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 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 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:
seats = round ( votes divisor ) {\displaystyle {\text{seats}}=\operatorname {round} \left({\frac {\text{votes}}{\text{divisor}}}\right)}
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:
average := votes post ( seats ) {\displaystyle {\text{average}}:={\frac {\text{votes}}{\operatorname {post} ({\text{seats}})}}}
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.
D'Hondt (Jefferson) method
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