Baldwin's method: Difference between revisions
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{{Wikipedia}}
Under '''Baldwin's method''', candidates are voted for on [[Ranked voting]] 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 never on the ballot.
It was systematized by Joseph M. Baldwin<ref>{{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|via=}}</ref> in 1926, who incorporated [[Condorcet method|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|series=|volume=46|issue=|pages=304–308|via=}}</ref> extending it to support incomplete ballots and equal rankings. Baldwin's method has been confused with [[Nanson's method]] 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}}</ref> This method predates but is related to [[Nanson's method]]. Nanson noted Baldwin's method was already in use by the Trinity College at the University of Melbourne Dialectic Society when he invented his method.<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|via=}}</ref>{{Rp|217}}
This system was rechristened as '''Total Vote Runoff''' by Nobel Memorial Prize laureate economists [https://en.wikipedia.org/wiki/Edward_B._Foley Edward B. Foley] and [https://en.wikipedia.org/wiki/Eric_Maskin Eric Maskin], who proposed it as a way to fix problems in the [https://en.wikipedia.org/wiki/Instant-runoff_voting instant-runoff method]. Maskin and Foley note that unlike instant-runoff, TVR ensures majority support for the winner and typically elects more broadly-acceptable candidates.<ref>{{Cite news|last=Foley|first=Edward B.|url=https://www.washingtonpost.com/opinions/2022/11/01/alaska-final-four-primary-begich-palin-peltola/|title=Alaska’s ranked-choice voting is flawed. But there’s an easy fix.|date=November 1, 2022|work=Washington Post|access-date=2022-11-09|last2=Maskin|first2=Eric S.|language=en-US|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.|author-link=Edward B. Foley|author-link2=Eric Maskin}}</ref><ref>{{Cite journal|last=Foley|first=Edward B.|date=2023-01-18|title=Total Vote Runoff: A Majority-Maximizing Form of Ranked Choice Voting|url=https://papers.ssrn.com/abstract=4328946|language=en|location=Rochester, NY}}</ref><ref>{{Cite web|url=https://electionlawblog.org/?p=132792|title=“Total Vote Runoff” tweak to Ranked Choice Voting|last=Foley|first=Ned|author-link=Edward B. Foley|date=November 1, 2022|website=Election Law Blog|language=en-US|access-date=2022-11-09|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|url=https://electionlawblog.org/?p=132963|title=An Additional Detail about “Total Vote Runoff”|last=Foley|first=Ned|author-link=Edward B. Foley|date=November 8, 2022|website=Election Law Blog|language=en-US|access-date=2022-11-09|quote=Begich and Peltola each get half a vote by being tied for second place on this ballot}}</ref>
== Satisfied and failed criteria ==
[[Baldwin's method]] satisfies 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. Furthermore it satisfies the [[majority criterion]], the [[mutual majority criterion]], the [[Condorcet loser criterion]] and the [[Smith set|Smith criterion]].
[[Baldwin's method]] does not satisfy the [[independence of irrelevant alternatives]] criterion, the [[monotonicity criterion]], the [[participation criterion]], the [[consistency criterion]] and the [[independence of clones criterion]]. [[Baldwin's method]] violates [[reversal symmetry]] (unlike [[Nanson's method]]).<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>
[[Baldwin's method]] can be run in polynomial time to obtain a single winner, but at each stage, there may be several candidates with the lowest Borda score. In fact, it is NP-complete to decide whether a given candidate is a potential Baldwin winner, i.e. whether there exists an elimination sequence that leaves a given candidate uneliminated.<ref>{{Cite journal|last=Mattei|first=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}}</ref> This implies that this method is computationally more difficult to compute than Borda's method.<ref>{{Cite journal|last=Davies|first=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}}</ref>
In practice, the computational bottleneck can be resolved easily enough by adopting some tiebreaking method (like eliminating all tied candidates simultaneously). However, the high frequency of near-ties leaves these methods open to lawsuits (similarly to [[Instant-runoff voting|plurality-with-elimination]]) and can lead to chaotic results.
