Simplify Your Workflow: Search MiniWebtool.
Add Extension
Home Page > Math > Advanced Math Operations > Stable Marriage Problem Solver

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.

Stable Marriage Problem Solver
Quick examples:
Classic 3×3 Residency match Textbook P/R 5×5 teams 4×4 (B proposes)
A
One line per member: Name: Pref1, Pref2, ... — list every Group B member in preference order.
B
Same format. Each Group B member ranks all Group A members.
In Gale-Shapley, the proposing side gets their best possible stable partner; the receiving side gets their worst.

Embed Stable Marriage Problem Solver Widget

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:

  1. Every unengaged proposer proposes to the highest-ranked receiver on their list who has not yet rejected them.
  2. 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.
  3. Rejected proposers become free again and move on to their next choice in the following round.
  4. 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:

How to Use This Calculator

  1. Enter preferences for Group A in the left text area — one line per member, full ranked list.
  2. Enter preferences for Group B in the right text area — one line per member, same format.
  3. Pick the proposing side. Choose Group A or Group B. Try both to compare the A-optimal vs B-optimal outcomes.
  4. Click "Solve Stable Matching." The calculator runs Gale-Shapley and produces the stable pairs, statistics, animation, and proof of stability.
  5. Scrub the animation with the play / step / reset controls to see every proposal, acceptance, swap, and rejection in order.
  6. 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.

Further Reading

Reference this content, page, or tool as:

"Stable Marriage Problem Solver" at https://MiniWebtool.com/stable-marriage-problem-solver/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Apr 22, 2026

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

Related MiniWebtools:

Adjacency Matrix CalculatorNew
Age Word Problem SolverNew
Angle Between Vectors CalculatorNew
Coin Word Problem SolverNew
Distance Between Two Points Calculator
Factor Calculator
Fraction Calculator
Long Division CalculatorNew
Mixture Problem SolverNew
Percentage Calculator

Advanced Math Operations:

Top & Updated:

