Hamilton method

The Hamilton method (or Largest Remainder-Hare/LR-Hare) is a version of the largest remainder method for allocating seats proportionally for representative assemblies with party list voting systems. It uses the Hare quota as the initial quota.

Example

In 1790, the U.S. had 15 states. For the purpose of allocating seats in the House of Representatives, the state populations were as follows:

State	Population
Virginia	630 560
Massachusetts	475 327
Pennsylvania	432 879
North Carolina	353 523
New York	331 589
Maryland	278 514
Connecticut	236 841
South Carolina	206 236
New Jersey	179 570
New Hampshire	141 822
Vermont	85 533
Georgia	70 835
Kentucky	68 705
Rhode Island	68 446
Delaware	55 540
Total	3 615 920

Suppose that there were to be 60 seats in the House.

The quota is 3 615 920 ÷ 60 = 60 265.333. Dividing the state populations by the quota gives

State	Quotient	Remainder
Virginia	10	27 906.7
Massachusetts	7	53 469.7
Pennsylvania	7	11 021.7
North Carolina	5	52 196.3
New York	5	30 262.3
Maryland	4	37 452.7
Connecticut	3	56 045.0
South Carolina	3	25 440.0
New Jersey	2	59 039.3
New Hampshire	2	21 291.3
Vermont	1	25 267.7
Georgia	1	10 569.7
Kentucky	1	8 439.7
Rhode Island	1	8 180.7
Delaware	0	55 540.0
Total	52

Each state receives a number of seats equal to the integer part of the quotient. The remaining 8 seats are given to the states with the largest remainders: New Jersey, Connecticut, Delaware, Massachusetts, North Carolina, Maryland, New York, and Virginia. The final apportionment is:

State	Seats	District size	Rel. rep.
Virginia	11	57 324	1.0513
Massachusetts	8	59 416	1.0143
Pennsylvania	7	61 840	0.9745
North Carolina	6	58 920	1.0228
New York	6	55 265	1.0905
Maryland	5	55 703	1.0819
Connecticut	4	59 210	1.0178
South Carolina	3	68 745	0.8766
New Jersey	3	59 857	1.0068
New Hampshire	2	70 911	0.8499
Vermont	1	85 533	0.7046
Georgia	1	70 835	0.8562
Kentucky	1	68 705	0.8772
Rhode Island	1	68 446	0.8805
Delaware	1	55 540	1.0851
Total	60

Extensions of theory

Several cardinal PR methods reduce to Hamilton if certain divisors are used. Some of which are:

• Sequential Monroe voting
• Sequentially Spent Score
• Monroe's method

Notes

Hamilton doesn't guarantee that a majority of voters will always win at least half of the seats (though LR-Hagenbach-Bischoff does, since a majority always has more votes than a majority of Hagenbach-Bischoff quotas).

35-seat example

	Votes	Votes %	Fraction	Automatic seats	Remainders	Additional seats	Final seats	Seats %
A	503	50.3%	17.605	17	0.605		17	48.57%
B	304	30.4%	10.640	10	0.640	+1	11	31.43%
C	193	19.3%	6.755	6	0.755	+1	7	20%
Total seats awarded				33		+2	35

Party A, with 50.3% of the votes, only gets 17 out of 35 seats, which is 48.57% of the seats, a minority.

See also

• Thomas Hare
• Alabama paradox

This page uses Creative Commons Licensed content from Wikipedia (view authors).