Highest averages method

In mathematics, economics, and social choice theory, the highest averages method, also called the divisor method, is an apportionment algorithm most well-known for its common use in proportional representation. Divisor algorithms seek to fairly divide a legislature between several groups, such as political parties or states. More generally, divisor methods are used for rounding a set of real numbers to a whole number of objects.

Divisor methods aim to treat voters equally by ensuring every legislator represents an equal number of voters, as nearly as possible.

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, the procedures are equivalent and give the same answer.

Signposts and rounding

Main article: Signpost sequence

Divisor methods are based on rounding rules defined using a signpost sequence of real numbers, where each signpost marks the boundary between two natural numbers. Values above the signpost are rounded up, while those below are rounded down. We call this sequence 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 approximately equal to the number of voters represented by each legislator. If each 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:

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

This divisor can be thought of as the number of votes a party needs to earn one additional seat in the legislature (sometimes called an electoral quota).

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 must 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 vote average, 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. These approaches give different apportionments. We can define a generalized average using a signpost sequence:

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

Specific divisor methods

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

Divisor formulas

Method	Signposts	Rounding of Seats	Approx. first values
Adams	k	Up	0.00 1.00 2.00 3.00
Dean	2÷(1⁄k + 1⁄k+1)	Harmonic	0.00 1.33 2.40 3.43
Huntington–Hill	√k(k + 1)	Geometric	0.00 1.41 2.45 3.46
Power mean	p√(kp + (k+1)p)/2	Power mean	-
Stationary	k + r	Weighted	0+r  1+r  2+r  3+r
Webster/Sainte-Laguë	k + 1⁄2	Arithmetic	0.50 1.50 2.50 3.50
Jefferson/D'hondt	k + 1	Down	1.00 2.00 3.00 4.00

Jefferson (D'Hondt) method

Main article: D'Hondt method

Thomas Jefferson's method was the first divisor method to be proposed. It assigns the representative to the state that would be most underrepresented at the end of the round.

Jefferson's method uses the sequence $\operatorname {post} (k)=k+1$, i.e. (1, 2, 3, ...), which implies it always rounds a party's apportionment down.

Jefferson's method has the advantages of guaranteeing the lower quota rule and minimizes the size of the most overrepresented party in the legislature. However, it generally gives large parties a share of seats substantially exceeding their share of the vote. Jefferson's method performs poorly when judged by most metrics of proportionality.

Adams' (Cambridge) method

Adams' method was conceived of by John Quincy Adams after noticing Jefferson's method allocated too few seats to smaller states. 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.

Adams' method can only violate the upper quota rule, and minimizes the worst-case underrepresentation. However, upper quota violations in the pure Adams method are very common. Like Jefferson, Adams' method performs poorly according to most metrics of proportionality.

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.

Webster (Sainte-Laguë) method

Main article: 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 produces more proportional apportionments than D'Hondt's by almost every metric of misrepresentation. As such, it is typically recommended over D'Hondt by political scientists and mathematicians. It is also notable for minimizing seat bias even when dealing with parties that win very small numbers of seats. Webster's method can (very rarely) break the quota rule, although it has never done so in any congressional apportionment.

In very small districts, parties can manipulate Webster by splitting into many lists, each winning one seat with less than a quota. This is often addressed by modifying the first divisor to be slightly larger, which creates an implicit threshold.

Hill's (Huntington–Hill) method

Main article: 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, the difference betwen 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 used for congressional apportionment, the two methods differed only in whether they assigned a single seat to Michigan or Arkansas.

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 will also receive at least one seat. This property can be desirable (as when apportioning seats to states) or undesirable, in which case the first divisor may be adjusted to create a natural threshold.

Bias

There are many different metrics that can be used to measure seat bias. While Webster's method is sometimes described as unbiased, this relies on a technical definition of bias 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, while Huntington-Hill exhibits a mild bias towards smaller states. However, other researchers have noted that slightly different definitions of bias find the opposite result (a small bias of Webster towards large states).

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 suggested Hill's method, while courts have ruled that the choice between the two constitutes a political question.

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%.

Jefferson's method								Webster's method
Party	Yellow	White	Red	Green	Purple	Total		Party	Yellow	White	Red	Green	Purple	Total
Votes	46,000	25,100	12,210	8,350	8,340	100,000		Votes	46,000	25,100	12,210	8,350	8,340	100,000
Seats	11	6	2	1	1	21		Seats	9	5	3	2	2	21
Quota	9.66	5.27	2.56	1.75	1.75	21		Quota	9.66	5.27	2.56	1.75	1.75	21
Votes/Seat	4182	4183	6105	8350	8340	4762		Votes/Seat	5111	5020	4070	4175	4170	4762
% Error	13.0%	13.0%	-24.8%	-56.2%	-56.0%	(100.%)		(% Range)	-7.1%	-5.3%	15.7%	13.2%	13.3%	(22.6%)
Seats	Averages					Signposts		Seats	Averages					Signposts
1	46,000	25,100	12,210	8,350	8,340	1.00		1	92,001	50,201	24,420	16,700	16,680	0.50
2	23,000	12,550	6,105	4,175	4,170	2.00		2	30,667	16,734	8,140	5,567	5,560	1.50
3	15,333	8,367	4,070	2,783	2,780	3.00		3	18,400	10,040	4,884	3,340	3,336	2.50
4	11,500	6,275	3,053	2,088	2,085	4.00		4	13,143	7,172	3,489	2,386	2,383	3.50
5	9,200	5,020	2,442	1,670	1,668	5.00		5	10,222	5,578	2,713	1,856	1,853	4.50
6	7,667	4,183	2,035	1,392	1,390	6.00		6	8,364	4,564	2,220	1,518	1,516	5.50
7	6,571	3,586	1,744	1,193	1,191	7.00		7	7,077	3,862	1,878	1,285	1,283	6.50
8	5,750	3,138	1,526	1,044	1,043	8.00		8	6,133	3,347	1,628	1,113	1,112	7.50
9	5,111	2,789	1,357	928	927	9.00		9	5,412	2,953	1,436	982	981	8.50
10	4,600	2,510	1,221	835	834	10.00		10	4,842	2,642	1,285	879	878	9.50
11	4,182	2,282	1,110	759	758	11.00		11	4,381	2,391	1,163	795	794	10.50

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.

