Benham's method: Difference between revisions
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When voters are allowed to equally rank candidates, Benham's method can either be implemented by equally splitting each voter's vote between each candidate they equally ranked highest, or giving each equally-highest-ranked candidate one vote. See the [[Equal-ranking methods in IRV]] page for more information. |
When voters are allowed to equally rank candidates, Benham's method can either be implemented by equally splitting each voter's vote between each candidate they equally ranked highest, or giving each equally-highest-ranked candidate one vote. See the [[Equal-ranking methods in IRV]] page for more information. |
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== Example == |
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(This doesn't feature the actual [[ballot]]<nowiki/>s, but shows the IRV round-by-round breakdown and the pairwise table; for each IRV round, a column next to it is created to show how many votes all candidates from the bottom going up have combined; this allows you to see how candidates were [[Batch elimination|batch eliminated]]. The winner is the candidate who pairwise beats all other candidates remaining in the round): |
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{| class="wikitable" |
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|+IRV round-by-round breakdown (candidates to be eliminated are ''italicized'', while candidates who just barely avoided elimination are <u>underlined</u>) |
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!Initial ordering of |
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candidates (1st) |
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!1st (shows candidates' % of |
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1st choice votes and |
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<nowiki>#</nowiki> of other candidates |
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they pairwise beat) |
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!Cumulative |
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votes (1st) |
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!Reordering of |
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candidates (2nd) |
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!2nd |
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|- |
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|B |
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|35% (2/4) |
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| |
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|A |
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|40% (1/2) |
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|A |
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|25% (3/4) |
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| |
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|B |
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|32% (0/2) |
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|- |
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|C |
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|<u>17% (3/4)</u> |
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|<s>32%</s> |
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(>= 25%) |
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|C |
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|'''28% (2/2)''' |
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|- |
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|D |
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|''8% (1/4)'' |
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|''15% (<17%)'' |
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| |
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| |
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|- |
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|E |
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|''7% (1/4)'' |
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|7% (<8%) |
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| |
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| |
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|} |
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{| class="wikitable" |
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|+Pairwise table (not reorganized) |
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(wins are bolded, and eventually eliminated candidates' matchups are <s>struck-out</s>) |
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! |
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!A |
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!B |
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!C |
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!<s>D</s> |
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!<s>E</s> |
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|- |
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|A |
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| --- |
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|'''163''' |
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|297 |
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|'''<s>405</s>''' |
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|'''<s>389</s>''' |
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|- |
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|B |
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|105 |
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| --- |
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|156 |
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|'''<s>378</s>''' |
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|'''<s>401</s>''' |
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|- |
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|C |
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|'''328''' |
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|'''157''' |
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| --- |
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|<s>305</s> |
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|'''<s>250</s>''' |
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|- |
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|<s>D</s> |
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|<s>120</s> |
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|<s>167</s> |
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|'''<s>307</s>''' |
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|<s>---</s> |
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|<s>250</s> |
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|- |
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|<s>E</s> |
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|<s>57</s> |
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|<s>61</s> |
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|<s>127</s> |
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|'''<s>278</s>''' |
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|<s>---</s> |
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|} |
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Note that in regular IRV, it's possible C could've lost because he had the fewest 1st choices, which would've resulted in A and B continuing to the next round and one of them winning. |
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== Notes == |
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Benham's method is a [[Generalized Condorcet criterion|Smith-efficient]] [[Condorcet method]]. This is because there will always be a point in the count where at least one [[Smith set]] member is uneliminated, and that candidate must beat all other candidates by virtue of being in the Smith Set. (It is also possible that once some members of the Smith set are eliminated, one of the multiple remaining members wins by virtue of beating every remaining candidate, including those in the Smith set.) Benham's method passes [[Monotonicity|mono-add-plump]], unlike several Condorcet-IRV hybrids. It fails [[ISDA]], however.<ref>{{Cite web|url=http://www.votingmatters.org.uk/ISSUE29/I29P1.pdf|title=Four Condorcet-Hare Hybrid Methods for Single-Winner Elections|date=|access-date=|website=|last=Green-Armytage|first=James|archive-url=|archive-date=|url-status=live}}</ref> |
Benham's method is a [[Generalized Condorcet criterion|Smith-efficient]] [[Condorcet method]]. This is because there will always be a point in the count where at least one [[Smith set]] member is uneliminated, and that candidate must beat all other candidates by virtue of being in the Smith Set. (It is also possible that once some members of the Smith set are eliminated, one of the multiple remaining members wins by virtue of beating every remaining candidate, including those in the Smith set.) Benham's method passes [[Monotonicity|mono-add-plump]], unlike several Condorcet-IRV hybrids. It fails [[ISDA]], however.<ref>{{Cite web|url=http://www.votingmatters.org.uk/ISSUE29/I29P1.pdf|title=Four Condorcet-Hare Hybrid Methods for Single-Winner Elections|date=|access-date=|website=|last=Green-Armytage|first=James|archive-url=|archive-date=|url-status=live}}</ref> |
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