View source for Nanson's method

{{Short description|Single-winner electoral system}}
{{Electoral systems}}
The [[Borda count]] [[electoral system]] can be combined with an [[Instant-runoff voting|instant-runoff]] procedure to create hybrid election methods that are called '''Nanson method''' and '''Baldwin method''' (also called '''Total Vote Runoff''' or '''TVR'''). Both methods are designed to satisfy the [[Condorcet criterion]], and allow for incomplete ballots and equal rankings.

== Nanson method ==
[[File:Ranked ballot Nanson.jpg|thumb|A ranked ballot with incomplete preferences, as illustrated by Nanson.<ref name=":0" />{{Rp|37}}]]
The Nanson method is based on the original work of the [[mathematician]] [[Edward J. Nanson]] in 1882.<ref name=":0">{{Cite journal|last=Nanson|first=E. J.|date=1882|title=Methods of election|url=https://archive.org/details/transactionsproc1719roya/page/197|journal=Transactions and Proceedings of the Royal Society of Victoria|volume=19|pages=197–240}}</ref>

Nanson's method eliminates those choices from a Borda count tally that are at or below the average Borda count score, then the ballots are retallied as if the remaining candidates were exclusively on the ballot. This process is repeated if necessary until a single winner remains.

If a [[Condorcet winner]] exists, they will be elected.  If not, (there is a [[Condorcet cycle]]) then the preference with the smallest majority will be eliminated.<ref name=":0" />{{Rp|214}}

Nanson's method can be adapted to handle incomplete ballots (including "[[Bullet voting|plumping]]") and equal rankings ("bracketing"), though he describes two different methods to handle these cases: a theoretically correct method involving fractions of a vote, and a practical method involving whole numbers (which has the side effect of diminishing the voting power of voters who plump or bracket).<ref name=":0" />{{Rp|231, 235}}  This then allows the use of [[Approval voting|Approval]]-style voting for uninformed voters who merely wish to approve of some candidates and disapprove of others.<ref name=":0" />{{Rp|236}}

The method can be adapted to multi-winner elections by removing the name of a winner from the ballots and re-calculating, though this just elects the highest-ranked ''n'' candidates and does not result in proportional representation.<ref name=":0" />{{Rp|240}}

Schwartz in 1986 studied a slight variant of Nanson's rule, in which candidates less than ''but not equal to'' the average Borda count score are eliminated in each round.<ref name=":1" />

== Baldwin method ==

Candidates are voted for on ranked ballots as in the Borda count. Then, the points are tallied in a series of rounds. In each round, the candidate with the fewest points is eliminated, and the points are re-tallied as if that candidate were not on the ballot.

This method actually predates Nanson's, who notes it was already in use by the [[Trinity College, Cambridge|Trinity College]] Dialectic Society.<ref name=":0" />{{Rp|217}}

It was systematized by [[Joseph M. Baldwin]]<ref name=":2">{{Cite journal |last=Baldwin |first=J. M. |date=1926 |title=The technique of the Nanson preferential majority system of election |url=https://archive.org/details/proceedingsroyaxxxvroyaa/page/42 |journal=Proceedings of the Royal Society of Victoria |volume=39 |pages=42–52 |quote=In each case where on a voting paper no preference is expressed as between two candidates, half a preference is to be credited to each of the two candidates … For each paper where any number, p, of candidates are placed equal with a preference ranking as first, 1/p is to be credited to each of the candidates so placed.}}</ref> in 1926, who incorporated [[Condorcet method#Pairwise counting and matrices|a more efficient matrix tabulation]]<ref>{{Cite journal|last=Hogben|first=G.|date=1913|title=Preferential Voting in Single-member Constituencies, with Special Reference to the Counting of Votes|url=http://rsnz.natlib.govt.nz/volume/rsnz_46/rsnz_46_00_005780.html|journal=Transactions and Proceedings of the Royal Society of New Zealand|volume=46|pages=304–308}}</ref> and extended it to support incomplete ballots and equal rankings, by counting fractional points in such cases.

The two methods have been confused with each other in some literature.<ref name=":1">{{Cite journal|last=Niou|first=Emerson M. S.|date=1987|title=A Note on Nanson's Rule|journal=Public Choice|volume=54|issue=2|pages=191–193|issn=0048-5829|citeseerx=10.1.1.460.8191|doi=10.1007/BF00123006|s2cid=154538772}}</ref>

