Tideman's Alternative methods

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Tideman's Alternative methods are voting methods which all repeatedly loop between the following two steps until only one candidate remains, who is then the winner:

1. Eliminate all candidates not in a particular set of candidates, and then:
2. Eliminate the candidate preferred the most by the fewest voters (IRV-style elimination, essentially).

These methods are named after Nicolaus Tideman, though they are not his invention.

Benham's method

When the Condorcet winner is used as the "set" (the set of all candidates such that it includes all candidates if there is no Condorcet winner, but otherwise includes only the Condorcet winner), it is known as Benham's method.

Tideman's Alternative Smith

When the Smith set is used as the particular set, it is known as Tideman's Alternative Smith, and likewise for the Schwartz set.

Example:

6 D>A>B>C

5 B>C>A>D

4 C>A>B>D

A beats B beats C beats A, but all three beat D, so D is the only one not in the Smith set, and is eliminated. Redistributing D voters' votes, C now has the fewest votes and is eliminated next. Since A beats B, A wins.

Tideman's Alternative Smith passes ISDA, since to begin with the voting method eliminates everyone not in the Smith set. It is equivalent to Smith//IRV when there are only 3 candidates in the Smith set, since both methods will eliminate everyone outside of the Smith set, then eliminate one of the 3 candidates in the Smith set, resulting in either a pairwise tie between the two remaining candidates (thus both are tied in IRV/ in the Smith set) or one pairwise beating the other (and thus receiving a majority under IRV/being the only member of the shrunken Smith set). An example where they would differ goes as follows: suppose the Smith set has 7 candidates. Both methods would eliminate everyone but these 7, then would eliminate the same candidate out of the 7. Now suppose there are 3 candidates who form their own Smith set when ignoring their pairwise matchups against the just-eliminated candidate; Smith//IRV might eliminate all 3 of them and elect someone else, whereas Tideman's Alternative Smith would eliminate everyone but the 3 and thus guarantee one of them wins.

Notes

Tideman's Alternative methods can be generalized to work with any voting method by modifying the second step to instead eliminate the loser of that voting method. For example, one could eliminate the Score voting loser (the candidate with the fewest points) instead of the FPTP loser. This is one way of making any voting method Smith-efficient which may be better than the simpler "eliminate everyone outside the Smith set and then run that voting method" procedure.