S_m$ such that $s' \geq_m s$, or </li>
<li> $w$ is in a marriage that is at least as good for her as $s$, that is, there is some $s' \in X \cap S_w$ such that $s' \geq_w s$. </li> </li>
</ul>

<p>The classical stable marriage problem is equivalent to the restricted version of this problem where for each possibe marriage $(m, w, x) \in S$, the "extra information" $x$ is required to be some fixed dummy value.</p>

<p><strong>Has anybody studied this generalization and given it some name?</strong> If not, is there some naming convention for "this kind of generalization"?</p>

<reference-request><game-theory>
Is there a name for this generalization of the stable marriage problem?
dankness
# Is there a name for this generalization of the stable marriage problem?

(Asked by: username_0 on 2017-09-11)

I'm thinking of the following problem: There's a set $M$ of men and a set $W$ of women; furthermore there's a set $S$ of "possible marriages". Each possible marriage is a triple $(m, w, x)$, consisting of a man $m \in M$, a woman $w \in W$ and some arbitrary piece of extra information $x$.

For a man $m$, let $S\_m = \{(m', w', x) \in S \mid m = m'\}$ be the set of possible marriages involving him, and for any woman $w$, let $S\_w = \{(m', w', x) \in S \mid w = w'\}$ be the set of possible marriages involving her.

Instead of ranking the women he would marry, a man $m$ ranks the possible marriages: There is a total ordering $\leq\_m$ defined on $S\_m$. Similarly, for any woman $w$, there is a total ordering $\leq\_w$ defined on $S\_w$.

The problem is to find a stable matching. A stable matching is now a subset $X \subseteq S$ such that

* every person is matched at most once, that is, for each man $m$, the intersection $X \cap S\_m$ has at most one element, and for each woman $w$, the intersection $X \cap S\_w$ has at most one element, and
* $X$ is *stable* in the sense that for each possible marriage $s \in S$ between a man $m$ and a woman $w$, either

- $m$ is in a marriage that is at least as good for him as $s$, that is, there is some $s' \in X \cap S\_m$ such that $s' \geq\_m s$, or
- $w$ is in a marriage that is at least as good for her as $s$, that is, there is some $s' \in X \cap S\_w$ such that $s' \geq\_w s$.

The classical stable marriage problem is equivalent to the restricted version of this problem where for each possibe marriage $(m, w, x) \in S$, the "extra information" $x$ is required to be some fixed dummy value.

**Has anybody studied this generalization and given it some name?** If not, is there some naming convention for "this kind of generalization"?

**Tags:** reference-request, game-theory

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## 1 answers

### Answer by username_1 on 2017-09-11. Score: 7 | Accepted answer

Yes, this has been studied. Look up "matching with contracts" and the earlier papers by Crawford & Knoer (1981), and Kelso & Crawford (1982) in Econometrica.

> **Comments:**
> **username_0**: Thanks a lot! Crawford & Knoer (1981) and Kelso & Crawford (1982) seem to be more restricted, but "matching with contracts" (which seems to be introduced in Hatfield & Milgrom (2005)) seems to be exactly what I was looking for!  
> **username_1**: @dankness It is actually possible to embed the Hatfield & Milgrom model in the Kelso and Crawford model. See [this](http://people.hss.caltech.edu/~fede/published/embed.pdf) paper.  

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