Stable Marriage Problem Solver

Solve the stable marriage / stable matching problem using the Gale-Shapley algorithm. Paste ranked preference lists for two equally-sized groups and get the guaranteed-stable pairing, with an animated proposal-by-proposal trace, satisfaction statistics, blocking-pair verification, and an interactive bipartite visualization.

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About Stable Marriage Problem Solver

The Stable Marriage Problem Solver is an interactive implementation of the Gale-Shapley deferred-acceptance algorithm, the 1962 matching algorithm that David Gale and Lloyd Shapley proved always produces a stable pairing between two equally-sized groups given each member's full ranking of the other side. This tool takes your preference lists, runs the algorithm step by step, and shows the result as a stable matching, an animated bipartite visualization, preference heatmaps, and a verified proof that no blocking pair exists.

What Is the Stable Marriage Problem?

Given two disjoint sets of equal size — say n men and n women, or n applicants and n positions — and a complete preference list from every member ranking every member of the other side, a matching is a one-to-one pairing between the two sets. The matching is called stable if no pair (a, b) outside the matching would both rather be together than stay with their assigned partners.

Formally, a matching M is stable if there is no blocking pair — a pair (a, b) with a matched to b' and b matched to a' such that:

a prefers b to b' AND b prefers a to a'

If both conditions hold, both a and b would abandon their current partners, which destabilises the matching. A stable matching is one where no such pair exists.

The Gale-Shapley Algorithm

Gale and Shapley proved — constructively — that a stable matching always exists for any set of preferences, and they gave an efficient algorithm to find one. The algorithm runs in rounds:

Every unengaged proposer proposes to the highest-ranked receiver on their list who has not yet rejected them.
Each receiver who got one or more proposals picks the one they prefer most (compared against any current tentative fiancé) and tentatively accepts; everyone else is rejected.
Rejected proposers become free again and move on to their next choice in the following round.
The algorithm terminates when every proposer is engaged — which is guaranteed to happen in at most n² proposals.

Time complexity: O(n²) Space complexity: O(n²) Proposals before termination: at most n²

Key Theoretical Properties

Existence & Uniqueness

A stable matching always exists (Gale & Shapley, 1962), but is not necessarily unique. For a given preference set there can be multiple stable matchings, and they form a lattice ordered by joint preference.

Proposer Optimality

When one side proposes, Gale-Shapley produces the proposer-optimal stable matching: every proposer receives the best partner they could possibly be paired with in any stable matching. By a symmetric argument, this is also the receiver-pessimal matching — the receiving side gets their worst stable partner. Switching the proposing side in this calculator often changes the result.

Strategy Proofness for Proposers

Under Gale-Shapley, proposers have no incentive to misreport their preferences: telling the truth is a dominant strategy for them. Receivers, however, can sometimes benefit from strategic misreporting — one of the reasons the hospital-residents market in the US is designed with students as the proposing side.

Rural Hospitals Theorem

The set of unmatched agents is identical across all stable matchings. So if your instance has unbalanced sizes (beyond the scope of this classic tool) the same people will be unmatched in every stable solution.

Input Format

This calculator expects one line per member, with name followed by a colon and a full ranked preference list separated by commas:

Name1: first choice, second choice, third choice, ..., last choice Name2: first choice, second choice, ... ...

Requirements:

Both groups must have exactly the same number of members.
Every member must rank every member of the other group — no partial lists.
Names can use letters, digits, underscores, hyphens, spaces, and common Unicode letters.
Up to 15 members per side are supported for fast interactive animation.

How to Use This Calculator

Enter preferences for Group A in the left text area — one line per member, full ranked list.
Enter preferences for Group B in the right text area — one line per member, same format.
Pick the proposing side. Choose Group A or Group B. Try both to compare the A-optimal vs B-optimal outcomes.
Click "Solve Stable Matching." The calculator runs Gale-Shapley and produces the stable pairs, statistics, animation, and proof of stability.
Scrub the animation with the play / step / reset controls to see every proposal, acceptance, swap, and rejection in order.
Inspect the heatmap. Each cell shows the ranking; the yellow-outlined cells form the final matching — look at how high or low those cells sit to see how "happy" each side is.

