Baldwin's method: Difference between revisions
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{{Wikipedia}}
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}}
== 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]] also 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>
[[Baldwin's method]] can be run in polynomial time to obtain a single winner, however, at each stage, there might be several candidates with lowest Borda score. In fact, it is 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|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>
==Cardianal Variant==
A [[Cardinal Voting]] variant of this system can be made by simply taking the scores initially rather than taking ranks and converting them with [[Borda count]]. In this context the motivation for the normalization at each round is derived by considering an affine transformation. 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 will always maximize effective vote power which is the issue attempted to be equalized by this method.
More explicitly. Let MAX and MIN be the extreme available grades. Let u_c be a voters score for candidate c, let u_min and u_max be their score for her worst and best candidates in the considered election round. The rescaled utility is:
<math>\begin{equation}
v_c(u_c) = MIN + (MAX– MIN) \frac{(u_c – u_{min})}{(u_{max} – u_{min})}
\end{equation}</math>
For example, in a [0,10] system the translation is
<math>\begin{equation}
v_c(u_c) = 10 \frac{(u_c – u_{min})}{(u_{max} – u_{min})}
\end{equation}</math>
It would transform [1,3,5] to [0,5,10]
===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.
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* [[Minet Ranked-Choice Voting]]
* [[Nanson's method]]
==References==
[[Category:Smith-efficient Condorcet methods]]
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