In math terms, which apportionment method (Hamilton, Jefferson, Webster, Adams) is best? include reasons to support your choice including advantages & disadvantages.
This is a difficult question as it has been shown that no apportionment method can exist that is paradox free -- that is there is no method with the following characteristics:
(1) It follows the quota rule -- each district gets one of the numbers closest to its fair share. (I.e. if the district should get 7.3 delegates, then it is given either 7 or 8 delegates.)
(2) Avoids the Alabama paradox -- if the number of seats is increased then no district loses seats.
(3) Avoids the population paradox -- if district A increases in population and district B decreases in population and a seat is transferred from A to B.
Historically, the U.S. has used five apportionment systems -- Jefferson's, Hamilton's, Webster's (twice), and the current method Huntington-Hill. Hamilton's was approved by the first congress, but suffered the first presidential veto, so Jefferson's method was first. Adams' method was never used.
Adams, Webster, and Jefferson used a divisor method, while Hamilton used a quota system.
Of the three divisor systems, Adams' favors small districts, Jefferson favors large districts, while Webster's is relatively unbiased. By favoring, we mean that small(large) districts gain seats under that method. So by this criteria, Webster's method might be considered best. However, Webster violates the quota rule.
In each of the divisor methods, a divisor D is calculated as `"totalpopulation"/"numseats" ` . Then a correction factor d is introduced. For Adams, find d such that the state allocation `"statepop"/(D+d) ` when rounded up for each state yields a sum that is a whole number. For Jefferson we change the denominator to D-d and round down. For Webster, d is positive or negative, and round using normal rounding rules.
The Hamiltonian system is a quota system. Again you calculate the divisor D; each state will have an initial apportionment that might include fractional delegates. These fractional parts add up to a whole number -- assign delegates one at a time for this whole number to districts with the largest fractional part.
The issue with Hamilton's method (used 1852-1901) is it is subject to the Alabama paradox, the new state paradox, and the population paradox. These paradoxes mainly arise as you change the number of delegates/representatives.
An example from the link below: three districts with 603,249, 148.(Total = 1000). Proportion for each district -- 6.03,2.49,1.48.
Adams -- 5,3,2
Webster -- 6,3,1
Hamilton -- 6,3,1
Jefferson -- 7,2,1
It has been shown that Hamilton will lie between Adams and Jefferson, but direct comparison with Webster is impossible as one uses a divisor method and the other a quotient method.
Our current method is a modification of Webster's method -- now the rounding rule used is called the method of equal proportions.
I would choose Webster's from the four given, as it is relatively unbiased towards the size of the district. The major drawback is the trial and error method to find d and that it violates the quota rule.