Concept

Condorcet cycle

A set of candidates each of which loses to another, so following the majorities round arrives back where it started.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Also named here as pairwise majority, voting rule — the same set of essays touches all of them, so they are one junction rather than several.

the majority tournament100 voters, every pair decidedevery profile of the space216 of them, one cell each+34+36+30ABCno Condorcet winner (12)a winner exists (204)the 3 arcs of the ring are the majority in each pair, and following them returns to A: A → B → C → A12 of the 216 profiles of 3 voters over 3 candidates have no Condorcet winner — 5.6% of the space, every oneof them built and tested

The majority that goes in a circle

Every voter hands in a ranking, and a ranking is transitive by construction. Compare the candidates two at a time and let the majority decide each pair, and the verdicts need not fit together into a ranking at all.

applied · voting rules
the profileone column per group, size abovethe rulesand what each returns7655221st2nd3rd4th5thDACEACEBDBBEBEBCDACDADCDACEAEBwinnerpluralityBordainstant runoffCondorcetCoombsABCDEthe deciding count8 first places63 points19 of 27 at the end4 of 4 pairs14 of 27 at the endthe candidate that rule returnsthe count it was decided on27 voters in 6 groups over 5 candidates; a majority is more than 13.5the five rules return 5 different winners: plurality A, Borda B, instant runoff C, Condorcet D, Coombs Einstant runoff eliminates B, E, D; Coombs eliminates A, C, D

Five rules and five winners

Twenty-seven ranked ballots, five entirely reasonable ways of counting them, and five different candidates declared the winner. Every count is correct, every rule is defensible, and the answer turns out to be a property of the rule rather than of the ballots.

applied · voting rules
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.

Pairwise majorityPreference profileVoting ruleCounterexampleIndependence of irrelevant alternativesArrows theoremBorda countCounting argumentDecisive setGraphInstant runoffProof by contradiction

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