Matpic
mathematics, in pictures
Concept

Apportionment

The problem of turning shares that are not whole numbers into whole numbers adding to the same total. It arises whenever seats are divided between regions in proportion to their populations, and every rule for doing it breaks one of three obviously desirable properties — which one is a decision rather than a calculation.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

Also named here as divisor method — the same set of essays touches all of them, so they are one junction rather than several.

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

Named alongside it

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

Divisor methodQuotaCounting argumentGeometric meanMonotonicityRoundingAlabama paradoxAxiomBiasCounterexampleExpectationImpossibility

All concepts