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Concept

Monotonicity

The property that a rule's output moves in one direction when its input moves in one direction — more of something in never gives less of something out. It is the property most often quietly assumed and most often violated by rules that look reasonable.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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 Inscribed polygons in a quarter circle, and the length they climb towards. Four polygons inscribed in a quarter circle with increasing numbers of corners, each drawn over the curve, with its length beneath it — the lengths increase towards the curve's own.

Which curves have a length at all

A length is defined as a supremum over inscribed polygons, which behaves because every refinement is longer than the last. It is also sometimes infinite — and the condition separating the two cases is a sum of absolute differences that either settles or does not.

analysis · Arc length Luxembourg's votes and Luxembourg's power, 1958 to 1995. Paired bars for Luxembourg at five enlargements of the Council: vote share falling and power share rising from nothing in 1958 to 0.95% in 1973 and 3.02% in 1981.

A vote worth nothing until it was outnumbered

From 1958 to 1973 Luxembourg held one vote of seventeen in the Council of the European Communities and could never once change an outcome. When the Council grew, Luxembourg's share of the votes fell and its share of the power rose from nothing. Power is not a quantity a member holds; it is a property of the whole assembly, and changing the assembly moves it in directions nobody would guess.

applied · Shapley value

Named alongside it

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

QuotaApportionmentCounterexampleDivisor methodGeometric meanRoundingApproximationArc lengthAxiomBiasCoalitionContinuity

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