Sequential elimination method

The sequential elimination methods are a class of voting systems that repeatedly eliminate the last-place finisher of another voting method until a single candidate remains.^[1] The method used to determine the loser is called the base method. Common are the two-round system, instant-runoff voting, and some primary systems.

Instant-runoff voting is a sequential loser method based on plurality voting, while Baldwin's method is a sequential loser method based on the Borda count.^[2]

Properties

Proofs of criterion compliance for loser-elimination methods often use mathematical induction, and so can be easier than proving such compliance for other method types. For instance, if the base method passes the majority criterion, a sequential loser-elimination method based on it will pass mutual majority. Loser-elimination methods are also not much harder to explain than their base methods.^[2]

However, loser-elimination methods often fail monotonicity due to chaotic effects (sensitivity to initial conditions): the order in which candidates are eliminated can create erratic behavior.^[1]

If the base method passes independence from the weakest alternative, the loser-elimination method is equivalent to the base method.^[1] In other words, methods that are immune to weak spoilers are already "their own" elimination methods, because eliminating the weakest candidate does not affect the winner.

If the base method satisfies a criterion for a single candidate (e.g. the majority criterion or the Condorcet criterion), then a sequential loser method satisfies the corresponding set criterion (e.g. the mutual majority criterion or the Smith criterion), so long as eliminating a candidate can't remove another candidate from the set in question. This is because when all but one of the candidates of the set have been eliminated, the single-candidate criterion applies to the remaining candidate.^[1]

References

1. Xia, Lirong; Lang, Jérôme; Ying, Mingsheng (2007-06-25). "Sequential voting rules and multiple elections paradoxes". Proceedings of the 11th conference on Theoretical aspects of rationality and knowledge - TARK '07. New York, NY, USA: Association for Computing Machinery. pp. 279–288. doi:10.1145/1324249.1324286. ISBN 978-1-4503-7841-3.
2. Bag, PK; Sabourian, H; Winter, E. "Sequential Elimination vs. Instantaneous Voting" (PDF). Mimeo.

Further reading

• Chowdhury, Subhasish M.; Kim, Sang-Hyun (July 2017). "'Small, yet Beautiful': Reconsidering the optimal design of multi-winner contests" (PDF). Games and Economic Behavior. 104: 486–493. doi:10.1016/j.geb.2017.05.012. S2CID 43411835.
• Klunover, Doron (December 2023). "Bureaucracy and labor market inefficiency: A contest model". European Journal of Political Economy. 80: 102472. doi:10.1016/j.ejpoleco.2023.102472. S2CID 262024392.