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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 (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.
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 (as it had since 1840), the 1876 election would have gone to Tilden instead of Hayes.
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.
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.
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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 (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.
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 (as it had since 1840), the 1876 election would have gone to Tilden instead of Hayes.
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.
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.
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