Worked Example — The Classic 3×3

Men: Alex, Bryan, Chris. Women: Bea, Claire, Diana. Preferences:

Alex: Bea, Claire, Diana Bryan: Claire, Bea, Diana Chris: Diana, Bea, Claire Bea: Bryan, Alex, Chris Claire: Alex, Bryan, Chris Diana: Chris, Bryan, Alex

Running Gale-Shapley with men proposing yields Alex–Bea, Bryan–Claire, Chris–Diana in a single round (every man's first choice matches a woman whose first choice is someone else — no conflict). The matching is stable: no man-woman pair would both be better off swapping partners, so there is no blocking pair.

Real-World Applications

Application	Group A	Group B	Who proposes
NRMP Residency Match (US)	Medical students	Hospital programs	Students — designed to be student-optimal since 1998
NYC / Boston School Choice	Families	Public schools	Families — replaced a strategic-gaming mechanism in the 2000s
College Admissions	Applicants	Universities	Originally Gale-Shapley's motivating example
Kidney Exchange	Donor-recipient pairs	Other donor-recipient pairs	Specialized cycle-finding extension of matching theory
Dating & Roommate Matching	Users	Potential partners	Consumer apps often use simplified versions of the same idea

Why Lloyd Shapley Won a Nobel Prize

In 2012 the Royal Swedish Academy of Sciences awarded the Nobel Prize in Economics to Lloyd Shapley (for the theory, alongside David Gale who had passed away) and Alvin Roth (for applying the theory to real markets, including redesigning the US medical residency match and kidney exchange networks). The award cited "the theory of stable allocations and the practice of market design."

Frequently Asked Questions

What is the stable marriage problem?

The stable marriage problem asks: given two equally-sized groups where each member ranks all members of the other group from most to least preferred, can we pair everyone up so that no two people would both prefer to leave their current partners for each other? Such a pairing is called a stable matching. The Gale-Shapley algorithm solves this problem in O(n²) time and always finds a stable matching.

How does the Gale-Shapley algorithm work?

The Gale-Shapley deferred-acceptance algorithm proceeds in rounds. In each round, every currently-unengaged proposer proposes to the highest-ranked receiver on their list who has not yet rejected them. Each receiver tentatively accepts the best proposal received so far and rejects the rest; any displaced proposer becomes free again. The algorithm terminates when no proposer is free, which happens in at most n² proposals.

Is the stable matching unique?

No. A stable matching instance can have many stable matchings. However, when one side proposes, Gale-Shapley always produces the proposer-optimal stable matching: every proposer gets the best partner they could possibly get in any stable matching. By symmetry, this is also the receiver-pessimal matching for the other side. Switching the proposing side often yields a different stable matching.

What is a blocking pair?

A blocking pair is a pair (a, b) that is not currently matched where a prefers b to their current partner AND b also prefers a to their current partner. If any blocking pair exists, the matching is unstable because those two would rather pair up with each other. A stable matching has no blocking pairs, which this calculator verifies automatically after each solve.

What are real applications of stable matching?

The Gale-Shapley algorithm powers the National Resident Matching Program that assigns medical students to residencies in the United States, school-choice systems in Boston and New York, college admissions in several countries, organ-donor kidney exchange chains, and roommate-assignment systems. Lloyd Shapley and Alvin Roth won the 2012 Nobel Prize in Economics partly for this work.

Do both groups need to be the same size?

In the classical stable marriage formulation, yes. Both sides must have the same number of members and each must provide a complete ranking of the other side. Unbalanced variants (such as the stable matching with incomplete lists or the hospital-residents problem) exist but require modified algorithms. This calculator enforces equal sizes and complete preference lists.

Stable Marriage Problem Solver

About Stable Marriage Problem Solver

What Is the Stable Marriage Problem?

The Gale-Shapley Algorithm

Key Theoretical Properties

Existence & Uniqueness

Proposer Optimality

Strategy Proofness for Proposers

Rural Hospitals Theorem

Input Format

How to Use This Calculator

Worked Example — The Classic 3×3

Real-World Applications

Why Lloyd Shapley Won a Nobel Prize

Frequently Asked Questions

What is the stable marriage problem?

How does the Gale-Shapley algorithm work?

Is the stable matching unique?

What is a blocking pair?

What are real applications of stable matching?

Do both groups need to be the same size?

Further Reading

Advanced Math Operations:

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