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In mathematics, economics, and political science, the highest averages methods, also called divisor methods, are a class of apportionment algorithms for proportional representation. Divisor algorithms seek to fairly divide a legislature between agents (such as political parties or federal states). More generally, divisor methods are used to divide or round a whole number of objects being used to represent (non-whole) shares of a total.^[1]

Divisor methods aim to treat voters equally by ensuring every legislator represents an equal number of voters, as nearly as possible.^[2]: 30

Definitions

The two names for these methods reflect two different ways to ways of thinking about them, and their two independent inventions (first in the context of United States congressional apportionment, and later in proportional representation of parties in Europe). Nevertheless, both procedures are equivalent, and yield the same answers.

Signposts and rounding

Apportionment methods are a kind of rounding rule: in proportional representation (or apportionment), every party (or state) has an ideal number of legislators, which is not a whole number. To make these into whole numbers, we use a rounding rule given by a signpost sequence of real numbers, where each signpost marks the boundary between two natural numbers: numbers larger than the signpost are rounded up, while those less than or equal to it are rounded down. This sequence is post(k), where k ≤ post(k) ≤ k+1.

Divisor method

The divisor procedure apportions seats by searching for a divisor or ideal district size, which is roughly equal to the number of voters represented by each legislator. If every legislator represents an equal number of voters, then the number of seats for each state can be found by dividing the population by the divisor.

However, seat allocations must be whole numbers, so to find the apportionment for a given state we must round after dividing. Thus, each party's apportionment is given by:

${\displaystyle {\text{seats))=\operatorname {round} \left({\frac {\text{votes)){\text{divisor))}\right)}$

This divisor is equal to the number of votes required to earn one extra seat in the legislature.

However, if this divisor is chosen incorrectly, this procedure may assign too many or too few seats, and the apportionments for each state will not add up to the total legislature size. A feasible divisor can therefore be found by trial and error.

Highest averages

With the highest averages procedure, every party begins with 0 seats. Then, at each iteration, we allocate a seat to the party with the highest average number of votes, i.e. the party with the most votes per seat. This method proceeds until all seats are allocated.

However, a reasonable question is whether we should look at the vote average before we assign the seat, what the average will be after assigning the seat, or if we should compromise between them with some kind of continuity correction. All these approaches give a different apportionment. We can define a generalized average using a signpost sequence:

${\displaystyle {\text{average)):={\frac {\text{votes)){\operatorname {post} ({\text{seats)))))}$

Specific divisor methods

While all divisor methods share the same general procedure, they differ in the choice of signpost sequence and therefore rounding rule. These rounding rules give each method its unique properties.

Note that for methods where the first signpost is zero, every party with at least one vote will receive a seat before any party receives a second seat; in practice, this typically means that every party must receive at least one seat, unless disqualified by some electoral threshold.

Jefferson (D'Hondt) method

Jefferson's method was the first divisor method to be invented or used. For every state, the method finds what the average size of a congressional district would be if it were given an extra legislator. It then assigns the representative to the state that would be most underrepresented at the end of the round.

Jefferson's method uses the sequence ${\displaystyle \operatorname {post} (k)=k+1}$ (1, 2, 3, ...),^[3] which implies it always rounds each party's apportionment down.

Jefferson's method has the advantages of guaranteeing the lower quota rule and minimizes the worst-case overrepresentation. However, it generally gives large parties a share of seats exceeding their share of the vote,^[4] and thus encourages dishonest voting and two-party systems (though not as strongly as systems like plurality). Jefferson's method performs poorly when judged by most misapportionment metrics.^[5]

Adams' (Cambridge) method

Adams' method was conceived of by John Quincy Adams after noticing Jefferson's method allocated too few seats to smaller states.^[6] It can be described as the inverse of Jefferson's method; it awards a seat to the party that has the most votes per seat before the seat is added. The divisor function is post(k) = k, which is equivalent to always rounding up.^[5]

Adams' method can only violate the upper quota rule,^[7] and minimizes the worst-case underrepresentation. However, upper quota violations in the pure Adams method are very common.^[8] Like Jefferson, Adams' method performs poorly by common misapportionment metrics.^[5]

Adams' method was suggested as part of the Cambridge compromise for apportionment of European parliament seats to the member states, with the aim of satisfying degressive proportionality.^[9]

Webster (Sainte-Laguë) method

Webster's method uses the fencepost sequence post(k) = k+.5, i.e. 0.5, 1.5, 2.5, which corresponds to the standard rounding rule. Equivalently, the odd integers, i.e. 1, 3, 5, etc. can be used as divisors.

Webster's method is more proportional than D'Hondt's by many metrics of misrepresentation, and as such is typically recommended over D'Hondt by political scientists and mathematicians.^[10][11] It is also notable for being the least biased method in historical data of United States congressional apportionments (though the section on bias), and for being unbiased even when dealing with parties that win very small numbers of seats. Webster's method can occasionally break the quota rule; in rare situations, this can result in a party being given a .

