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An apportionment method always encourages schisms if the coalition receives at most + seats (in other words, it is merge-proof - two parties cannot gain a seat by merging). Among the divisor methods: [7]: Thm.9.1, 9.2, 9.3 Jefferson's method is the unique split-proof divisor method;
Huntington-Hill uses a continuity correction as a compromise, given by taking the geometric mean of both divisors, i.e.: [4] A n = P n ( n + 1 ) {\displaystyle A_{n}={\frac {P}{\sqrt {n(n+1)}}}} where P is the population of the state, and n is the number of seats it currently holds before the possible allocation of the next seat.
[1] [2] 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). [ 2 ] 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 ).
For illustration, continue with the above example of four parties. The advantage ratios of the four parties are 1.2 for A, 1.1 for B, 1 for C, and 0 for D. The reciprocal of the largest advantage ratio is 1/1.15 = 0.87 = 1 − π *. The residuals as shares of the total vote are 0% for A, 2.2% for B, 2.2% for C, and 8.7% for party D.
The number of allocated seats for a given region increases from s to s + 1 exactly when the divisor equals the population of the region divided by s + 1/2, so at each step the next region to get a seat will be the one with the largest value of this quotient. That means that this successive adjustment method for implementing Webster's method ...
The goal is to find an apportionment method is a vector of integers , …, with = =, called an apportionment of , where is the number of seats allocated to party i. An apportionment method is a multi-valued function M ( t , h ) {\displaystyle M(\mathbf {t} ,h)} , which takes as input a vector of entitlements and a house-size, and returns as ...
Tbl.A7.2 Every quota-capped divisor method satisfies house monotonicity. Moreover, quota-capped divisor methods satisfy the quota rule. [5]: Thm.7.1 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. [5]:
Biproportional apportionment is a proportional representation method to allocate seats in proportion to two separate characteristics. That is, for two different partitions each part receives the proportional number of seats within the total number of seats.