Difference between revisions of "Hamilton method"

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{{Wikipedia|Largest remainder method}}
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The '''Hamilton method''' is a version of the [[largest remainder method]] for allocating seats proportionally for representative assemblies with [[Party-list proportional representation|party list]] [[voting systems]]. It uses the [[Hare quota]] as the initial quota.
  
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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.
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==Example==
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{{US_House_apportionment_example}}
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The quota is 3 615 920 ÷ 60 = 60 265.333. Dividing the state populations by the quota gives
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<table class="wikitable" border="">
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<tr>
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<th>State</th>
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<th>Quotient</th>
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<th>Remainder</th>
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</tr>
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<tr>
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<td>Virginia</td>
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<td align="right">10</td>
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<td align="right">27 906.7</td>
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</tr>
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<tr>
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<td>Massachusetts</td>
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<td align="right">7</td>
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<td align="right">53 469.7</td>
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</tr>
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<tr>
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<td>Pennsylvania</td>
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<td align="right">7</td>
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<td align="right">11 021.7</td>
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</tr>
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<tr>
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<td>North Carolina</td>
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<td align="right">5</td>
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<td align="right">52 196.3</td>
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</tr>
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<tr>
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<td>New York</td>
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<td align="right">5</td>
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<td align="right">30 262.3</td>
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</tr>
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<tr>
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<td>Maryland</td>
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<td align="right">4</td>
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<td align="right">37 452.7</td>
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</tr>
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<tr>
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<td>Connecticut</td>
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<td align="right">3</td>
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<td align="right">56 045.0</td>
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</tr>
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<tr>
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<td>South Carolina</td>
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<td align="right">3</td>
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<td align="right">25 440.0</td>
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</tr>
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<tr>
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<td>New Jersey</td>
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<td align="right">2</td>
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<td align="right">59 039.3</td>
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</tr>
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<tr>
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<td>New Hampshire</td>
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<td align="right">2</td>
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<td align="right">21 291.3</td>
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</tr>
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<tr>
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<td>Vermont</td>
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<td align="right">1</td>
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<td align="right">25 267.7</td>
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</tr>
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<tr>
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<td>Georgia</td>
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<td align="right">1</td>
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<td align="right">10 569.7</td>
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</tr>
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<tr>
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<td>Kentucky</td>
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<td align="right">1</td>
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<td align="right">8 439.7</td>
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</tr>
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<tr>
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<td>Rhode Island</td>
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<td align="right">1</td>
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<td align="right">8 180.7</td>
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</tr>
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<tr>
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<td>Delaware</td>
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<td align="right">0</td>
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<td align="right">55 540.0</td>
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</tr>
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<tr>
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<th>Total</th>
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<th align="right">52</th>
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</tr>
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</table>
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  +
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:
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<table class="wikitable" border="">
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<tr>
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<th>State</th>
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<th>Seats</th>
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<th>District size</th>
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<th>Rel. rep.</th>
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</tr>
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<tr>
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<td>Virginia</td>
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<td align="right">11</td>
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<td align="right">57 324</td>
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<td align="right">1.0513</td>
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</tr>
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<tr>
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<td>Massachusetts</td>
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<td align="right">8</td>
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<td align="right">59 416</td>
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<td align="right">1.0143</td>
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</tr>
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<tr>
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<td>Pennsylvania</td>
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<td align="right">7</td>
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<td align="right">61 840</td>
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<td align="right">0.9745</td>
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</tr>
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<tr>
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<td>North Carolina</td>
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<td align="right">6</td>
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<td align="right">58 920</td>
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<td align="right">1.0228</td>
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</tr>
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<tr>
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<td>New York</td>
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<td align="right">6</td>
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<td align="right">55 265</td>
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<td align="right">1.0905</td>
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</tr>
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<tr>
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<td>Maryland</td>
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<td align="right">5</td>
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<td align="right">55 703</td>
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<td align="right">1.0819</td>
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</tr>
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<tr>
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<td>Connecticut</td>
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<td align="right">4</td>
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<td align="right">59 210</td>
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<td align="right">1.0178</td>
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</tr>
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<tr>
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<td>South Carolina</td>
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<td align="right">3</td>
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<td align="right">68 745</td>
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<td align="right">0.8766</td>
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</tr>
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<tr>
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<td>New Jersey</td>
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<td align="right">3</td>
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<td align="right">59 857</td>
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<td align="right">1.0068</td>
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</tr>
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<tr>
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<td>New Hampshire</td>
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<td align="right">2</td>
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<td align="right">70 911</td>
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<td align="right">0.8499</td>
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</tr>
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<tr>
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<td>Vermont</td>
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<td align="right">1</td>
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<td align="right">85 533</td>
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<td align="right">0.7046</td>
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</tr>
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<tr>
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<td>Georgia</td>
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<td align="right">1</td>
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<td align="right">70 835</td>
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<td align="right">0.8562</td>
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</tr>
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<tr>
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<td>Kentucky</td>
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<td align="right">1</td>
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<td align="right">68 705</td>
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<td align="right">0.8772</td>
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</tr>
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<tr>
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<td>Rhode Island</td>
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<td align="right">1</td>
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<td align="right">68 446</td>
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<td align="right">0.8805</td>
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</tr>
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<tr>
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<td>Delaware</td>
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<td align="right">1</td>
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<td align="right">55 540</td>
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<td align="right">1.0851</td>
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</tr>
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<tr>
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<th>Total</th>
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<th align="right">60</th>
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</tr>
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</table>
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== Extensions of theory ==
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Several [[cardinal PR]] methods reduce to Hamilton if certain divisors are used. Some of which are:
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* [[Sequential Monroe voting]]
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* [[Sequentially Spent Score]]
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* [[Monroe's method]]
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== Notes ==
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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]]).
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{| class="wikitable"
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|+35-seat example
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!
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!Votes
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!Votes %
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!Fraction
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!Automatic seats
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!Remainders
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!Additional seats
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!Final seats
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!Seats %
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|-
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|A
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|'''503'''
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|'''50.3%'''
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|17.605
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|17
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|0.605
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|
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|17
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|'''48.57%'''
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|-
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|B
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|304
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|30.4%
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|10.640
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|10
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|0.640
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| +1
  +
|11
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|31.43%
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|-
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|C
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|193
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|19.3%
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|6.755
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|6
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|0.755
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| +1
  +
|7
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|20%
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|-
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|Total seats awarded
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|
  +
|
  +
|
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|33
  +
|
  +
| +2
  +
|35
  +
|
  +
|}
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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
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UNDER PROPORTIONAL REPRESENTATION|last=|first=|date=|website=|page=16|url-status=live|archive-url=|archive-date=|access-date=}}</ref>
   