Theoretically, Sainte-Laguë can encourage parties to split into many small lists, aiming to win one seat for each list, although such coordination becomes extremely difficult for districts that have more than a few seats. Such a strategy has been attempted as part of the elections in Hong Kong^{[citation needed]}. This problem can easily be fixed by introducing a small quota for parties to win a seat, most often equal to the exact Droop quota.

Hill's (Huntington–Hill) method

In the Huntington–Hill method, the signpost sequence is post(k) = √k (k+1), the geometric mean of the neighboring numbers. Conceptually, this method rounds to the integer that has the smallest relative (percent) difference. For example, 2.47 and 3 is about 19%, while the difference from 2 is about 21%, so 2.47 is rounded up. This method is used for allotting seats in the US House of Representatives among the states.

Hill's method tends to produce very similar results to Webster's method; when first adopted for use in congressional apportionment, the two methods differed only in whether they assigned a single seat to Michigan or Arkansas.^[2]: 58

Comparison of properties

Zero-seat apportionments

Huntington-Hill, Dean, and Adams' method all have a value of 0 for the first fencepost, giving an average of ∞. Thus, without a threshold, all parties that have received at least one vote also receive at least one seat. This property can be desirable, for example when apportioning seats to electoral districts or states. Otherwise, small parties may need to be eliminated using an electoral threshold (such as the Droop quota).

Bias

The concept of "bias" in seat shares turns out to be complex, as "bias" can be defined in several different ways. While Webster's method is often argued to be "unbiased" or said to have the "least bias" of any system,^[12] this relies on a technical definition of bias that is specific to the field of statistics: the bias is defined as the expected difference between a state's number of seats and its quota. In other words, a method is called unbiased if the average number of seats a state receives is equal to its average quota.

By this definition, Webster's method is the unique unbiased apportionment method,^[12] while Huntington-Hill exhibits a mild bias towards smaller states. However, other researchers have shown that alternative definitions of bias based on relative differences show the opposite result.^[13]

In practice, the difference between these definitions is small when handling parties or states with at least one seat. Thus, both Huntington-Hill and Webster's method can be considered unbiased or low-bias methods (unlike Jefferson or Adams' methods). A 1930 report to Congress by the National Academy of Sciences recommended the adoption of Hill's method, but other arguments have since been put forward by supporters of both sides, leading most mathematicians to consider the choice of divisor method a matter of opinion.^[13]

Comparison and examples

Example: Jefferson

The following example shows how Jefferson's method can differ substantially from less-biased methods such as Webster's. In this election, the largest party wins 46% of the vote, but takes 52.5% of the seats, enough to win a majority outright against a coalition of all other parties (which together reach 54% of the vote). Moreover, it does this in violation of quota: the largest party is entitled only to 9.7 seats, but it wins 11 regardless. The largest congressional district for Jefferson's method is roughly twice the size of the smallest district here. Webster's method shows none of these properties, with a maximum error of 22.6%.

Example: Adams

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.

Example: All systems

The following shows a worked-out example for all voting systems. Notice how Huntington-Hill and Adams' methods give every party one seat before assigning any more, unlike Webster's or Jefferson's.

Properties

Monotonicity

Divisor methods are generally preferred by mathematicians to largest remainder methods because they are less likely to exhibit paradoxical behavior. In particular, divisor methods satisfy the principle of population monotonicity, which states that if a party receives additional votes, while holding all other parties' vote totals constant, this should not cause them to lose seats. With quota methods, voting for a party can cause them to lose a seat by increasing the vote quota (the average number of votes per seat); this effect can result in their remainder shrinking and the state losing a seat.

Divisor methods also satisfy resource monotonicity (unlike quota rules), which says that increasing the number of seats in a legislature should not cause a state to lose a seat.

Min-Max inequality

Every divisor method can be defined using the min-max inequality. Letting brackets denote array indexing, an allocation is valid if-and-only-if:^{[1]: 78–81}

max votes[party]/ post(seats[party]) ≤ min votes[party]/ post(seats[party]+1)

In other words, it is impossible to lower the highest vote average by reassigning a seat from one party to another. Every number in this range is a possible divisor. If the inequality is strict, the solution is unique; otherwise (if the inequality is an equality), there is a "tie" between multiple apportionments (traditionally broken in favor of the largest party)^[1]: 83.

Method families

The divisor methods described above can be generalized into families.

Generalized average

In general, it is possible to construct an apportionment method from any generalized average function, by defining the signpost function as post(k) = avg(k, k+1).

Stationary family

A divisor method is called stationary^[14]: 68 if its signposts are of the form ${\displaystyle d(k)=k+r}$ for some real number ${\displaystyle r\in [0,1]}$. The methods of Adams, Webster, and Jefferson are stationary; those of Dean and Huntington-Hill are not. A stationary method corresponds to rounding numbers up if they exceed a weighted arithmetic mean of k, k+1.

The Danish method is used in Danish elections to allocate each party's compensatory seats (or levelling seats) at the electoral province level to individual multi-member constituencies. It divides the number of votes received by a party in a multi-member constituency by 0.33, 1.33, 2.33, 3.33 etc. The fencepost sequence is given by post(k) = k+1⁄3. This system purposely tries to allocate seats equally rather than exactly proportionally.^[15]

The Imperiali method is a stationary pseudo-apportionment algorithm with divisors 2, 3, 4, 5, etc., corresponding to a divisor function ${\displaystyle d(k)=k+2}$. It is designed to disfavor the smallest parties, akin to a cutoff. It is used in Belgian municipal elections. This highest-averages method is not a proportional representation method; even if a perfectly proportional allocation exists, it is not guaranteed to find it.

