Stable Matching, explained.
Stable matching assigns partners so that no unmatched pair would both prefer each other to their current assignments. Deferred acceptance is one way to construct such a matching under specified preference assumptions.
Why it happens
An applicant proposes in preference order. A team keeps its favourite proposal received so far and releases a less-preferred applicant. A rejected applicant never needs to try that team again: the team can only replace its held offer with one it prefers more. The finite proposal process eventually stops.
A stable matching has no unmatched pair who both prefer each other to their assigned partners. Deferred acceptance maintains tentative matches while rejected proposers move down their preference lists.
Read the result
Choose any enabled applicant button and inspect the tentative assignments. A held offer is not final until everyone is matched. Read the history when an applicant is released. At completion, the blocking-pair count checks the actual rankings; zero does not mean everyone received a first choice.
A worked example
Three applicants, one place per team
Ari prefers Maple, and Bo also proposes to Maple, which ranks Bo above Ari.
Maple holds Bo's offer and releases Ari, who moves to the next team on Ari's own list.
Continue until all places are held. Check every unmatched applicant-team pair: stability rules out a mutually preferred deviation, not disappointment.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Each side has complete strict rankings of three partners and one place per team. Every click sends a free applicant's next untried proposal. Teams retain the applicant they rank higher. The final board explicitly checks all unmatched pairs for blocking pairs. New round changes both preference profiles using a seeded shuffle.
Where this idea is useful
A practical use
Matching-market design can help explain placement systems where both sides have preferences, rather than treating assignments as a single score ranking.
A common misconception
“A stable assignment maximises everybody's happiness.”
Stability excludes blocking pairs under the stated preferences. It does not maximise summed satisfaction, guarantee equal outcomes, or resolve a contested definition of fairness.
What this explanation leaves out
- This small one-to-one complete-preference model omits capacities, couples, priorities, and unacceptable matches. Stability does not mean fairness, happiness, or maximum total satisfaction; proposer advantage requires the usual model assumptions.
Why are the matches tentative?
A team may receive a more-preferred proposal later. Keeping offers tentative allows displaced applicants to continue while the team retains its best proposal so far.
In an allocation you know, whose preferences are represented—and what would count as a mutually preferred change?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Explore the original research or the teaching reference behind this experiment.