 
==See also==
  
==See also==
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* [[List of democracy and elections-related topics]]
 
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*[[Alexander Hamilton]]
  
 
*[[Thomas Hare]]
  
*[[Thomas Hare]]
 
*[[Alabama paradox]]
  
*[[Alabama paradox]]
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*[[Largest remainder method]]-details on the Hamilton method
  
   
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[[Category:Voting systems]]
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[[Category:Apportionment methods]]
   
 
{{fromwikipedia}}
  
{{fromwikipedia}}

Latest revision as of 04:36, 18 April 2020

Wikipedia has an article on:

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[edit | edit source]

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:

StatePopulation
Virginia630 560
Massachusetts475 327
Pennsylvania432 879
North Carolina353 523
New York331 589
Maryland278 514
Connecticut236 841
South Carolina206 236
New Jersey179 570
New Hampshire141 822
Vermont85 533
Georgia70 835
Kentucky68 705
Rhode Island68 446
Delaware55 540
Total3 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[edit | edit source]

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

Notes[edit | edit source]

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.[1]

See also[edit | edit source]

This page uses Creative Commons Licensed content from Wikipedia (view authors).
  1. ↑ "ELECTION INVERSIONS UNDER PROPORTIONAL REPRESENTATION" (PDF). p. 16. line feed character in |title= at position 20 (help)
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