The Borda count electoral system can be combined with an instant-runoff procedure to create hybrid election methods that are called Nanson's method and Baldwin's method. Both methods are designed to satisfy the Condorcet criterion, and allow for incomplete ballots and equal rankings.
Nanson's method
[[File:Ranked ballot Nanson.jpg|thumb|A ranked ballot with incomplete preferences, as illustrated by Nanson. 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 only the remaining candidates had been on the ballot. This process is repeated if necessary until a single winner remains.
If a Condorcet winner exists, they will be elected. If not (that is, if there is a Condorcet cycle), then the preference with the smallest majority will be eliminated. in 1926, who incorporated a more efficient matrix tabulation 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.
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 what they perceived as problems with instant-runoff voting in U.S. jurisdictions that use it.
Satisfied and failed criteria
Nanson's method and Baldwin's method satisfy the Condorcet criterion:
Both Nanson's and Baldwin's methods can be run in polynomial time to obtain a single winner. For Baldwin's method, however, at each stage, there might be several candidates with lowest Borda score, and different choices of which candidate to eliminate can result in different ultimate outcomes. 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.
Both methods are computationally more difficult to manipulate than Borda's method.
Use of the methods
Nanson's method was used in city elections in Marquette, Michigan, US in the 1920s. It was formerly used by the Anglican Diocese of Melbourne and in the election of members of the University Council of the University of Adelaide. It was used by the University of Melbourne until 1983.
References
- Duncan Sommerville (1928) "Certain hyperspatial partitionings connected with preferential voting", Proceedings of the London Mathematical Society 28(1):368–82.