Highest averages method

In mathematics, economics, and social choice theory, the highest averages or divisor methods, sometimes called divide-and-round, are a family of apportionment algorithms that aim to fairly divide a legislature between several groups, such as political parties or states.   More generally, divisor methods can be used to round shares of a total, e.g. percentage points (which must add up to 100). The two names for these methods—highest averages and divisors—reflect two different ways of thinking about them, and their two independent inventions. However, both procedures are equivalent and give the same answer.

Divisor methods aim to treat voters equally by ensuring every legislator represents an equal number of voters, as nearly as practicable, by iteratively assigning additional representatives to whichever party is most underrepresented. In doing so, the method maintains proportional representation, i.e. a party with twice as many votes should win twice as many seats.

History
The highest averages method was first described by the American Founding Father Thomas Jefferson, as a way to guarantee compliance with a constitutional requirement for states to have at most one representative per 30,000 people. He proposed an apportionment by dividing each state's population by 33,000 people/representative, then rounding down. Apportionment would become a major topic of debate, especially after the discovery of pathologies in many superficially-reasonable rounding rules. These changes often had substantial results, with the use of an improper rounding procedure ultimately determining the 1876 presidential election. Similar debates would appear in Europe after the adoption of proportional representation.

Definitions
Divisor methods are based on rounding rules, defined using a signpost sequence $post(k)$, where $k ≤ post(k) ≤ k+1$. Each signpost marks the boundary between natural numbers, with numbers being rounded down if and only if they are less than the signpost. 

Divisor method
The divisor procedure apportions seats by searching for a divisor or electoral quota. This divisor can be thought of as the number of votes a party needs to earn one additional seat in the legislature, the ideal population of a congressional district, or the number of voters represented by each legislator.

If each legislator represented an equal number of voters, the number of seats for each state could 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 (using the signpost sequence) after dividing. Thus, each party's apportionment is given by:

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

Usually, the divisor is initially set to equal the Hare quota. However, this procedure may assign too many or too few seats. In this case the apportionments for each state will not add up to the total legislature size. A feasible divisor can be found by trial and error.

Highest averages
With the highest averages algorithm, 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.

It is reasonable to ask whether we should look at the vote average before assigning the seat, what the average will be after assigning the seat, or if we should compromise with a continuity correction. These approaches each give different apportionments. In general, we can define the averages using the signpost sequence:

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

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.

Specific 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.

Jefferson (D'Hondt) method
Thomas Jefferson proposed the first divisor method in 1792. It assigns the representative to the state that would be most underrepresented at the end of the round. It remains the most-common method for proportional representation to this day.

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

apportionment never falls below the lower end of the ideal frame, and it minimizes the worst-case overrepresentation in the legislature. However, Jefferson's method performs poorly when judged by most metrics of proportionality. The rule typically gives large parties an excessive number of seats, with the apportionment frequently exceeding the ideal share rounded up.

This pathology led to widespread mockery of Jefferson's method when it was realized it would "round" New York's apportionment of 40.5 up to 42, with Senator Mahlon Dickerson saying the extra seat must come from the "ghosts of departed representatives".

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 new seat is added. The divisor function is $k$, which is equivalent to always rounding up.

Adams' apportionment never exceeds the upper end of the ideal frame, and minimizes the worst-case underrepresentation. However, violations of the lower seat quota are 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's (Sainte-Laguë) method
Daniel Webster's method uses the fencepost sequence $0.00 1.00 2.00 3.00$ (i.e. 0.5, 1.5, 2.5); this corresponds to the standard rounding rule. Equivalently, the odd integers (1, 3, 5…) can be used to calculate the averages instead.

Webster's method produces more proportional apportionments than D'Hondt's by almost every metric of misrepresentation. 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). It is also notable for minimizing seat bias even when dealing with parties that win very small numbers of seats. Webster's method can theoretically violate the ideal share rule, although this is extremely rare for even moderately-large parliaments; it has never been observed to violate quota in any United States congressional apportionment.

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.

Hill's (Huntington–Hill) method
In the Huntington–Hill method, the signpost sequence is $2÷(1/k + 1/k+1)$, 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 between 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.

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 metrics of seat bias. While Webster's method is sometimes described as "uniquely" unbiased, this uniqueness property relies on a technical definition of bias as the expected difference between a state's number of seats and its ideal share. In other words, a method is called unbiased if the number of seats a state receives is, on average across many elections, equal to its ideal share.

By this definition, Webster's method is the least-biased apportionment method, while Huntington-Hill exhibits a mild bias towards smaller states. However, other researchers have noted that slightly different definitions of bias, generally based on percent errors, find the opposite result (Hill's method is unbiased, while Webster's method is slightly biased towards large states).

In practice, the difference between these definitions is small when handling parties or states with more than one seat. Thus, both Huntington-Hill and Webster's method can be considered unbiased or low-bias methods (unlike Jefferson or Adams' methods). A 1929 report to Congress by the National Academy of Sciences recommended Hill's method, while the Supreme Court has ruled the choice of bias metric to be a matter of opinion.

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 is nearly twice the size of the smallest district. 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.

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: "$0.00 1.33 2.40 3.43$"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 $√k(k + 1)$.

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 the weighted arithmetic mean of $0.00 1.41 2.45 3.46$ and $r = 1/3$. Smaller values of $k + r$ 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 $0.33 1.33 2.33 3.33$; 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 $k + 1/2$, the power mean method has signpost function $0.50 1.50 2.50 3.50$. The Huntington-Hill method corresponds to the limit as $p = 2$ 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 $0.71 1.58 2.55 3.54$, 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 $k + 1$, 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.

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 (the latter being called the full-seat modification).

Majority-preservation clause
A majority-preservation clause guarantees any party winning a majority of the vote will receive at least half the seats in a legislature. Without such a clause, it is possible for a party with slightly more than half the vote to receive just barely less than half the seats (if using a method other than D'Hondt). This is typically accomplished by adding seats to the legislature until an apportionment that preserves the majority for a parliament is found.

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. 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.