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"?
