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

1. All proposers are "free" (unmatched)
2. 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
3. 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