Matpic
mathematics, in pictures
Concept

Independence of irrelevant alternatives

The condition that a verdict between two candidates depends only on how the voters ranked those two.

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

first profile4 voters, one column eachsecond profilethe same voters on A and Bv1v2v3v41st2nd3rdAABBBCAACBCCv1v2v3v41st2nd3rdAABBBBACCCCAA 6 · B 5 pointsA 5 · B 6 pointsBorda: A ≻ BBorda: B ≻ AflipsABthe candidates that movedevery voter ranks A against B the same way in both profiles; only the third candidate moves — and Bordareverses its verdictno pair of profiles flips Borda at 3 voters; at 4 voters 3456 of the 104976 ordered pairs inside a class do

Four conditions, and no rule that has all of them

The rung below shows five reasonable rules returning five different winners, which invites the obvious question of which one is right. The answer is that the conditions anybody would write down cannot all hold at once — and here each named rule's own violation is found by search rather than quoted.

applied · voting rules the electoratethe first column is the manipulating voterevery ballot that voter could submitand the winner it producestrue1121st2nd3rdAABCBBCBCCAAelectsfor the voterA ≻ B ≻ CA ≻ C ≻ BB ≻ A ≻ CB ≻ C ≻ AC ≻ A ≻ BC ≻ B ≻ AChonestCno gainBbetterBbetterCno gainCno gainthe honest ballota misreport that paysthe control: the same voters, A and B only0 of 2 ballots payelectsfor the voterA ≻ BB ≻ ABhonestBno gainthe voter's true ranking is A ≻ B ≻ C; the honest ballot elects C under instant runoff2 of the 6 ballots the voter could submit elect somebody the voter ranks higher: B ≻ A ≻ C; B ≻ C ≻ Athe control runs the identical search with only A and B left: 0 of the 2 ballots pay, which is what astrategy-proof contest looks like

A lie that pays

Three rungs of this ladder have read a ballot as a report of a preference. This one reads it as a move, and walks every move one voter has — all six rankings, the winner each produces, and the ones that beat honesty.

applied · voting rules

Named alongside it

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

Condorcet cyclePairwise majorityPreference profileVoting ruleArrows theoremCounterexampleDecisive setProof by contradictionStrategic votingStrategy proofness

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