Matpic
mathematics, in pictures
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. One matching that is not stable, and all 24 counted by blocking pairs. An unstable matching with its blocking pair ringed and both members' rankings marked, above an exhaustive census of every matching of the instance by how many blocking pairs it has.

    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. The 4 stable matchings of the instance, ordered by side one's preference. A Hasse diagram of the stable matchings with the best for side one at the top, beside a table giving the join and the meet of every pair.

    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. Every ranking 4 could submit, and the 4 that pay. One participant's true ranking, a cell for every ranking they could submit instead labelled with the partner it returns, the profitable misreports listed, and the same search run on the proposing side finding none.

    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

All ladders