How it works, precisely
Two distinct groups rank each other. Side 1 proposes, side 2 disposes — and each entry on side 2 may have a capacity of several places.
Each proposer applies to their first choice. Each receiver provisionally holds the best candidates up to its capacity and turns the rest away. Nothing is final: that is the whole point of DEFERRED acceptance.
Rejected candidates apply to their next choice, which may displace a provisionally held candidate, who applies again in turn. It ends when nobody has a proposal left to make.
The result is stable: no candidate-programme pair would prefer each other to their assignment. It is also OPTIMAL for the proposing side — among all stable matchings, every proposer gets the best one possible.
A worked example
Three candidates, three programmes with one place each, cross rankings.
| Candidate | Wishes | Programme | Ranking |
|---|---|---|---|
| Chloé | A, B, C | A | Ali, Chloé, Bruno |
| Ali | A, C, B | B | Chloé, Bruno, Ali |
| Bruno | B, A, C | C | Bruno, Chloé, Ali |
- Round 1: Chloé and Ali apply to A, Bruno to B. A prefers Ali and drops Chloé; B holds Bruno.
- Round 2: Chloé applies to B. B prefers Chloé to Bruno and swaps — Bruno is displaced.
- Round 3: Bruno applies to A, which keeps Ali; then to C, which accepts. Final: Ali→A, Chloé→B, Bruno→C.
Bruno was held and then displaced: that is the deferred mechanism, and why Parcoursup results keep moving for weeks. The final outcome is stable.
Where it comes from
David Gale and Lloyd Shapley published « College Admissions and the Stability of Marriage » in the American Mathematical Monthly in 1962. They proved that a stable matching between two groups ALWAYS exists, and gave a simple algorithm to build it: deferred acceptance.
A historical twist: the National Resident Matching Program, which has assigned American medical residents to hospitals since 1952, was already using an equivalent algorithm — found empirically, ten years before it was theorised. Alvin Roth demonstrated this in 1984 and led its 1998 redesign to handle couples.
Roth and Shapley received the 2012 Nobel Prize in Economics for the theory of stable allocations and market design. In France, Parcoursup has applied the principle since 2018, replacing APB, with an automatic responder playing the role of deferred acceptance.
Where it is used
- Parcoursup and French post-secondary admissions.
- The American medical residency match (NRMP) since 1952.
- School assignment in New York and Boston, redesigned by Roth and colleagues.
- Mentoring programmes, internships and project allocation between two distinct populations.
Limits and pitfalls
- Always a stable matching
- Handles capacities (several spots per side-2 entry)
- Ranking sincerely is optimal for side 1
- Structurally favours the proposing side
- Side 2 may be tempted to rank tactically
- Some stay unmatched if spots run out
Frequently asked questions
Why does my assignment keep changing?
Because acceptance is deferred: a place is held for you provisionally, and a better-ranked candidate can displace you — just as you may displace someone elsewhere. The process only settles at the end, and that is what guarantees a stable result.
Should I rank my wishes tactically?
If you are on the proposing side — the candidates, in Parcoursup — no: ranking sincerely is provably optimal. Putting a « realistic » wish above the one you actually want can only hurt you.
What does « stable » mean?
That no candidate-programme pair would both prefer each other to what they got. Without that property, side deals form outside the system — exactly what happened in the United States before 1952.
How does this differ from serial dictatorship?
Serial dictatorship has only one side ranking: the options have no opinion. Here both sides rank, and the assignment must satisfy both — hence stability, a notion that has no meaning in serial dictatorship.