# FpA-fpC

fpA-fpC (for first preference A minus first preference C) is a three-candidate Condorcet method based on first preference Copeland. It was first proposed by Kristofer Munsterhjelm in 2016.[1] Its election cases are:

• If there's a Condorcet winner, then that candidate wins.
• If the Smith set is size two, then the winner is according to majority rule.
• If the Smith set is size three, then for each candidate, assume without loss of generality that the candidate is A in an A>B>C>A cycle. A's score is A's first preferences minus C's first preferences. The candidate with the highest score wins.

This method shares the strategy resistance of Smith-IRV hybrids, such as chicken dilemma compliance and dominant mutual third burial resistance; yet, unlike them, is monotone. It is open (not obvious) how to extend the method to more than three candidates in a way that retains both monotonicity and strategy resistance.

## Chicken dilemma

Any generalization will preserve its chicken dilemma compliance, as that criterion is only defined on three-candidate elections. However, this is not true of dominant mutual third burial resistance.

It produces similar results to electing the winner according to Carey's Improved First Past the Post method if there is a Condorcet cycle, and electing the Condorcet winner otherwise.

## Partial generalizations

In trying to extend fpA-fpC to pass DMTBR for any number of candidates, methods have been found that are monotone and pass the desired criteria when the dominant mutual set consists of a single candidate. Call these weaker criteria DMTC (elects the DMT candidate in this case) and DMTCBR (dominant mutual third candidate burial resistance).

The following ranked voting methods are monotone and pass DMTC and DMTCBR:

• fpA - sum fpC: Let A's score be A's first preferences minus the sum of first preferences of everybody who beats A pairwise. Highest score wins.
• fpA - max fpC: Let A's score be A's first preferences minus the first preferences of the candidate with the most first preferences who beats A pairwise. Highest score wins.
• IFPP-like generalization: If the two candidates with the most first preferences have more than a third of the first preferences each, elect the one who beats the other pairwise. Otherwise elect the Plurality winner.[nb 1]
• Friendly Cover: Let A's friends be every candidate A weakly covers, and A's defeaters every candidate who defeats A pairwise. A's score is the sum of friends' first preferences minus the sum of defeaters' first preferences. Highest score wins. With the right type of weak covering relation, this passes a restricted clone independence criterion.[2]

fpA - sum fpC is less vulnerable to cloning than fpA - max fpC, but has more serious Plurality failures.

These generalizations are not automatically Condorcet, but can be made to pass Condorcet without failing any of the three criteria. E.g. Smith,(fpA - sum fpC) passes all of Smith, DMTC, DMTCBR, and monotonicity.

## Historical emails

This is a list of emails to the EM-list, originally compiled by User:Kristomun in a comment on the "Talk:" page for this article, but possibly augmented by others after February 2022:

• "Strategy-resistant monotone methods" 2016-02-07 (User:Kristomun)
• "Re: MJ -- The easiest method to 'tolerate'" 2016-09-22 (User:Kristomun)
• "Resolvable weighted positional systems all fail independence of clones" 2017-12-03 (User:Kristomun)
• "Fwd: What are some simple methods that accomplish the following conditions?" 2019-06-01 (Forest Simmons)
• "Re: Improved Copeland" 2019-06-07 (User:Kristomun, also Forest Simmons)
• "Unmanipulable majority and Condorcet" 2020-01-03 (User:Kristomun, also Forest Simmons)
• "Minimally manipulable methods: preliminary results" 2020-11-02 (User:Kristomun; first post about the MIP minimally manipulable method search)
• "Re: A Metric for Issue/Candidate Space" 2020-12-23 (Kevin Venzke)
• "Re: Best Ranked Preference Deterministic Method?" 2020-12-24 (User:Kristomun, Kevin Venzke, Forest Simmons)
• "Re: extending fpA-fpC" 2020-12-26 (Kevin Venzke)
• "Re: Agenda Based Banks" 2021-08-04 (Forest Simmons)
• "Re: Defeat Strength Demystified" 2021-09-17 (Forest Simmons)
• "Re: 'Independence of cycles' and a possible new method" 2021-12-12 (User:Kristomun, Toby Pereira, Colin Champion, Forest Simmons)
• "Re: Quick and Clean Burial Resistant Smith, compromise" 2022-01-09 (User:Kristomun, Kevin Venzke, Daniel Carrera, Forest Simmons)

## Notes

1. If an order of finish is required, then in the first case rank the pairwise winner first, the loser second, and everybody else by Plurality order. In the second case, rank everybody in Plurality order.

## References

1. Munsterhjelm, K. (2016-02-07). "Strategy-resistant monotone methods". Election-methods mailing list archives.
2. Munsterhjelm, K. (2022-09-12). "Friendly Cover, a burial-resistant monotone almost cloneproof method". Election-methods mailing list archives.