Defeat strength: Difference between revisions

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Condorcet methods require the resolution of cycles. Typically, this is done by ignoring "weak" pairwise defeats in favor of "strong" ones. The metric used to distinguish these is called defeat strength.

Most election experts agree that, out of the standard ways to measure defeat strength, winning votes are the best, with margins in second.

List of measures

Standard

winning votes (wv) = number of votes for W>L if greater than the number of votes for L>W, otherwise zero.
margins = (number of votes for W>L) - (number of votes for L>W)
- Gives more strategic incentive than wv but may be easier to understand
pairwise (non)opposition: number of votes for W≥L, or equivalently 1 - votes for L > W
- Strong intuitive appeal (pick the candidate opposed by the least voters)
- Example method: MMPO
  - Gives even less strategic incentive than wv (satisfies later-no-help and favorite betrayal).
  - Violates plurality criterion. A complete unknown can win with no real support, just because everyone forgot to rank them on their ballot.
Relative margins: Defeat strength = margin ÷ (votes for W≠L)

Cardinal

Approval-based support = no. of voters approving of the winner but not of the loser of the defeat. Gives special influence to preferences which cross the approval cutoff and thus helps diminish certain strategies. Useful when one assumes that only these voters will support the corresponding "majority complaint"
Cardinal rated strength = sum of difference in the candidates' cardinal ratings on all ballots which rate the winner over the loser of the defeat. Even more strategy-resistant than wv, but involves interpersonal comparisons of cardinal ratings.
Winning approval = approval score of the winner of the defeat. Using this as defeat strength leaves only one immune candidate: the least approved of those who beat all more approved ones. Similar for other scores.

Kinds of defeats

(Pairwise) defeat = more voters prefer A over B than B over A
Majority-strength defeat = pairwise defeat which has a wv-strength of more than half the no. of voters. Using only such defeats can reduce incentive to truncate by reducing the likelihood that additional preferences will harm earlier ones. Voters adding a preference can create a majority-strength win, but they can't reverse the direction of one.

References

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 === Standard ===
 * '''winning votes (wv)''' = number of votes for W>L if greater than the number of votes for L>W, otherwise zero.
-** Example method: the [[Schulze method]] is usually taken to use winning votes.{{cn|date=May 2024}}
 * '''margins''' = (number of votes for W>L) - (number of votes for L>W)
-** Gives more strategic incentive than wv but may be easier to understand.
+** Gives more strategic incentive than wv but may be easier to understand
-** Example method: Tideman originally defined [[Ranked Pairs]] as a margins method.<ref name="Tideman2">{{Cite journal |last=Tideman |first=T. N. |date=1987-09-01 |title=Independence of clones as a criterion for voting rules |url=https://doi.org/10.1007/BF00433944 |journal=Social Choice and Welfare |language=en |volume=4 |issue=3 |pages=185–206 |doi=10.1007/BF00433944 |issn=1432-217X}}</ref>
 * '''pairwise (non)opposition''': number of votes for W≥L, or equivalently 1 - votes for L > W
-** Gives even less strategic incentive than wv, and has a strong theoretical appeal (pick the candidate opposed by the fewest voters)
+** Strong intuitive appeal (pick the candidate opposed by the least voters)
+** Example method: [[MMPO]]
-** Violates [[plurality criterion]].
+*** Gives even less strategic incentive than wv (satisfies later-no-help and favorite betrayal).
-*** A complete unknown can win with no real support, just because everyone forgot to rank them on their ballot.
+*** Violates [[plurality criterion]]. A complete unknown can win with no real support, just because everyone forgot to rank them on their ballot.
-** Very little strategic incentive
-** Example method: [[MMPO]].
 * '''Relative margins''': Defeat strength = margin &divide; (votes for W&ne;L)