Matpic
mathematics, in pictures
Generator

A profile of 100 ranked ballots, and the majority in every pair

A generator in the applied library, called 57 times across 10 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

ballot is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

A profile of 100 ranked ballots, and the majority in every pair. The voter groups as columns with the ranking down each, beside the pairwise majority matrix whose cells are the margins.

show: "manipulate"

Every ballot one voter could submit under instant runoff. One voter's true ranking beside every ranking that voter could submit instead, with the winner each produces and the profitable misreports marked.

show: "iia"

Independence of irrelevant alternatives, broken by Borda. Two profiles that agree on every voter's ranking of two candidates and differ only in where the others sit, with the rule's verdict between the two reversed.

show: "rules"

Five rules on one profile of 27 ballots, and 5 different winners. The ballot groups as columns beside a table of five voting rules with the winner each returns and the count that decided it.

show: "cycle"

A majority cycle over 3 candidates, and how often 3 voters produce one. The majority tournament as a directed polygon with each arc's margin, beside one cell for every profile of the stated size, filled where no Condorcet winner exists.

show: "doctrinal"

Where the two procedures part company. A table of 3 judges' verdicts on two premises and the conclusion each is committed to, with the two majorities at the foot disagreeing about the conclusion.

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Applied

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

A share of the votes is not a share of the power

Give three members four, four and one vote, with five needed to pass. Every winning coalition needs exactly two of them, so all three have equal power — and one of them holds a ninth of the votes.

 Applied

Deciding the premises or the conclusion

A body that cannot be both decisive and coherent has to choose which. The two live options are to vote on the reasons and let the verdict follow, or to vote on the verdict and let the reasons look after themselves — and they reach opposite answers on exactly the profiles the impossibility identifies.

 Applied

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

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

Four ways out, and what each costs

An impossibility theorem lists conditions and says no rule has them all. That leaves exactly as many escapes as there are conditions, each of them a real institution — a dictator, a two-stage procedure, a supermajority, a restricted agenda — and each escape's price can be counted rather than argued about.

 Applied

No rule escapes the doctrinal paradox

A court whose members each hold a consistent position can reach an inconsistent verdict by majority. The anchor's first rung exhibits one such case, which invites the hope that a better rule would avoid it — and every rule that responds to the votes at all fails somewhere.

 Applied

None of the four conditions is spare

Four conditions pick out one sharing rule. The half that is usually shown is that they are enough; the other half is that each is needed — drop any one and a different rule satisfies the rest, so the list cannot be shortened.

 Applied

The court that contradicts itself

Three judges each answer three questions, and each answers them consistently. Take the majority on each question separately and the answers no longer hang together — the body as a whole asserts a combination no member of it holds, and no rearrangement of the procedure removes the problem.

 Applied

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.

The whole library · What the figures prove