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Mathematics · unit version 1

Stable matching and preference guarantees

Find mutually preferred deviations, trace deferred acceptance, and prove exactly what its stability and optimality guarantees require.

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Readiness and preparation

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Does pairing everyone prove that an assignment is stable?

A matching can have maximum cardinality while two participants prefer each other to their assigned partners.

Compare reasoning and find a refresher

No. Stability additionally requires no unmatched pair whose two members both prefer one another to their current partners. A perfect matching only certifies that each participant has one partner.

Review perfect matchings and feasibility

Does proving a process stops also prove its output is correct?

Deferred acceptance needs separate termination and stability arguments.

Compare reasoning and find a refresher

No. A bound on distinct proposals proves termination, while a preserved best-seen/rejection invariant supports the final stability proof.

Review invariants and termination

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Stable outcomes, changed rules and uniqueness

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Lesson sequence

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  1. Identify blocking pairs and model assumptions

    Lesson details and objectives

    Distinguish a preference-stable outcome from a matching that merely pairs everyone.

    • Translate a pair of preference tables into a precise blocking-pair test.
    • Certify stability by checking all unmatched pairs or refute it with one mutually preferred pair.
    • Explain how incomplete lists or ties change the model and its existence claims.

    6 practice tasks.

  2. Trace deferred acceptance and prove stability

    Lesson details and objectives

    Use provisional holds and permanent rejection history to prove termination, completeness and stability.

    • Execute individual proposals while keeping each participant matched at most once.
    • Prove the rejection invariant, finite proposal bound and perfect completion.
    • Use rejection history to rule out every possible blocking pair.

    6 practice tasks.

  3. Prove proposer-optimality and interpret its limits

    Lesson details and objectives

    Use the first rejected stable pair to prove optimality among stable outcomes, without claiming universal fairness.

    • Prove that deferred acceptance never rejects a pair belonging to any stable matching.
    • Deduce each proposer’s best stable partner and order-independent final matching.
    • Analyze role reversal, receiver outcomes and changed assumptions without confusing stability with social optimality.

    6 practice tasks.