Baldwin's method

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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[1] in 1926, who incorporated a more efficient matrix tabulation,[2] extending it to support incomplete ballots and equal rankings. Baldwin's method has been confused with Nanson's method in some literature.[3] 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.[4]:217

Satisfied and failed criteria

Baldwin's method satisfies the Condorcet criterion.[3] 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 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.[5]

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.[6]. This implies that this method is computationally more difficult to compute than Borda's method.[7]

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 ${\displaystyle u_c}$ be a voters score for candidate c, let ${\displaystyle u_{min}}$ and ${\displaystyle u_{max}}$ be their score for her worst and best candidates in the considered election round. The rescaled utility is:

${\displaystyle $$v_c(u_c) = MIN + (MAX– MIN) \frac{(u_c – u_{min})}{(u_{max} – u_{min})}$$}$

For example, in a [0,10] system the translation is

${\displaystyle $$v_c(u_c) = 10 \frac{(u_c – u_{min})}{(u_{max} – u_{min})}$$}$

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 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 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.

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.

References

1. Baldwin, J. M. (1926). "The technique of the Nanson preferential majority system of election". Proceedings of the Royal Society of Victoria. 39: 42–52.
2. Hogben, G. (1913). "Preferential Voting in Single-member Constituencies, with Special Reference to the Counting of Votes". Transactions and Proceedings of the Royal Society of New Zealand. 46: 304–308.
3. a b Niou, Emerson M. S. (1987). "A Note on Nanson's Rule". Public Choice. 54 (2): 191–193. CiteSeerX 10.1.1.460.8191. doi:10.1007/BF00123006. ISSN 0048-5829.
4. Nanson, E. J. (1882). "Methods of election". Transactions and Proceedings of the Royal Society of Victoria. 19: 197–240.
5. "Re: [Election-Methods] Borda-elimination, a Condorcet method for public elections?". www.mail-archive.com. Retrieved 2019-06-19.
6. Mattei, Nicholas; Narodytska, Nina; Walsh, Toby (2014-01-01). "How Hard is It to Control an Election by Breaking Ties?". Proceedings of the Twenty-first European Conference on Artificial Intelligence. ECAI'14. Amsterdam, The Netherlands, The Netherlands: IOS Press. 263 (ECAI 2014): 1067–1068. doi:10.3233/978-1-61499-419-0-1067. ISBN 9781614994183.
7. Davies, Jessica; Katsirelos, George; Narodytska, Nina; Walsh, Toby; Xia, Lirong (2014-12-01). "Complexity of and algorithms for the manipulation of Borda, Nanson's and Baldwin's voting rules". Artificial Intelligence. 217: 20–42. doi:10.1016/j.artint.2014.07.005. ISSN 0004-3702.