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Ladder

Stable matching — the ladder

3 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. a matching that is not stableA1B2C3D4why that pair blocksD13would rather4has now23ADwould ratherChas nowBhow many of the 24 matchings have each number of blocking pairs40315283241516blocking pairsevery matching, sorted — the 4 stable ones first000011122222333333334456A1 B2 C3 D4 has 1 blocking pair: D and 3 would each rather have the otherall 24 matchings were formed and each put all 16 blocking questions; 4 of them came back with nonethe strip holds one cell per matching, sorted by that count with the 4 zeroes at the left

    Nobody has a reason to run away

    A matching is stable when no two people on opposite sides would both rather have each other than what they have — a condition that names nothing to build and everything to rule out. The surprise is that something always satisfies it, however perverse the rankings are made.

    rung 1 · applied
  2. better for side one, worse for side twoM1 A4 B2 C1 D3side one's ranks 2, 2, 2, 2M2 A3 B2 C1 D4side one's ranks 3, 2, 2, 3M3 A4 B1 C2 D3side one's ranks 2, 4, 3, 2M4 A3 B1 C2 D4side one's ranks 3, 4, 3, 3join and meet, on every pairpairjoinmeetM1, M2M1M2M1, M3M1M3M1, M4M1M4M2, M3M1M4M2, M4M2M4M3, M4M3M44 of the 24 matchings are stable, drawn with the best for side one at the topevery one of the 16 ordered pairs was joined and met, and all 32 results were themselves stablethe top A4 B2 C1 D3 is what side one proposing returns; the bottom A3 B1 C2 D4 is what side two proposing returns

    The side that proposes wins

    An instance usually has several stable matchings, and the set of them is not a heap — it is a lattice, closed under taking the better partner and under taking the worse. The two ends of that lattice are exactly what deferred acceptance returns from the two sides, so whoever proposes decides which end the instance lands on.

    rung 2 · applied
  3. 4's true ranking of side one, best firstCDAtruthfulBtruthfully: A4 B2 C1 D3one cell per ranking 4 could submit, labelled with the partner it returnsAAAAAABBBBBBAABBADAADDADthe true rankinga ranking that paysthe 4 profitable misreports, submitted ranking and partner obtainedsubmittedpartnertrue rankCDBAD2 of 4DBACD2 of 4DBCAD2 of 4DCBAD2 of 4the control: the same search, every member of both sides, 24 rankings eachside onepaysside twopaysA010B020C030D0444 truly ranks side one CDAB and gets A, its 3rd choiceof the 24 rankings it could submit instead, 4 return a partner it strictly prefersthe same sweep over every member of the proposing side searched 96 rankings and found none, which is whatmakes the 4 a finding

    No stable rule is safe from a lie

    A stable matching always exists, and the side that proposes gets the best one it could hope for. This essay closes the ladder with the result that spoils it — one participant's whole strategy space searched, four submissions found that beat the truth, and a theorem saying no rule anywhere escapes.

    rung 3 · applied

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