Nanson's method
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The
Nanson method
The Nanson method is based on the original work of the mathematician Edward J. Nanson in 1882.[1]
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
Nanson's method can be adapted to handle incomplete ballots (including "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).[1]: 231, 235 This then allows the use of Approval-style voting for uninformed voters who merely wish to approve of some candidates and disapprove of others.[1]: 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.[1]: 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.[2]
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 Dialectic Society.[1]: 217
It was systematized by Joseph M. Baldwin[3] in 1926, who incorporated a more efficient matrix tabulation[4] 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.[2]
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 method in U.S. jurisdictions that use it.[5][6][7][8][9]
Satisfied and failed criteria
The Nanson method and the Baldwin method satisfy the
They do not satisfy the
Both the Nanson and the Baldwin methods can be run in 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-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.[11]
Both methods are computationally more difficult to manipulate than Borda's method.[12]
Use of Nanson and Baldwin
Nanson's method was used in city elections in the
References
- ^ a b c d e f g Nanson, E. J. (1882). "Methods of election". Transactions and Proceedings of the Royal Society of Victoria. 19: 197–240.
- ^ S2CID 154538772.
- ^ Baldwin, J. M. (1926). "The technique of the Nanson preferential majority system of election". Proceedings of the Royal Society of Victoria. 39: 42–52.
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.
- ^ 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.
- 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.
- ISSN 2325-7318.
- ^ 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
- ^ 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
- ^ Foley, Ned. ""Total Vote Runoff" & Baldwin's method". Election Law Blog. Retrieved 2022-11-20.
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
- ^ "Re: [Election-Methods] Borda-elimination, a Condorcet method for public elections?". www.mail-archive.com. Retrieved 2019-06-19.
- S2CID 1399756.
- ISSN 0004-3702.
- ^ McLean, I. (2002). "Australian electoral reform and two concepts of representation" (PDF).
- Proceedings of the London Mathematical Society28(1):368–82.