Divisor methods

Every divisor method begins with the same move: pick a number, divide each state's (or party's) population by it, and round the result to get a whole number of seats. The divisor is adjusted up or down until the rounded totals add up to exactly the chamber size. What looks like a neutral calculation hides a structural choice inside the rounding step, because there are several defensible ways to convert a decimal into a whole number, and they do not treat large and small equally.

The simplest version is Jefferson's method — used in the United States from 1792 to 1840, and still the basis of the d'Hondt system used in proportional elections across much of Europe. Jefferson always rounds down, discarding the fractional remainder. Because large populations produce large raw quotients, rounding down costs them proportionally less than it costs small populations, so the method systematically favours larger states or parties. You can verify this by trying any example: a quotient of 4.9 becomes 4, losing 18 percent of its entitlement; a quotient of 1.9 also becomes 1, losing 47 percent. The larger the number before the decimal, the smaller the proportional loss from truncation.

Webster's method corrects for this by rounding at the conventional point: anything below 0.5 rounds down, anything 0.5 and above rounds up. The cut-off is the arithmetic mean of the two integers on either side of the decimal. This produces results closer to strict proportionality than Jefferson's, with no systematic tilt toward large or small. Congress used Webster's method for the 1842 and 1901 apportionments, and it is the basis of the Sainte-Laguë divisor sequence used in Scandinavian and several other proportional systems.

Huntington-Hill, the method the United States has used since 1941, places the rounding boundary at the geometric mean of the two adjacent integers rather than the arithmetic mean. The geometric mean of n and n+1 is always slightly below their midpoint, which means the boundary shifts downward compared with Webster. A state rounds up at a slightly lower decimal than it would under Webster — and because this effect is largest at small whole numbers, the method tends to favour smaller states. The mathematical justification offered at the time was that the geometric mean minimises the relative difference in representation between any two states, rather than the absolute difference. Both are defensible objectives; they simply reach different seats.

Adams's method, the mirror image of Jefferson's, always rounds up. Every fractional entitlement, however small, earns a seat, strongly favouring the smallest states — so much so that it is rarely used in practice, because it tends to produce absurd allocations when populations vary widely.

The relationship between these four methods is cleaner than it first appears: they differ only in where they place the rounding threshold between n and n+1. Adams uses 0, meaning always round up; Jefferson uses 1, meaning always round down; Webster uses the arithmetic mean; Huntington-Hill uses the geometric mean. Slide the threshold up or down and you move seats between large and small, in a direction that is entirely predictable before the count begins.

Because the divisor is adjusted to make totals fit the chamber size, the final divisor has no independent meaning — it is a computational tool, not a policy variable. The real work is done entirely by the rounding rule. Change the rule, keep every population figure identical, and some states gain seats while others lose them, with no change to any underlying reality. Highest averages methods in party-list elections follow exactly the same logic: each divisor sequence corresponds to one of these rounding rules, and the choice of sequence is also a choice about which party size the formula rewards.