# Baldwin's method

Wikipedia has an article on:

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[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

This system was rechristened as Total Vote Runoff by Nobel Memorial Prize laureate economists Edward B. Foley and Eric Maskin, who proposed it as a way to fix problems in the 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.[5][6][7][8]

## 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 violates reversal symmetry (unlike Nanson's method).[9]

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.[10] This implies that this method is computationally more difficult to compute than Borda's method.[11]

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 plurality-with-elimination) and can lead to chaotic results.

## Cardinal variant

Using scores instead of 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:

${\displaystyle v_{c}(u_{c})={\frac {u_{c}-u_{\min }}{u_{\max }-u_{\min }}}}$

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 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.

## Example

Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near the capital as possible.

The candidates for the capital are:

• Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
• Nashville, with 26% of the voters, near the center of Tennessee
• Knoxville, with 17% of the voters
• Chattanooga, with 15% of the voters

The preferences of the voters would be divided like this:

42% of voters
(close to Memphis)
26% of voters
(close to Nashville)
15% of voters
(close to Chattanooga)
17% of voters
(close to Knoxville)
1. Memphis
2. Nashville
3. Chattanooga
4. Knoxville
1. Nashville
2. Chattanooga
3. Knoxville
4. Memphis
1. Chattanooga
2. Knoxville
3. Nashville
4. Memphis
1. Knoxville
2. Chattanooga
3. Nashville
4. Memphis

This gives the following points table:

VotersCandidate Memphis Nashville Knoxville Chattanooga 42×3=126 0 0 0 126 42×2 = 84 26×3 = 78 17×1 = 17 15×1 = 15 194 0 26×1 = 26 17×3 = 51 15×2 = 30 107 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:

VotersCandidate Memphis Nashville Knoxville Chattanooga 42×2 = 84 0 0 0 84 42×1 = 42 26×2 = 52 17×1 = 17 15×1 = 15 126 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.

## 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. Foley, Edward B.; Maskin, Eric S. (November 1, 2022). "Alaska's ranked-choice voting is flawed. But there's an easy fix". Washington Post. ISSN 0190-8286. Retrieved 2022-11-09. 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.
6. Foley, Edward B. (2023-01-18). "Total Vote Runoff: A Majority-Maximizing Form of Ranked Choice Voting". Rochester, NY. Cite journal requires |journal= (help)
7. Foley, Ned (November 1, 2022). ""Total Vote Runoff" tweak to Ranked Choice Voting". Election Law Blog. Retrieved 2022-11-09. 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
8. Foley, Ned (November 8, 2022). "An Additional Detail about "Total Vote Runoff"". Election Law Blog. Retrieved 2022-11-09. Begich and Peltola each get half a vote by being tied for second place on this ballot
9. "Re: [Election-Methods] Borda-elimination, a Condorcet method for public elections?". www.mail-archive.com. Retrieved 2019-06-19.
10. 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.
11. 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.