# Sequential loser-elimination method

A **sequential loser-elimination method** is a method that works by repeatedly eliminating the loser of another voting method until a single candidate remains, and then electing that candidate. The method that is used to determine the loser is called the *base method*.

Instant runoff voting (without whole-votes-equal ranking) is a sequential loser-elimination method based on First past the post, and Baldwin is a sequential loser-elimination method based on the Borda count.

## Proportional systems

Some multiwinner methods determine the winners by finding the winner set that's optimal according to some criterion. Proportional approval voting and Ebert's method, among others, are of this class. Since it's usually very difficult to test every possible set by brute force, variant methods exist that guide the search by adding candidates one at a time, or removing them, one at a time, from the set of candidate winners, until the method arrives at the right number of winners.

Elimination methods work by starting with the set of every possible candidate, and then determining the quality of the solution if A is removed from the set, B is removed, etc., one at a time. The candidate whose removal would keep quality as high as possible is then removed, and the method repeats the process. These methods can be considered loser-elimination methods with an implicit base method, since the loser (the candidate who has the least effect on quality) is sequentially being eliminated.

Single transferable vote is related to elimination methods, but is not purely a sequential loser-elimination method since STV protects candidates who have reached the quota from being eliminated.

## Properties and criteria

Sequential loser-elimination methods may be ordinal, like instant-runoff voting, or cardinal, like Instant Runoff Normalized Ratings.

Loser-elimination methods often fail monotonicity due to chaotic effects (sensitivity to initial conditions). The chaotic nature of loser-elimination methods can also make it difficult to determine the winner if the base method runs into a tie at any step, since breaking the tie in a particular way can have unpredictable results down the line.

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.

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 or Condorcet criterion), then a sequential loser-elimination method satisfies the corresponding set criterion (e.g. mutual majority or Smith) if eliminating a candidate can't remove another candidate from the set in question. This 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.

## Notes

Note that IRV with whole-votes equal-ranking may not be a sequential-loser elimination method depending on which rules are used to determine the winner; see the STV#Ways of dealing with equal rankings section.

Note that though a voting method may be a sequential loser-elimination method in its single-winner case, it may not be so under certain generalizations of the criterion to the multi-winner case. Consider the following 2-winner example for STV with Droop quotas:

99 A>B 1 C

A and B would win. However, if the criterion for a multi-winner sequential loser-elimination method is that it must repeatedly eliminate until only (# of winners) candidates remain, with no surplus distribution being done, and with those remaining candidates winning, then A and C would win, since B is the candidate with the fewest 1st choices here. So for the multi-winner or proportional case, it may be required to allow surplus distribution or other steps in order to best generalize the sequential loser-elimination criterion for the multi-winner case.