==Cardinal variant==
Using [[Score voting|scores]] instead of [[Borda count|Borda counts]] gives the '''Cardinal Baldwin''' method; the lowest-scored candidate is eliminated and the ballots are rescaled (normalized) in each round. When the lowest scored candidate is removed such a rescaling would then rescale so that each voter has some candidate at the MAX and some at the MIN score. This maximizes each voter's effective power at each step; eliminating minor candidates in this way prevents them from substantially affecting the results.
Assuming the scores are all scaled to fall in the range [0, 1], ballots are rescaled as follows:
<math>v_c(u_c) = \frac{u_c - u_{\min}}{u_\max - u_\min}</math>
For example, we would transform [.1, .3, .5] to [0, .5, 1.0].
===Related systems===
[[STAR voting]] is a simplified version of this where instead of eliminating each candidate one by one all but the last two candidates are removed at once. This alteration recovers the [[monotonicity criterion]].
[[Distributed Voting]] is a [[cumulative voting]] variant.
==Notes==
Note that Baldwin's method is [[Smith-efficient]]; this is because [[Borda]] can [[Weighted positional method#Criterion compliances|never rank a Condorcet winner last]], and a Condorcet winner will always stay a Condorcet winner when losing candidates are removed/eliminated from an election. When all but one member of the [[Smith set]] is eliminated, the remaining member of the Smith set will [[Pairwise beat|pairwise beat]] all other candidates by definition, and thus will "become" a Condorcet winner at that point that can no longer be eliminated, and thus is guaranteed to be the final remaining candidate and win.
Line 5 ⟶ 38:
Example:
25 A>B>C
40 B>C>A
35 C>A>B
Borda scores are A 185, B 205, C 210. A beats B beats C beats A, so there is no Condorcet winner, and so A, the Borda loser, is eliminated. Since B beats C, B wins. Note that this is a different result than [[Black's method]], which would elect C. They are both related to [[Nanson's method]].
== Example ==
{{Tenn voting example}}This gives the following points table:
{| class="wikitable" style="border:none"
! {{diagonal split header|Candidate|Voters}}
!Memphis
!Nashville
!Knoxville
!Chattanooga
| rowspan="5" style="border: none; background: white;" |
!Score
|-
!Memphis
|42×3=126
|0
|0
|0
|126
|-
!Nashville
|42×2 = 84
|26×3 = 78
|17×1 = 17
|15×1 = 15
|194
|-
!Knoxville
|0
|26×1 = 26
|17×3 = 51
|15×2 = 30
|107
|-
!Chattanooga
|42×1 = 42
|26×2 = 52
|17×2 = 34
|15×3 = 45
|173
|}
Knoxville has the least amount of points, so it is eliminated.
We now have this table:
{| class="wikitable" style="border:none"
! {{diagonal split header|Candidate|Voters}}
!Memphis
!Nashville
!Knoxville
!Chattanooga
| rowspan="4" style="border: none; background: white;" |
!Score
|-
!Memphis
|42×2 = 84
|0
|0
|0
|84
|-
!Nashville
|42×1 = 42
|26×2 = 52
|17×1 = 17
|15×1 = 15
|126
|-
!Chattanooga
|0
|26×1 = 26
|17×2 = 34
|15×2 = 30
|90
|}
Now Memphis is eliminated.
This leaves us with Nashville and Chattanooga. Nashville has 42+26 points, giving it 68 points, while Chattanooga has 17+15 points giving it 32. This makes Nashville the winner.
== See also ==
* [[Minet Ranked-Choice Voting]]
* [[Nanson's method]]
==References==
[[Category:Smith-efficient Condorcet methods]]
[[Category:Sequential loser-elimination methods]]
[[Category:Condorcet-Borda hybrid methods]]
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