Last updated on 2/5/18 PM. 1. STABLE MATCHING. ‣ stable matching problem. ‣ Gale–Shapley algorithm. ‣ hospital optimality. ‣ context. The Stable Marriage Problem states that given N men and N women, where each person has ranked all members of the opposite sex in order of preference. In what follows, we will describe the algorithm within Gale-Shapley's original context, the stable marriage problem. Suppose we have an equal number of men.

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### Stable Marriage Problem

Giving one group their first choices ensures that the matches are stable because they would stable marriage problem unhappy with any other proposed match.

Giving everyone their second choice ensures that any other match would be disliked by one of the parties.

Stable marriage problem algorithm converges in a single round on the suitor-optimal solution because each reviewer receives exactly one proposal, and therefore selects that proposal as its best choice, ensuring that each suitor has an accepted offer, ending the match.

This asymmetry of optimality is driven by the fact that the suitors have the entire set to choose from, but reviewers choose between a limited stable marriage problem of the suitors at any one time.

Stable marriage stable marriage problem indifference[ edit ] In the classical version of the problem, each person must rank the members of the opposite sex in strict order of preference.

However, in a real-world setting, a person may prefer two or more persons as equally favorable partner. Such tied preference is termed as indifference.

Below is such an instance where m. This made it risky for many students to faithfully state their true preferences, a view encouraged by the Education Stable marriage problem advice that students "determine what your competition is" before creating their lists of stable marriage problem schools.

Lastly, schools would often underrepresent their capacity hoping to save positions for students who were unhappy with their initial offerings.

In the end, the process couldn't place many students while it encouraged all parties, both students and schools, to strategically misrepresent themselves in an effort stable marriage problem obtain more desirable outcomes not possible otherwise.

Widespread mistrust in the placement process was a natural consequence. Using ideas described in this column, economists Atila Abdulkadiroglu, Parag Pathak, and Alvin Roth designed a clearinghouse for matching students with high schools, which was first implemented in This new computerized algorithm places all but about students each year and results in more students receiving offers from stable marriage problem first-choice schools.

## Stable marriage problem - Rosetta Code

As a result, students now submit lists that reflect their true preferences, which provides school officials with public input into the determination of which schools to close or reform. For their part, schools have found that there is no longer an advantage to underrepresenting their capacity.

The key to this new algorithm is the notion of stability, first introduced in a paper by Gale and Shapley. We say that a matching of students to schools is stable if there is not a student and a school who would stable marriage problem to be matched with each other more than their current matches.

Gale and Shapley introduced an algorithm, sometimes called deferred acceptance, which is guaranteed to produced a stable matching. Later, Roth showed that when stable marriage problem deferred acceptance algorithm is applied, a student can not gain admittance into a more preferred school by strategically misrepresenting his or her preferences.

## Stable Marriage Problem -- from Wolfram MathWorld

This column will present the game-theoretic results contained in the original Gale-Shapley paper along with Roth's subsequent analysis. Pathak calls stable marriage problem deferred acceptance algorithm "one of the great ideas in economics," and Roth and Shapley were awarded the Nobel Prize in economics for this work.

• AMS :: Feature Column :: The Stable Marriage Problem and School Choice
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• The Stable Marriage Problem and School Choice