Highest averages method
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.' More generally, divisor methods are used to round shares of a total to a fraction with a fixed denominator.'
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. 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 a constitutional 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 Congress, especially after the discovery of pathologies in many superficially-reasonable rounding rules. Similar debates would appear in Europe 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 1870 reapportionment, when Congress used an ad-hoc apportionment to favor Republican states. Had each state's electoral vote total been exactly equal to its entitlement, or had Congress used Webster's method or a largest remainders method, the 1876 election would have gone to Tilden instead of 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, 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 after dividing. Thus, each party's apportionment is given by:
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 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. In general, we can define the averages using the signpost sequence:
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.| Method | Signposts | Rounding of Seats | Approx. first values |
| Adams | Up | ||
| Dean | Harmonic | ||
| Huntington–Hill | Geometric | ||
| Stationary | Weighted | ||
| Webster/Sainte-Laguë | Arithmetic | ||
| Power mean | Power mean | ||
| Jefferson/D'Hondt | Down |
Jefferson (D'Hondt) method
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.Jefferson's method uses the sequence, i.e., which means it will always round a party's apportionment down.
Jefferson's apportionment 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 Jefferson's method when it was learned Jefferson's 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 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, 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 Jefferson's 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) method
The Sainte-Laguë or Webster method, first described in 1832 by American statesman and senator Daniel Webster and later independently in 1910 by the French mathematician André Sainte-Lague, uses the fencepost sequence ; this corresponds to the standard rounding rule. Equivalently, the odd integers can be used to calculate the averages instead.The Webster method produces more proportional apportionments than Jefferson'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. It is also notable for minimizing seat bias even when dealing with parties that win very small numbers of seats. The Webster 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 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, 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 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 method, except that it guarantees every state or party at least one seat. When first used to assign seats in the House, 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 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 the Webster 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 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.
In practice, the difference between these definitions is small when handling parties or states with more than one seat. Thus, both the Huntington–Hill and Webster methods can be considered unbiased or low-bias methods. A 1929 report to Congress by the National Academy of Sciences recommended the Huntington–Hill method, while the Supreme Court has ruled the choice to be a matter of opinion.