# Stable marriage problem

## Pairing where no unchosen pair prefers each other over their choice / From Wikipedia, the free encyclopedia

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In mathematics, economics, and computer science, the **stable marriage problem** (also **stable matching problem** or **SMP**) is the problem of finding a stable matching between two equally sized sets of elements given an ordering of preferences for each element. A matching is a bijection from the elements of one set to the elements of the other set. A matching is *not* stable if:

- There is an element
*A*of the first matched set which prefers some given element*B*of the second matched set over the element to which*A*is already matched, and *B*also prefers*A*over the element to which*B*is already matched.

In other words, a matching is stable when there does not exist any pair (*A*, *B*) which both prefer each other to their current partner under the matching.

The stable marriage problem has been stated as follows:

Given

nmen andnwomen, where each person has ranked all members of the opposite sex in order of preference, marry the men and women together such that there are no two people of opposite sex who would both rather have each other than their current partners. When there are no such pairs of people, the set of marriages is deemed stable.

The existence of two classes that need to be paired with each other (heterosexual men and women in this example) distinguishes this problem from the stable roommates problem.