Tideman alternative method

Tideman's Alternative Methods, including Alternative Smith and Alternative Schwartz, are two electoral systems developed by Nicolaus Tideman which select a single winner using votes that express preferences. These methods can also create a sorted list of winners.

These methods are Smith- and Schwartz-efficient, respectively, and thus are Condorcet methods. They operate by using instant-runoff voting for cycle resolution.

Procedure

Tideman's Alternative Smith with three in the Smith set

Tideman's Alternative procedure is as follows:

Identify the Smith or Schwartz set.
If the set consists of one candidate, elect that candidate.
Eliminate all candidates outside the set and redistribute ballots.
Eliminate the plurality loser.
Repeat the procedure.

To create a sorted list of preferred candidates, select a winner, remove that winner from the list of candidates, and repeat.

Features

Tideman's Alternative Methods are easier to understand than other methods, such as Ranked Pairs and Schulze, owing to the simplicity of explaining both the Smith set (the smallest set of all candidates who each defeat every non-Smith candidate) and Instant run-off voting (eliminating the candidate with the fewest votes). This increases the likelihood of voter acceptance.

This method strongly resists both tactical voting and tactical nomination, reducing the amount of political manipulation possible or favorable in large elections. They inherit this resistance from instant run-off voting, as both methods resolve a Condorcet winner from the Smith set by eliminating non-Smith (or non-Schwartz) candidates and performing instant run-off voting on the result.

Although IRV itself faces criticism for theoretical and historical failures, all Smith- and Schwartz-efficient voting methods attempt to resolve a candidate from these respective sets. Unlike IRV, these methods invariably elect a Condorcet winner; when there is none, they elect different winners based on arbitrary criteria. Ranked Pairs elects the winner with the strongest overall ranking, while the Schulze method attempts to elect a winner without the worst pairwise loss. Tideman's Alternative Method elects a candidate in a manner strongly resisting tactical nomination and voting.

Tideman's Alternative Methods fail independence of irrelevant alternatives. However, the methods adhere to a less strict property, sometimes called independence of Smith-dominated alternatives (ISDA). It says that if one candidate (X) wins an election, and a new alternative (Y) is added, X will win the election if Y is not in the Smith set. ISDA implies the Smith criterion and Condorcet criterion.

Note that the Condorcet winner can be used as the "set" (if there is a Condorcet winner, they are the only member of the set. Otherwise, all candidates are in the set). This variation is known as Benham's method.

Comparison table

The following table compares Tideman's Alternative Methods with other preferential single-winner election methods:

