The Borda count can be combined with an instant-runoff procedure to create hybrid election methods that are called Nanson method and Baldwin method.
The Nanson method is based on the original work of the mathematician Edward J. 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 the remaining candidates were exclusively on the ballot. This process is repeated if necessary until a single winner remains.
This variant was devised by Joseph M. Baldwin and works like this:
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.
Satisfied and failed criteria
The Nanson method and the Baldwin method satisfy the Condorcet criterion: since Borda always gives any existing Condorcet winner more than the average Borda points, the Condorcet winner will never be eliminated. They do not satisfy the independence of irrelevant alternatives criterion, the monotonicity criterion, the participation criterion, the consistency criterion and the independence of clones criterion, while they do satisfy the majority criterion, the mutual majority criterion, the Condorcet loser criterion, and the Smith criterion. The Nanson method satisfies and the Baldwin method violates reversal symmetry.
Both the Nanson and the Baldwin method 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.
Use of Nanson and Baldwin
Nanson's method was used in city elections in the U.S. town of Marquette, Michigan 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.