The highest averages, divisor, or divide-and-round methods 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). 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).

The methods aim to treat voters equally by ensuring legislators represent an equal number of voters by ensuring every party has the same seats-to-votes ratio (or divisor). Such methods divide the number of votes by the number of votes per seat to get the final apportionment. By doing so, the method maintains proportional representation, as a party with e.g. twice as many votes will win about twice as many seats.

The divisor methods are generally preferred by social choice theorists and mathematicians to the largest remainder methods, as they produce more-proportional results by most metrics and are less susceptible to apportionment paradoxes. In particular, divisor methods avoid the population paradox and spoiler effects, unlike the largest remainder methods.

History

Divisor methods were first invented by Thomas Jefferson to comply with the United States Constitution's requirement that states have at most one representative per 30,000 people. His solution was to divide each state's population by 30,000 before rounding down.

Apportionment would become a major topic of debate in the US Congress, especially after the discovery of pathologies in many superficially-reasonable rounding rules. Similar debates would appear in Europe and Latin America after the adoption of proportional representation, typically as a result of large parties attempting to introduce thresholds and other barriers to entry for small parties. Such apportionments often have substantial consequences, as in the redistribution of seats in the US Congress following the 1870 United States census, which employed an ad-hoc calculation favoring Republican-leaning states. Had each state's electoral vote total been exactly equal to its entitlement, or had this redistribution used the Webster/Sainte-Laguë method or a largest remainders method, the 1876 United States presidential election would have been won by Samuel J. Tilden rather than its eventual winner Rutherford B. Hayes.

Definitions 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 are based on rounding rules, defined using a signpost sequence , where . Each signpost marks the boundary between natural numbers, with numbers being rounded down if and only if they are less than the signpost.

Divisor procedure 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, or the number of voters represented by each individual legislator (in systems using single-member constituencies, this is coterminous with the ideal population of the constituency).

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

Usually, it 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 party will not add up to the total legislature size. A feasible divisor can be found by trial and error. Given that the divisor is some approximation of the ratio of total votes to total seats, this operation can be viewed as varying that total number of seats and letting the rounding compensate it back to the actual legislature size.

Highest averages procedure 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.

However, it is unclear whether it is better to 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 slightly different apportionments. Many electoral laws use the average before assigning the seat. In general, we can define the averages using the signpost sequence:

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. +Divisor formulasMethodSignposts (divisor procedure)Signposts (highest averages procedure)Roundingof SeatsApprox. first values (divisor procedure)Approx. first values (highest averages procedure)AdamsUpDeanHarmonicHuntington–HillGeometricStationary(e.g. )WeightedWebster/Sainte-LaguëArithmeticPower mean(e.g. )Power meanD'HondtDownImperialiMacanese D'HondtElectoral law of Macau, Article 17(2): ""

D'Hondt (Jefferson) method

Thomas Jefferson was the first to propose a divisor method, in 1792; it was later independently developed by Belgian political scientist Victor d'Hondt in 1878. It assigns the representative to the list that would be most underrepresented at the end of the round. It remains the most-common method for proportional representation to this day.

The D'Hondt method uses the sequence , i.e. (1, 2, 3, ...), which means it will always round a party's apportionment down.

An apportionment under the D'Hondt method never falls below the lower end of the ideal frame, and it minimizes the worst-case overrepresentation in the legislature. However, it performs poorly when judged by most other metrics of proportionality. The rule typically gives large parties an excessive number of seats, with their seat share often exceeding their entitlement rounded up.

This pathology led to widespread mockery of the D'Hondt method when it was learned the D'Hondt method could "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' method Adams' method was conceived of by John Quincy Adams after noticing the D'Hondt method allocated too few seats to smaller states. It can be described as the inverse of the D'Hondt 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 , 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, like the D'Hondt method, Adams' method performs poorly according to most metrics of proportionality. It also often violates the lower seat quota.

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 (Schepers) method

The Webster/Sainte-Laguë method was first described in 1832 by American statesman and senator Daniel Webster and later independently in 1910 by the French mathematician André Sainte-Laguë. In 1980 the German physicist Hans Schepers, at the time Head of the Data Processing Group of the Bundestag, suggested that the distribution of seats according to the D'Hondt method be modified to avoid putting smaller parties at a disadvantage. German media started using the term Schepers Method and later German literature usually calls it Sainte-Laguë/Schepers.

The Webster/Sainte-Laguë method uses the fencepost sequence (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; the odd-number expression is more common in electoral laws.

The Webster/Sainte-Laguë method shows a more equal seats-to-votes ratio for different sized parties among apportionment methods; it produces more proportional apportionments than D'Hondt 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. The Webster/Sainte-Laguë method can theoretically violate the ideal frame, 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 the Webster/Sainte-Laguë method 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 instead of 0.5, equal to 1.4 or 2 instead of 1), which creates an implicit threshold.

Huntington–Hill method

In the Huntington–Hill method, the signpost sequence is , 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.

The Huntington–Hill method tends to produce very similar results to the Webster/Sainte-Laguë method, except that it guarantees every state or party at least one seat (see ). When first used to assign seats in the US House of Representatives, the two methods produced identical results; in their second use, they differed only in assigning 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 assigning seats to constituencies in a reapportionment operation) or undesirable (as when assigning seats to party lists in an election), in which case the first divisor may be adjusted to create a natural threshold.

Bias There are many metrics of seat bias. While the Webster/Sainte-Laguë method is sometimes described as "uniquely" unbiased, this uniqueness property relies on a technical definition of bias, which is defined as the average difference between a state's number of seats and its seat entitlement. 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 seat entitlement.

By this definition, the Webster/Sainte-Laguë method is the least-biased apportionment method, while Huntington–Hill exhibits a mild bias towards smaller parties. 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/Sainte-Laguë method is slightly biased towards large parties).