Comparison of single-winner voting systems
Criterion Method	Majority winner	Majority loser	Mutual majority	Condorcet winner^{[Tn 1]}	Condorcet loser	Smith^{[Tn 1]}	Smith-IIA^{[Tn 1]}	IIA/LIIA^{[Tn 1]}	Cloneproof	Monotone	Participation	Later-no-harm^{[Tn 1]}	Later-no-help^{[Tn 1]}	No favorite betrayal^{[Tn 1]}	Ballot type
First-past-the-post voting	Yes	No	No	No	No	No	No	No	No	Yes	Yes	Yes	Yes	No	Single mark
Anti-plurality	No	Yes	No	No	No	No	No	No	No	Yes	Yes	No	No	Yes	Single mark
Two round system	Yes	Yes	No	No	Yes	No	No	No	No	No	No	Yes	Yes	No	Single mark
Instant-runoff	Yes	Yes	Yes	No	Yes	No	No	No	Yes	No	No	Yes	Yes	No	Ranking
Coombs	Yes	Yes	Yes	No	Yes	No	No	No	No	No	No	No	No	Yes	Ranking
Nanson	Yes	Yes	Yes	Yes	Yes	Yes	No	No	No	No	No	No	No	No	Ranking
Baldwin	Yes	Yes	Yes	Yes	Yes	Yes	No	No	No	No	No	No	No	No	Ranking
Tideman alternative	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	No	No	No	No	No	Ranking
Minimax	Yes	No	No	Yes^{[Tn 2]}	No	No	No	No	No	Yes	No	No^{[Tn 2]}	No	No	Ranking
Copeland	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	No	Yes	No	No	No	No	Ranking
Black	Yes	Yes	No	Yes	Yes	No	No	No	No	Yes	No	No	No	No	Ranking
Kemeny–Young	Yes	Yes	Yes	Yes	Yes	Yes	Yes	LIIA Only	No	Yes	No	No	No	No	Ranking
Ranked pairs	Yes	Yes	Yes	Yes	Yes	Yes	Yes	LIIA Only	Yes	Yes	No^{[Tn 3]}	No	No	No	Ranking
Schulze	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	Yes	No^{[Tn 3]}	No	No	No	Ranking
Borda	No	Yes	No	No	Yes	No	No	No	No	Yes	Yes	No	Yes	No	Ranking
Bucklin	Yes	Yes	Yes	No	No	No	No	No	No	Yes	No	No	Yes	No	Ranking
Approval	Yes	No	No	No	No	No	No	Yes^{[Tn 4]}	Yes	Yes	Yes	No	Yes	Yes	Approvals
Majority Judgement	No	No^{[Tn 5]}	No^{[Tn 6]}	No	No	No	No	Yes^{[Tn 4]}	Yes	Yes	No^{[Tn 3]}	No	Yes	Yes	Scores
Score	No	No	No	No	No	No	No	Yes^{[Tn 4]}	Yes	Yes	Yes	No	Yes	Yes	Scores
STAR	No	Yes	No	No	Yes	No	No	No	No	Yes	No	No	No	No	Scores
Random ballot^{[Tn 7]}	No	No	No	No	No	No	No	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Single mark
Sortition^{[Tn 8]}	No	No	No	No	No	No	No	Yes	No	Yes	Yes	Yes	Yes	Yes	None
Table Notes	^ ^a ^b ^c ^d ^e ^f ^g Condorcet's criterion is incompatible with the consistency, participation, later-no-harm, later-no-help, and sincere favorite criteria. ^ ^a ^b A variant of Minimax that counts only pairwise opposition, not opposition minus support, fails the Condorcet criterion and meets later-no-harm. ^ ^a ^b ^c In Highest median, Ranked Pairs, and Schulze voting, there is always a regret-free, semi-honest ballot for any voter, holding all other ballots constant and assuming they know enough about how others will vote. Under such circumstances, there is always at least one way for a voter to participate without grading any less-preferred candidate above any more-preferred one. ^ ^a ^b ^c Approval voting, score voting, and majority judgment satisfy IIA if it is assumed that voters rate candidates independently using their own absolute scale. For this to hold, in some elections, some voters must use less than their full voting power despite having meaningful preferences among viable candidates. ^ Majority Judgment may elect a candidate uniquely least-preferred by over half of voters, but it never elects the candidate uniquely bottom-rated by over half of voters. ^ Majority Judgment fails the mutual majority criterion, but satisfies the criterion if the majority ranks the mutually favored set above a given absolute grade and all others below that grade. ^ A randomly chosen ballot determines winner. This and closely related methods are of mathematical interest and included here to demonstrate that even unreasonable methods can pass voting method criteria. ^ Where a winner is randomly chosen from the candidates, sortition is included to demonstrate that even non-voting methods can pass some criteria.

References

Green-Armytage, James. Four Condorcet-Hare Hybrid Methods for Single-Winner Elections.

[condorcet-iia-incompatibility-1] ^ ^a ^b ^c ^d ^e ^f ^g Condorcet's criterion is incompatible with the consistency, participation, later-no-harm, later-no-help, and sincere favorite criteria.

[MinimaxVarient-2] A variant of Minimax that counts only pairwise opposition, not opposition minus support, fails the Condorcet criterion and meets later-no-harm.

[semipc-3] In Highest median, Ranked Pairs, and Schulze voting, there is always a regret-free, semi-honest ballot for any voter, holding all other ballots constant and assuming they know enough about how others will vote. Under such circumstances, there is always at least one way for a voter to participate without grading any less-preferred candidate above any more-preferred one.

[IIA_rating_methods-4] Approval voting, score voting, and majority judgment satisfy IIA if it is assumed that voters rate candidates independently using their own absolute scale. For this to hold, in some elections, some voters must use less than their full voting power despite having meaningful preferences among viable candidates.

[5] Majority Judgment may elect a candidate uniquely least-preferred by over half of voters, but it never elects the candidate uniquely bottom-rated by over half of voters.

[mjmmc-6] Majority Judgment fails the mutual majority criterion, but satisfies the criterion if the majority ranks the mutually favored set above a given absolute grade and all others below that grade.

[7] A randomly chosen ballot determines winner. This and closely related methods are of mathematical interest and included here to demonstrate that even unreasonable methods can pass voting method criteria.

[8] Where a winner is randomly chosen from the candidates, sortition is included to demonstrate that even non-voting methods can pass some criteria.

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