This system has been proposed for use in the United States under the name "Total Vote Runoff", by [[Edward B. Foley]] and [[Eric Maskin]], as a way to fix problems with the [[Instant-runoff voting|instant-runoff method]] in U.S. jurisdictions that use it.<ref>{{Cite news |last=Foley |first=Edward B. |author-link=Edward B. Foley |last2=Maskin |first2=Eric S. |author-link2=Eric Maskin |date=November 1, 2022 |title=Alaska’s ranked-choice voting is flawed. But there’s an easy fix. |language=en-US |work=Washington Post |url=https://www.washingtonpost.com/opinions/2022/11/01/alaska-final-four-primary-begich-palin-peltola/ |access-date=2022-11-09 |issn=0190-8286 |quote=the way Alaska uses ranked-choice voting also caused the defeat of Begich, whom most Alaska voters preferred to Democrat Mary Peltola … A candidate popular only with the party’s base would be eliminated early in a Total Vote Runoff, leaving a more broadly popular Republican to compete against a Democrat.}}</ref><ref>{{Cite journal |last=Edward B. Foley |date=2023-03-01 |title=Total Vote Runoff: A Majority-Maximizing Form of Rank Choice Voting |url=https://scholars.unh.edu/unh_lr/vol21/iss2/5 |journal=The University of New Hampshire Law Review |volume=21 |issue=2 |pages=323 |issn=2325-7318}}</ref><ref>{{Cite web |last=Foley |first=Ned |author-link=Edward B. Foley |date=November 1, 2022 |title=“Total Vote Runoff” tweak to Ranked Choice Voting |url=https://electionlawblog.org/?p=132792 |access-date=2022-11-09 |website=Election Law Blog |language=en-US |quote=a small but significant adjustment to the “instant runoff” method … equivalent to a candidate’s Borda score, and eliminating sequentially the candidate with the lowest total votes}}</ref><ref>{{Cite web |last=Foley |first=Ned |author-link=Edward B. Foley |date=November 8, 2022 |title=An Additional Detail about “Total Vote Runoff” |url=https://electionlawblog.org/?p=132963 |access-date=2022-11-09 |website=Election Law Blog |language=en-US |quote=Begich and Peltola each get half a vote by being tied for second place on this ballot}}</ref><ref>{{Cite web |last=Foley |first=Ned |title=“Total Vote Runoff” & Baldwin’s method |url=https://electionlawblog.org/?p=133027 |access-date=2022-11-20 |website=Election Law Blog |language=en-US |quote=In this respect, TVR differs from Baldwin’s method, which without checking whether any candidate has more than 50% of first-place votes would immediately recalculate Borda scores}}</ref>

== Satisfied and failed criteria ==

The Nanson method and the Baldwin method satisfy the [[Condorcet criterion]].<ref name=":1" /> Because Borda always gives any existing Condorcet winner more than the average Borda points, the Condorcet winner will never be eliminated.

They do not satisfy the [[independence of irrelevant alternatives]] criterion, the [[monotonicity criterion]], the [[participation criterion]], the [[consistency criterion]] and the [[independence of clones criterion]], while they do satisfy the [[Majority favorite criterion|majority criterion]], the [[mutual majority criterion]], the [[Condorcet loser criterion]] and the [[Smith set|Smith criterion]]. The Nanson method satisfies and the Baldwin method violates [[reversal symmetry]].<ref>{{Cite web|url=https://www.mail-archive.com/election-methods@lists.electorama.com/msg00625.html|title=Re: [Election-Methods] Borda-elimination, a Condorcet method for public elections?|website=www.mail-archive.com|access-date=2019-06-19}}</ref>

Both the Nanson and the Baldwin methods can be run in [[Time complexity|polynomial time]] to obtain a single winner. For the Baldwin method, however, at each stage, there might be several candidates with lowest Borda score. In fact, it is [[NP-completeness|NP-complete]] to decide whether a given candidate is a Baldwin winner, i.e., whether there exists an elimination sequence that leaves a given candidate uneliminated.<ref>{{Cite journal|last1=Mattei|first1=Nicholas|last2=Narodytska|first2=Nina|last3=Walsh|first3=Toby|date=2014-01-01|title=How Hard is It to Control an Election by Breaking Ties?|journal=Proceedings of the Twenty-First European Conference on Artificial Intelligence|volume=263|issue=ECAI 2014|series=ECAI'14|location=Amsterdam, the Netherlands, The Netherlands|publisher=IOS Press|pages=1067–1068|doi=10.3233/978-1-61499-419-0-1067|isbn=9781614994183|s2cid=1399756|arxiv=1304.6174}}</ref>

Both methods are computationally more difficult to manipulate than Borda's method.<ref>{{Cite journal|last1=Davies|first1=Jessica|last2=Katsirelos|first2=George|last3=Narodytska|first3=Nina|last4=Walsh|first4=Toby|last5=Xia|first5=Lirong|date=2014-12-01|title=Complexity of and algorithms for the manipulation of Borda, Nanson's and Baldwin's voting rules|journal=Artificial Intelligence|volume=217|pages=20–42|doi=10.1016/j.artint.2014.07.005|issn=0004-3702|doi-access=free}}</ref>

== Use of Nanson and Baldwin ==
Nanson's method was used in city elections in the [[United States|U.S.]] town of [[Marquette, Michigan]] in the 1920s.<ref name="Marquette">{{cite web |url=http://www.nuff.ox.ac.uk/Politics/papers/2002/w23/mclean.pdf |title=Australian electoral reform and two concepts of representation |last=McLean |first=I. |year=2002 }}</ref> It was formerly used by the [[Anglican]] Diocese of [[Melbourne]] and in the election of members of the University Council of the [[University of Adelaide]]. It was used by the [[University of Melbourne]] until 1983.

== References ==
{{Reflist}}
* [[Duncan Sommerville]] (1928) "Certain hyperspatial partitionings connected with preferential voting", [[Proceedings of the London Mathematical Society]] 28(1):368–82.

{{voting systems}}

[[Category:Single-winner electoral systems]]
[[Category:Instant-runoff voting]]