Why it matters

Stable matching solves residency, school assignment, employment. Understanding shapes mechanism design.

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The architecture

Proposers propose in preference order.

Receivers keep best proposal so far.

Rejected re-propose.

Gale-ShapleyProposetop preferenceAccept tentativeor replaceRejected re-proposenext choiceProposer-optimal stable matching; receivers get worst stable match
Gale-Shapley.
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How it works end to end

Terminates in O(n^2) proposals.

Result: proposer-optimal.

Applications: NRMP (medical residency), school choice.