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Concept

Divisor method

A rule for apportionment that divides every population by a common number and rounds the result. The divisor is adjusted until the rounded shares add to the right total, and the five classical methods differ only in where the rounding boundary sits between two whole numbers.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

Hamilton's method on 27 seats and 5 regions. A worksheet of populations, exact quotas, floors, remainders and the seats Hamilton's method awards to 5 regions.

The seat that vanishes when the house grows

Twenty-seven whole seats have to be divided between five regions whose exact shares are 15.417, 7.209, 1.755, 1.431 and 1.188. Every rule for rounding those five numbers breaks something, and the instance drawn here breaks all three of the classical ways at once.

applied · Apportionment Five rules, one dial. Seats for each of 5 regions at 21 settings of the rounding threshold, with the three settings that are the named methods marked; the largest region gains and the smallest loses as the threshold rises.

Five rules and one dial

Adams, Webster and Jefferson are usually taught as three rules for rounding a share. They are one rule with a number in it, and turning that number from nought to one moves seats from the smallest region to the largest, one at a time.

applied · Apportionment Which regions each rule favours. Average seats above or below exact quota for the largest and the smallest region, under each of the five methods, over 400 generated instances.

The rule with no favourites

Over four hundred instances, Jefferson's method gives the largest region a third of a seat more than its exact share and the smallest a third of a seat less. Adams reverses both. Webster's average is a hundredth of a seat, and that is not luck.

applied · Apportionment Two out of three, and never all three. A table of the five apportionment methods against three properties, each cell decided by a search over generated instances; no method has all three.

Two out of three, and never all three

Stay inside every region's quota, never take a seat away when the house grows, never take one from a region that grew faster. Each pair is achievable. All three together are not, and the proof is that no rule anywhere manages it.

applied · Apportionment What each rule is answering. A table of five apportionments against three measures of inequality between two regions, with a tick where no transfer of a seat reduces the measure; each measure certifies exactly one of the five.

Choosing what unfair means

Ask whether moving one seat between two regions would make them more equal, and the answer depends on what "equal" is measured in. Three measures, three different answers, and each of the classical methods is the one no transfer can improve for exactly one of them.

applied · Apportionment Seats to districts and to parties at once. A 4 by 3 table of seats, with every row total and every column total prescribed. The entries come from scaling the votes by one factor per row and one per column and rounding, and all the totals come out exactly right.

Seats to parties and places at once

Seats can be given to regions in proportion to one list of populations, and no rule does it perfectly. Ask for seats to regions and to parties simultaneously and the object stops being a list — and the impossibility that closed the subject does not apply.

applied · Apportionment Biproportional seats against their fair shares. A table of votes for 3 districts and 4 parties beside the seats the biproportional method gives, each with the fair share from the continuous fit, and the cell whose seats fall outside its quota marked.

The table inside every quota

Give seats to districts and parties at once, and every cell of the table has a fair share it ought to round from. A table rounding every cell to its floor or its ceiling, with every total exact, always exists. The biproportional method does not always choose one: here it gives a party 2 seats where its fair share is 3.088.

applied · Apportionment

Named alongside it

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

ApportionmentQuotaRoundingCounting argumentGeometric meanImpossibilityMonotonicityCounterexampleFairnessMatrixProportionalityAlabama paradox

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