Adams' Method								Webster's Method
Party	Yellow	White	Red	Green	Purple	Total		Party	Yellow	White	Red	Green	Purple	Total
Votes	55,000	17,290	16,600	5,560	5,550	100,000		Votes	55,000	17,290	16,600	5,560	5,550	100,000
Seats	10	4	3	2	2	21		Seats	11	4	4	1	1	21
Quota	11.55	3.63	3.49	1.17	1.17	21.		Quota	11.55	3.63	3.49	1.17	1.17	21.
Votes/Seat	5500	4323	5533	2780	2775	4762		Votes/Seat	4583	4323	5533	5560	5550	4762
% Error	-14.4%	9.7%	-15.0%	53.8%	54.0%	(99.4%)		(% Range)	3.8%	9.7%	-15.0%	-15.5%	-15.3%	(28.6%)
Seats	Averages					Signposts		Seats	Averages					Signposts
1	545,060	171,347	164,509	55,101	55,002	0.10		1	110,001	34,580	33,200	11,120	11,100	0.50
2	55,001	17,290	16,600	5,560	5,550	1.00		2	36,667	11,527	11,067	3,707	3,700	1.50
3	27,500	8,645	8,300	2,780	2,775	2.00		3	22,000	6,916	6,640	2,224	2,220	2.50
4	18,334	5,763	5,533	1,853	1,850	3.00		4	15,714	4,940	4,743	1,589	1,586	3.50
5	13,750	4,323	4,150	1,390	1,388	4.00		5	12,222	3,842	3,689	1,236	1,233	4.50
6	11,000	3,458	3,320	1,112	1,110	5.00		6	10,000	3,144	3,018	1,011	1,009	5.50
7	9,167	2,882	2,767	927	925	6.00		7	8,462	2,660	2,554	855	854	6.50
8	7,857	2,470	2,371	794	793	7.00		8	7,333	2,305	2,213	741	740	7.50
9	6,875	2,161	2,075	695	694	8.00		9	6,471	2,034	1,953	654	653	8.50
10	6,111	1,921	1,844	618	617	9.00		10	5,790	1,820	1,747	585	584	9.50
11	5,500	1,729	1,660	556	555	10.00		11	5,238	1,647	1,581	530	529	10.50
Seats	10	4	3	2	2			Seats	11	4	4	1	1

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.

	Jefferson method							Webster method							Huntington–Hill method							Adams method
party	Yellow	White	Red	Green	Blue	Pink		Yellow	White	Red	Green	Blue	Pink		Yellow	White	Red	Green	Blue	Pink		Yellow	White	Red	Green	Blue	Pink
votes	47,000	16,000	15,900	12,000	6,000	3,100		47,000	16,000	15,900	12,000	6,000	3,100		47,000	16,000	15,900	12,000	6,000	3,100		47,000	16,000	15,900	12,000	6,000	3,100
seats	5	2	2	1	0	0		4	2	2	1	1	0		4	2	1	1	1	1		3	2	2	1	1	1
votes/seat	9,400	8,000	7,950	12,000	∞	∞		11,750	8,000	7,950	12,000	6,000	∞		11,750	8,000	15,900	12,000	6,000	3,100		15,667	8,000	7,950	12,000	6,000	3,100
seat	seat allocation							seat allocation							seat allocation							seat allocation
1	47,000							47,000							∞							∞
2	23,500								16,000							∞							∞
3		16,000								15,900							∞							∞
4			15,900					15,667										∞							∞
5	15,667										12,000								∞							∞
6				12,000				9,400												∞							∞
7	11,750							6,714							33,234							47,000
8	9,400											6,000			19,187							23,500
9		8,000							5,333						13,567								16,000
10			7,950							5,300						11,314								15,900

Properties

Monotonicity

Divisor methods are generally preferred by mathematicians to largest remainder methods because they are less susceptible to apportionment paradoxes. In particular, divisor methods satisfy population monotonicity, i.e. voting for a party can never cause it to lose seats. Such population paradoxes occur by increasing the electoral quota, which can cause different states' remainders to respond erratically. Divisor methods also satisfy resource or house monotonicity, 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:

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, there is an exactly tied vote in the final apportionment stage.

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 if its signposts are of the form $d(k)=k+r$ for some real number $r\in [0,1]$. 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 and k+1. Smaller values of r tend to be friendlier to smaller parties.

Danish elections allocate leveling seats at the province level using-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 aims to allocate seats equally rather than exactly proportionally.

Power mean family

The power mean family of divisor methods includes the Adams, Huntington-Hill, Webster, Dean, and Jefferson methods (either directly or as limits). 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 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:

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: 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.

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. This family includes the logarithmic mean, geometric mean, and the identric mean. The Stolarsky means can be justified as minimizing these misrepresentation metrics, which are of major importance in the study of information theory.

Modifications

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. Other countries modify the first divisor to introduce a natural threshold; when using Webster's method, the first divisor is often set to 0.7 or 1.0 (called the full-seat modification).

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 add seats one-by-one to the state with the highest votes-per-seat average, so long as adding an additional seat does not result in the state exceeding its upper quota.

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

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

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

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

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.

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". 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.