Gale-Shapley Stable Matching
Overview
Gale-Shapley (deferred acceptance) finds a stable matching between two equally-sized sets where no unmatched pair prefers each other over their current match. Runs in O(n²) worst case. Proposer-optimal: the proposing side gets their best stable partner.
When to Use
Trigger conditions:
- Matching candidates to job positions based on mutual preferences
- Assigning students to schools or residents to hospitals
- Any two-sided matching where stability (no blocking pairs) is required
When NOT to use:
- For one-sided assignment (use Hungarian algorithm)
- When preferences are based on scores, not rankings (use optimization)
Algorithm
IRON LAW: The Proposing Side Gets Their BEST Stable Partner
Gale-Shapley is proposer-optimal and reviewer-pessimal. If employers
propose, they get their best stable match; candidates get their worst.
The CHOICE of who proposes determines which stable matching is found.Phase 1: Input Validation
Collect: preference rankings from both sides. Each participant ranks all members of the other side. Gate: Complete preference lists, equal-sized groups (or handle unequal with dummy entries).
Phase 2: Core Algorithm
- All proposers are "free" (unmatched)
- While any proposer is free and hasn't proposed to everyone:
- Free proposer proposes to their highest-ranked unproposed-to reviewer - Reviewer accepts if unmatched, or replaces current match if new proposer is preferred - Replaced proposer becomes free again
- Terminate when all proposers are matched
Phase 3: Verification
Check stability: for every unmatched pair (a,b), verify that at least one of them prefers their current match over the other. No blocking pairs = stable. Gate: Zero blocking pairs found.
Phase 4: Output
Return matching with stability confirmation.
Output Format
{
"matching": [{"proposer": "Candidate_A", "reviewer": "Company_X", "proposer_rank": 1, "reviewer_rank": 2}],
"metadata": {"pairs": 10, "rounds": 23, "blocking_pairs": 0, "proposer_side": "candidates"}
}Examples
Sample I/O
Input: 3 candidates, 3 companies, each with full preference rankings Expected: Stable matching with zero blocking pairs. Candidate-proposing gives candidate-optimal result.
Edge Cases
| Input | Expected | Why |
|---|---|---|
| All prefer same #1 | Still terminates, stable | Rejected proposers move to next choice |
| Identical preferences | Unique stable matching | Only one possibility |
| Unequal sides | Some unmatched on larger side | Add dummy entries or use many-to-one variant |
Gotchas
- Proposer advantage: If candidates propose, they get better matches than if companies propose. This is a design choice with equity implications.
- Incomplete preferences: If participants don't rank everyone, unmatched results are possible. Handle with acceptable-partner thresholds.
- Many-to-one: Hospital-resident matching uses the many-to-one variant (each hospital has multiple slots). Use the Roth-Peranson extension.
- Strategic manipulation: The reviewing side CAN benefit from misreporting preferences (truncating lists). The proposing side cannot — truthful reporting is dominant strategy for proposers.
- Preference elicitation: Getting honest, complete rankings is hard in practice. People satisfice rather than fully rank all options.
References
- For many-to-one matching (hospital-resident), see
references/many-to-one.md - For strategic behavior analysis, see
references/strategic-manipulation.md