Power mean family

The power mean family of divisor methods includes the Adams, Huntington-Hill, Webster, and Jefferson methods (either directly or as limits).^[14] For a given constant p, the power mean method has signpost function post(k) = _^p√k^p + (k+1)^p. The Huntington-Hill method corresponds to the limit as p tends to 0, while Adams and Jefferson represent the limits as p tends to negative or positive infinity.

The family also includes the less-common Dean's method for p=-1, which corresponds to using the harmonic mean. Dean's method is equivalent to rounding to the nearest average—every state has its seat count rounded in a way that minimizes the difference between the average district size and the ideal district size. For example:^[16]: 29

The 1830 representative population of Massachusetts was 610,408: if it received 12 seats its average constituency size would be 50,867; if it received 13 it would be 46,954. So, if the divisor were 47,700 as Polk proposed, Massachusetts should receive 13 seats because 46,954 is closer to 47,700 than is 50,867.

Rounding to the vote average with the smallest relative error once again yields the Huntington-Hill method because |log(x⁄y)| = |log(y⁄x)|, i.e. relative differences are reversible. This fact was central to Edward V. Huntington's use of relative (instead of absolute) errors in measuring misrepresentation, and to his advocacy for the Huntington-Hill technique:^[17] Huntington argued the choice of apportionment method should not depend on how the equation for equal representation is rearranged, and only relative errors (i.e. the Huntington-Hill technique) satisfy this property.^[16]: 53

Stolarsky mean family

Similarly, the Stolarsky mean can be used to define a family of divisor methods that minimizes the generalized entropy index of misrepresentation.^[18] This family includes the logarithmic mean, geometric mean, and the identric mean. The Stolarsky means can be justified by minimizing these inequality measures, which are of major importance in the study of information theory.^[19]

Modifications

In addition to pure divisor methods, there are many modified divisor methods that are closely related.

Thresholds

Many countries have electoral thresholds for representation, where parties must win a specified fraction of the vote in order to be represented; parties with fewer votes than the threshold requires for representation are eliminated.

In addition, many countries modify the first divisor to introduce an "effective threshold." One common modification is to use Webster's method, but to set the first divisor to 0.7 or 1.0 (full-seat modification) to keep parties from winning their first seat too easily.

Quota-capped divisor method

A quota-capped divisor method is an apportionment method where we begin by assigning every state its lower quota of seats. Then, we sequentially add seats one-by-one to the state with the highest votes-per-seat average, so long as adding an additional state does not result in the state exceeding its upper quota.

Formally, at each iteration ${\displaystyle h}$ (corresponding to allocating the ${\displaystyle h}$-th seat), the following sets are computed (see mathematics of apportionment for the definitions and notation):

• ${\displaystyle U(\mathbf {t} ,\mathbf {a} )}$ is the set of parties that can get an additional seat without violating their upper quota, that is, ${\displaystyle a_{i}<\lceil t_{i}\cdot h\rceil }$.
• ${\displaystyle L(\mathbf {t} ,\mathbf {a} )}$ is the set of parties whose number of seats might be below their lower quota in some future iteration, that is, ${\displaystyle a_{i}<\lfloor t_{i}\cdot (h+z)\rfloor }$ for the smallest integer ${\displaystyle z}$ for which ${\displaystyle \sum _{i}\lfloor t_{i}\cdot (h+z)\rfloor \geq h-1+z}$. If there is no such ${\displaystyle z}$ then ${\displaystyle L(\mathbf {t} ,\mathbf {a} )}$ contains all states.

The ${\displaystyle h}$-th seat is given to a party ${\displaystyle i\in U(\mathbf {t} ,\mathbf {a} )\cap L(\mathbf {t} ,\mathbf {a} )}$ for which the ratio ${\displaystyle {\frac {t_{i)){d(a_{i})))}$ is largest.

The Balinsky-Young quota method^[20] is the quota-capped variant of the D'Hondt method (also called: Quota-Jefferson). Similarly, one can define the Quota-Webster, Quota-Adams, etc.^[21]

Every quota-capped divisor method satisfies house-monotonicity. Moreover, quota-capped divisor methods satisfy upper quota by definition, and can be proved to satisfy lower quota as well.^{[16]: Thm.7.1}

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 "too many votes".^{[16]: Tbl.A7.2} This can happen when, due to party i getting more votes, the upper quota of some other party j decreases. Therefore, party j is not eligible to a seat in the current iteration, and some third party receives the next seat. But then, at the next iteration, party j is again eligible to a seat, and it beats party i.

Moreover, quota-capped versions of other algorithms frequently violate "true quota" in the presence of error (e.g. census miscounts). Jefferson's method frequently violates the true quota, even after being quota-capped, while Webster's method and Huntington-Hill perform well even without quota-caps.^[22]