Matpic
mathematics, in pictures
Concept

Quantifier order

The difference between one thing serving every case and each case having its own, which swapping two quantifiers changes.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

ji123456123456∀i ∃j : trueevery row carries at least one mark∃j ∀i : falsesome one column is marked all the way downthe relation "j is one more than i, counting round", on 6 rows and 6 columnsevery row has a mark: yes · some column is all marks: no

Every row, or one column

For every person there is someone who loves them, and there is someone who loves everyone, are the same six words in a different order. Draw the relation as a grid and they become two obviously different questions — one about rows, one about columns.

logic · quantifiers 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.

applied · stable matching

Named alongside it

The objects these essays reach for when they reach for this one.

CounterexampleBlocking pairDeferred acceptanceNegationOrder latticePredicate logicPreference profileQuantifierRelationStable matchingStrategy proofnessUniformity

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