Random PickerRandom Name PickerFPS ConverterInstagram User ID LookupLine CounterSort NumbersRelative Standard Deviation CalculatorBatting Average CalculatorMAC Address GeneratorRemove SpacesERA CalculatorJob FinderWord to Phone Number ConverterFeet and Inches to Cm ConverterMAC Address LookupRandom Truth or Dare GeneratorFacebook User ID LookupSum CalculatorPercent Off CalculatorSquare Root (√) CalculatorSun, Moon & Rising Sign Calculator 🌞🌙✨OPS CalculatorSHA256 Hash GeneratorLog Base 10 CalculatorImage ResizerMP3 LooperBitwise CalculatorNumber of Digits CalculatorSaturn Return CalculatorAudio SplitterPhone Number ExtractorSlope and Grade CalculatorRandom Credit Card GeneratorVertical Jump CalculatorRoman Numerals ConverterAI Text HumanizerRandom Sound Frequency GeneratorSlugging Percentage CalculatorRandom Activity GeneratorOn Base Percentage CalculatorSalary Conversion CalculatorCm to Feet and Inches ConverterRandom IMEI GeneratorRandom Movie PickerInvisible Text GeneratorMerge VideosNumber to Word ConverterWAR Calculator⬛ Aspect Ratio CalculatorOctal CalculatorCaffeine Overdose CalculatorRandom Fake Address GeneratorBinary to Gray Code ConverterRandom Superpower GeneratorRandom Poker Hand GeneratorDecimal to BCD ConverterFile Size ConverterRandom Loadout GeneratorMaster Number CalculatorText FormatterRandom Quote GeneratorVideo to Image ExtractorAdd Prefix and Suffix to TextRandom Writing Prompt GeneratorBCD to Decimal ConverterFirst n Digits of PiSteel Weight CalculatorRandom Birthday GeneratorWHIP CalculatorTime Duration CalculatorCompound Growth CalculatorLove Compatibility CalculatorWord Ladder GeneratorQuotient and Remainder CalculatorCompare Two StringsYouTube Channel StatisticsName Number CalculatorCM to Inches ConverterSHA512 Hash GeneratorOutlier CalculatorBattery Life CalculatorImage CompressorDMS to Decimal Degrees ConverterWhat is my Lucky Number?Remove AccentPercent Growth Rate CalculatorGray Code to Binary ConverterLeap Years ListRemove Line Breaks📅 Date CalculatorStair CalculatorAcreage CalculatorDay of Year CalendarVideo CompressorProportion CalculatorBinary to BCD ConverterSocial Media Username CheckerIP Subnet CalculatorRandom Number PickerEmail ExtractorURL ExtractorAI ParaphraserAI Punctuation AdderList of Prime NumbersDay of the Year Calculator - What Day of the Year Is It Today?IP Address to Hex ConverterSort Lines AlphabeticallyHex to BCD ConverterBCD to Binary ConverterLottery Number GeneratorBCD to Hex ConverterMedian CalculatorStandard Error CalculatorList RandomizerBreak Line by CharactersAverage CalculatorModulo CalculatorPVIFA CalculatorReverse VideoHypotenuse CalculatorRemove Audio from VideoActual Cash Value CalculatorScientific Notation to Decimal ConverterNumber ExtractorAngel Number CalculatorLog Base 2 CalculatorRoot Mean Square CalculatorSum of Positive Integers CalculatorSHA3-256 Hash GeneratorAI Sentence ExpanderLbs to Kg ConverterHex to Decimal ConverterRandom Group GeneratorConvolution CalculatorMAC Address AnalyzerRandom String GeneratorRemove Leading Trailing SpacesAmortization CalculatorMarkup CalculatorPVIF CalculatorDecimal to Hex ConverterInstagram Font GeneratorSocial Media Image Size GuideTikTok Money CalculatorTwitter/X Character CounterTwitter/X Timestamp ConverterYouTube Watch Time CalculatorTwitch Earnings CalculatorYouTube Shorts Monetization CalculatorFacebook Ad Cost CalculatorSocial Media ROI CalculatorSocial Media Post Time OptimizerCTR CalculatorROAS CalculatorInfluencer ROI CalculatorForce CalculatorAcceleration CalculatorVelocity CalculatorMomentum CalculatorProjectile Motion CalculatorKinetic Energy CalculatorPotential Energy CalculatorWork and Power CalculatorDensity CalculatorPressure CalculatorIdeal Gas Law CalculatorFree Fall CalculatorTorque CalculatorHorsepower CalculatorDilution CalculatorChemical Equation BalancerStoichiometry CalculatorPercent Yield CalculatorEmpirical Formula CalculatorBoiling Point CalculatorTitration CalculatorMole/Gram/Particle ConverterIrregular Polygon Area CalculatorFrustum CalculatorTorus Calculator3D Distance CalculatorGreat Circle Distance CalculatorCircumscribed Circle (Circumcircle) CalculatorInscribed Circle (Incircle) CalculatorAngle Bisector CalculatorTangent Line to Circle CalculatorHeron's Formula CalculatorCoordinate Geometry Distance CalculatorVolume of Revolution CalculatorSurface of Revolution CalculatorParametric Curve GrapherRiemann Sum CalculatorTrapezoidal Rule CalculatorSimpson's Rule CalculatorImproper Integral CalculatorL'Hôpital's Rule CalculatorMaclaurin Series CalculatorPower Series CalculatorSeries Convergence Test CalculatorInfinite Series Sum CalculatorAverage Rate of Change CalculatorInstantaneous Rate of Change CalculatorRelated Rates SolverOptimization Calculator (Calculus)Gradient Calculator (Multivariable)Divergence CalculatorCurl CalculatorLine Integral CalculatorSurface Integral CalculatorJacobian Matrix CalculatorNewton's Method CalculatorRREF Calculator (Row Echelon Form)Matrix Inverse CalculatorMatrix Multiplication CalculatorDot Product CalculatorCross Product CalculatorVector Magnitude CalculatorUnit Vector CalculatorAngle Between Vectors CalculatorNull Space CalculatorColumn Space CalculatorCramer's Rule CalculatorMatrix Diagonalization CalculatorQR Decomposition CalculatorCholesky Decomposition CalculatorMatrix Power CalculatorCharacteristic Polynomial CalculatorBayes' Theorem CalculatorF-Test / F-Distribution CalculatorHypergeometric Distribution