Sequential elimination method

The sequential loser (elimination) methods are a class of voting systems that repeatedly eliminate the last-place finisher of another voting method until a single candidate remains. The method used to determine the loser is called the base method.

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.

Criteria

Sequential loser methods are generally nonmonotonic.

Proving criterion compliances for loser-elimination methods often use inductive proofs, and can thus be easier than proving such compliances 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. 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.

When the base method passes local independence of irrelevant alternatives, the loser-elimination method is equivalent to the base method.

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.

If one base method is used as a tiebreaker for another, and both base methods pass the candidate criterion, then the sequential loser-elimination method satisfies the set criterion. If only one of them passes the candidate criterion, then the elimination method need not pass the set criterion.