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{{Wikipedia|Largest remainder method}}
The '''Hamilton method''' (or Largest Remainder-Hare/LR-Hare) is a version of the [[largest remainder method]] for allocating seats [[proportional representation|proportionally]] for representative assemblies with [[Party-list proportional representation|party list]] [[voting system]]s. It uses the [[Hare quota]] as the initial quota.
==Example==
{{US_House_apportionment_example}}
The quota is 3 615 920 ÷ 60 = 60 265.333. Dividing the state populations by the quota gives
<table class="wikitable" border="">
<tr>
<th>State</th>
<th>Quotient</th>
<th>Remainder</th>
</tr>
<tr>
<td>Virginia</td>
<td align="right">10</td>
<td align="right">27 906.7</td>
</tr>
<tr>
<td>Massachusetts</td>
<td align="right">7</td>
<td align="right">53 469.7</td>
</tr>
<tr>
<td>Pennsylvania</td>
<td align="right">7</td>
<td align="right">11 021.7</td>
</tr>
<tr>
<td>North Carolina</td>
<td align="right">5</td>
<td align="right">52 196.3</td>
</tr>
<tr>
<td>New York</td>
<td align="right">5</td>
<td align="right">30 262.3</td>
</tr>
<tr>
<td>Maryland</td>
<td align="right">4</td>
<td align="right">37 452.7</td>
</tr>
<tr>
<td>Connecticut</td>
<td align="right">3</td>
<td align="right">56 045.0</td>
</tr>
<tr>
<td>South Carolina</td>
<td align="right">3</td>
<td align="right">25 440.0</td>
</tr>
<tr>
<td>New Jersey</td>
<td align="right">2</td>
<td align="right">59 039.3</td>
</tr>
<tr>
<td>New Hampshire</td>
<td align="right">2</td>
<td align="right">21 291.3</td>
</tr>
<tr>
<td>Vermont</td>
<td align="right">1</td>
<td align="right">25 267.7</td>
</tr>
<tr>
<td>Georgia</td>
<td align="right">1</td>
<td align="right">10 569.7</td>
</tr>
<tr>
<td>Kentucky</td>
<td align="right">1</td>
<td align="right">8 439.7</td>
</tr>
<tr>
<td>Rhode Island</td>
<td align="right">1</td>
<td align="right">8 180.7</td>
</tr>
<tr>
<td>Delaware</td>
<td align="right">0</td>
<td align="right">55 540.0</td>
</tr>
<tr>
<th>Total</th>
<th align="right">52</th>
</tr>
</table>
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:
<table class="wikitable" border="">
<tr>
<th>State</th>
<th>Seats</th>
<th>District size</th>
<th>Rel. rep.</th>
</tr>
<tr>
<td>Virginia</td>
<td align="right">11</td>
<td align="right">57 324</td>
<td align="right">1.0513</td>
</tr>
<tr>
<td>Massachusetts</td>
<td align="right">8</td>
<td align="right">59 416</td>
<td align="right">1.0143</td>
</tr>
<tr>
<td>Pennsylvania</td>
<td align="right">7</td>
<td align="right">61 840</td>
<td align="right">0.9745</td>
</tr>
<tr>
<td>North Carolina</td>
<td align="right">6</td>
<td align="right">58 920</td>
<td align="right">1.0228</td>
</tr>
<tr>
<td>New York</td>
<td align="right">6</td>
<td align="right">55 265</td>
<td align="right">1.0905</td>
</tr>
<tr>
<td>Maryland</td>
<td align="right">5</td>
<td align="right">55 703</td>
<td align="right">1.0819</td>
</tr>
<tr>
<td>Connecticut</td>
<td align="right">4</td>
<td align="right">59 210</td>
<td align="right">1.0178</td>
</tr>
<tr>
<td>South Carolina</td>
<td align="right">3</td>
<td align="right">68 745</td>
<td align="right">0.8766</td>
</tr>
<tr>
<td>New Jersey</td>
<td align="right">3</td>
<td align="right">59 857</td>
<td align="right">1.0068</td>
</tr>
<tr>
<td>New Hampshire</td>
<td align="right">2</td>
<td align="right">70 911</td>
<td align="right">0.8499</td>
</tr>
<tr>
<td>Vermont</td>
<td align="right">1</td>
<td align="right">85 533</td>
<td align="right">0.7046</td>
</tr>
<tr>
<td>Georgia</td>
<td align="right">1</td>
<td align="right">70 835</td>
<td align="right">0.8562</td>
</tr>
<tr>
<td>Kentucky</td>
<td align="right">1</td>
<td align="right">68 705</td>
<td align="right">0.8772</td>
</tr>
<tr>
<td>Rhode Island</td>
<td align="right">1</td>
<td align="right">68 446</td>
<td align="right">0.8805</td>
</tr>
<tr>
<td>Delaware</td>
<td align="right">1</td>
<td align="right">55 540</td>
<td align="right">1.0851</td>
</tr>
<tr>
<th>Total</th>
<th align="right">60</th>
</tr>
</table>
== 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 quota|Hagenbach-Bischoff quotas]]).
{| class="wikitable"
|+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.<ref>{{Cite web|url=https://userpages.umbc.edu/~nmiller/RESEARCH/NRMILLER.PCS2013.pdf|title=ELECTION INVERSIONS
UNDER PROPORTIONAL REPRESENTATION|last=|first=|date=|website=|page=16|url-status=live|archive-url=|archive-date=|access-date=}}</ref>
==See also==
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*[[Alabama paradox]]
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