Defeat strength

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.
  • Example method: the Schulze method is usually taken to use winning votes.
• 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.
  • Example method: Tideman originally defined Ranked Pairs as a margins method.
• pairwise opposition: number of votes for W>L
  • Violates plurality criterion.
  • Example method: MMPO.
• 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 expressed to prefer A over B than expressed to prefer 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.