CalculatorNegative Binomial Distribution CalculatorGeometric Distribution CalculatorExponential Distribution CalculatorWeibull Distribution CalculatorBeta Distribution CalculatorSpearman Rank Correlation CalculatorFisher's Exact Test CalculatorContingency Table CalculatorOdds Ratio CalculatorRelative Risk CalculatorEffect Size CalculatorPermutations with Repetition CalculatorModular Exponentiation CalculatorPrimitive Root CalculatorPerfect Number CheckerAmicable Number CheckerTwin Prime FinderMersenne Prime CheckerGoldbach Conjecture VerifierMöbius Function CalculatorEgyptian Fraction CalculatorFibonacci Number CheckerDigital Root CalculatorPartition Function CalculatorBoolean Algebra SimplifierKarnaugh Map (K-Map) SolverLogic Gate SimulatorGraph Coloring CalculatorTopological Sort CalculatorAdjacency Matrix CalculatorRecurrence Relation SolverInclusion-Exclusion CalculatorLinear Programming SolverTraveling Salesman Solver (TSP)Hamiltonian Path CheckerPlanar Graph CheckerNetwork Flow Calculator (Max Flow)Stable Marriage Problem SolverFirst-Order ODE SolverSecond-Order ODE SolverDirection Field / Slope Field PlotterEuler's Method CalculatorBernoulli ODE SolverSystem of ODEs SolverGroup Theory Order CalculatorRing and Field CalculatorJordan Normal Form CalculatorMatrix Exponential CalculatorTensor Product CalculatorFast Fourier Transform (FFT) CalculatorZ-Transform CalculatorNumerical Integration CalculatorTOML to JSON ConverterJSON to CSV ConverterXML to JSON ConverterSQL to MongoDB Query ConverterCSS Flexbox PlaygroundCSS Grid GeneratorJWT GeneratorBcrypt Hash Generator / CheckerColor Code Converter (All Formats)Git Command Generator.env File GeneratorLorem Picsum / Placeholder Image GeneratorText to Binary/Hex/ASCII ConverterSyllable CounterSentence CounterParagraph CounterSpeaking Time CalculatorReading Time CalculatorWhitespace VisualizerStrikethrough Text GeneratorTorque Converter (Nm, ft-lb, kgf-cm)Data Transfer Rate ConverterFuel Efficiency ConverterAstronomical Unit ConverterRing Size ConverterPaper Size ReferenceClothing Size ConverterGas Mileage CalculatorEV Range CalculatorEV Charging Time Calculator0–60 / Quarter Mile CalculatorCar Lease CalculatorVehicle Towing Capacity CalculatorExposure Triangle CalculatorCrop Factor CalculatorMegapixel to Print Size CalculatorPhoto File Size EstimatorMusic BPM TapperMusic Key TransposerVideo Bitrate CalculatorSeed Germination Rate CalculatorFertilizer Calculator (NPK)Raised Bed Soil CalculatorFrost Date CalculatorLawn Fertilizer CalculatorCompost Calculator (C:N Ratio)Solar Panel CalculatorSolar ROI CalculatorHome Energy Audit CalculatorAppliance Energy Cost CalculatorWater Usage CalculatorElectricity Generation Cost CalculatorHeat Loss CalculatorFlight Distance CalculatorTravel Budget CalculatorJet Lag CalculatorPacking List GeneratorTip Splitter (Advanced)Lease vs Buy CalculatorHourly Rate Calculator (Freelancer)Invoice Late Fee CalculatorESPP CalculatorStock Split CalculatorOptions Probability CalculatorDollar to Gold ConverterBeam Load CalculatorPipe Flow CalculatorBolt Torque CalculatorGravel, Sand & Topsoil CalculatorRandom Sentence GeneratorRandom Paragraph GeneratorRandom Math Problem GeneratorRandom Bible Verse GeneratorRandom Cat/Dog Name GeneratorRandom Debate Topic GeneratorBody Recomposition CalculatorAlcohol Calorie CalculatorMedication Dosage CalculatorPace to Calories CalculatorHydration CalculatorTrain Meeting Problem SolverAge Word Problem SolverMixture Problem SolverWork Rate Problem SolverDistance-Speed-Time Triangle CalculatorCoin Word Problem SolverNumber Bonds GeneratorCarry and Borrow VisualizerTimes Tables QuizMental Math TrainerRoman Numeral Math SolverEgyptian Multiplication CalculatorVedic Math Tricks CalculatorRussian Peasant MultiplicationSoroban Abacus SimulatorAnnuity Payout CalculatorReverse Mortgage CalculatorVariable Annuity CalculatorFixed Indexed Annuity CalculatorBond Convexity CalculatorBond Duration Calculator (Macaulay & Modified)Forward Rate CalculatorMortgage Recast CalculatorTreasury Inflation-Protected Securities (TIPS) CalculatorStock Beta CalculatorTreynor Ratio CalculatorSortino Ratio CalculatorDoppler Effect CalculatorSpring Constant CalculatorPendulum Period CalculatorCentripetal Force CalculatorAngular Velocity CalculatorMoment of Inertia CalculatorSnell's Law CalculatorCoulomb's Law CalculatorElectric Field CalculatorMagnetic Field of Wire CalculatorLens Equation CalculatorA/B Test Significance CalculatorA/B Test Sample Size CalculatorConversion Rate CalculatorCustomer Lifetime Value (CLV) CalculatorCustomer Acquisition Cost (CAC) CalculatorChurn Rate CalculatorRetention Rate Cohort CalculatorNPS (Net Promoter Score) CalculatorPareto Chart GeneratorSix Sigma Process Capability CalculatorTessellation GeneratorSpirograph GeneratorVoronoi Diagram GeneratorDelaunay Triangulation GeneratorL-System Fractal GeneratorMandelbrot Set ExplorerJulia Set GeneratorPolar Equation Plotter3D Surface PlotterSierpinski Triangle GeneratorcURL Command BuilderHTTP Status Code ReferenceUUID Validator/DecoderURL ParserQuery String BuilderSVG to React/JSX ConverterSCSS to CSS CompilerLess to CSS CompilerTypeScript PlaygroundJSON Schema GeneratorImage to ASCII Art ConverterImage to SVG TracerLipogram CheckerPangram CheckerAcronym GeneratorBackronym GeneratorPig Latin TranslatorEXIF Data Viewer/RemoverROT13 Encoder/DecoderAtbash Cipher ToolVigenère Cipher ToolPronunciation IPA ConverterHemingway-Style Readability Editor