Highest averages method

Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results The highest averages, divisor, or divide-and-round methods[1] are a family of apportionment rules, i.e. algorithms for fair division of seats in a legislature between several groups (like political parties or states).

[1][2] More generally, divisor methods are used to round shares of a total to a fraction with a fixed denominator (e.g. percentage points, which must add up to 100).

In doing so, the method approximately maintains proportional representation, meaning that a party with e.g. twice as many votes will win about twice as many seats.

[5] Divisor methods were first invented by Thomas Jefferson to comply with a constitutional requirement, that states have at most one representative per 30,000 people.

[3]: 20 Apportionment would become a major topic of debate in Congress, especially after the discovery of pathologies in many superficially-reasonable rounding rules.

[1] However, seat allocations must be whole numbers, so to find the apportionment for a given state we must round (using the signpost sequence) after dividing.

[1] While all divisor methods share the same general procedure, they differ in the choice of signpost sequence and therefore rounding rule.

[2] Thomas Jefferson was the first to propose a divisor method, in 1792;[1] it was later independently developed by Belgian political scientist Victor d'Hondt in 1878.

[1] Jefferson's apportionment never falls below the lower end of the ideal frame, and it minimizes the worst-case overrepresentation in the legislature.

[12] Adams' apportionment never exceeds the upper end of the ideal frame, and minimizes the worst-case underrepresentation.

[14] Adams' method was suggested as part of the Cambridge compromise for apportionment of European parliament seats to member states, with the aim of satisfying degressive proportionality.

[17] As such, it is typically preferred to D'Hondt by political scientists and mathematicians, at least in situations where manipulation is difficult or unlikely (as in large parliaments).

[18] In small districts with no threshold, parties can manipulate Webster by splitting into many lists, each of which wins a full seat with less than a Hare quota's worth of votes.

This is often addressed by modifying the first divisor to be slightly larger (often a value of 0.7 or 1), which creates an implicit threshold.

[18] However, other researchers have noted that slightly different definitions of bias, generally based on percent errors, find the opposite result (The Huntington-Hill method is unbiased, while the Webster method is slightly biased towards large parties).

[19][22] A 1929 report to Congress by the National Academy of Sciences recommended the Huntington-Hill method,[23] while the Supreme Court has ruled the choice to be a matter of opinion.

The following example shows a case where Adams' method fails to give a majority to a party winning 55% of the vote, again in violation of their quota entitlement.

Notice how Huntington-Hill and Adams' methods give every party one seat before assigning any more, unlike Webster or Jefferson.

[5] In particular, divisor methods satisfy population monotonicity, i.e. voting for a party can never cause it to lose seats.

[5] Such population paradoxes occur by increasing the electoral quota, which can cause different states' remainders to respond erratically.

If the inequality is strict, the solution is unique; otherwise, there is an exactly tied vote in the final apportionment stage.

A stationary method corresponds to rounding numbers up if they exceed the weighted arithmetic mean of k and k+1.

The fencepost sequence is given by post(k) = k+1⁄3; this aims to allocate seats closer to equally, rather than exactly proportionally.

This fact was central to Edward V. Huntington's use of relative (instead of absolute) errors in measuring misrepresentation, and to his advocacy for Hill's rule:[27] Huntington argued the choice of apportionment method should not depend on how the equation for equal representation is rearranged, and only the relative error (minimized by Hill's rule) satisfies this property.

[3]: 53 Similarly, the Stolarsky mean can be used to define a family of divisor methods that minimizes the generalized entropy index of misrepresentation.

The Stolarsky means can be justified as minimizing these misrepresentation metrics, which are of major importance in the study of information theory.

[20] A majority-preservation clause guarantees any party winning a majority of the vote will receive at least half the seats in a legislature.

[20] This is typically accomplished by adding seats to the legislature until an apportionment that preserves the majority for a parliament is found.

[30] However, quota-capped divisor methods violate the participation criterion (also called population monotonicity)—it is possible for a party to lose a seat as a result of winning more votes.