[EM] AFB Ranked Pairs Attempt

Sun Apr 19 19:54:08 PDT 2026

Hi Gustav,

I think I understand your definition, a Ranked Pairs based on majorities only,
ranking them essentially by losing side votes, fewest being strongest. So here are
some scenarios for you to consider. Let me know if you disagree on the winners.

Regarding favorite betrayal:

0.411: A>B>C
0.285: B=C>A
0.210: C>A>B
0.091: B>C>A

Defeat order is B>C, A>B, C>A. A wins.

0.411: A>B>C
0.285: C>A=B  (i.e. B has been "betrayed")
0.210: C>A>B
0.091: B>C>A

This strengthens A>B and weakens B>C.
Defeat order is now A>B, C>A, B>C. C wins.

(MMPO elects C in both scenarios. ICA elects B in the first, A in the second.)

---

Regarding Later-no-harm:

0.297: C>A>B
0.294: A>B>C
0.289: B>C>A
0.117: A>B=C

Defeat order is A>B, B>C, C>A. A wins.

0.297: C>A>B
0.294: A>B>C
0.289: B>C>A
0.117: A>C>B  (i.e. adding a C lower preference)

This weakens B>C without reversing it.
Defeat order is now A>B, C>A, B>C. C wins.

(MMPO elects C in both scenarios.)

---

Regarding Later-no-help:

0.327: C>A=B
0.269: A>C>B
0.262: B>A>C
0.110: C>A=B
0.030: A>B=C

Defeat order is C>B, A>C. B has no majorities. So A wins.

0.327: C>B>A  (i.e. adding a B lower preference)
0.269: A>C>B
0.262: B>A>C
0.110: C>A=B
0.030: A>B=C

This creates a B>A defeat that is stronger than A>C.
Defeat order is now C>B, B>A, A>C. C wins.

---

I do think the method is monotone. But you have to be careful if you decide to have
a tiebreaker rule.

I hope I didn't make any errors in my math.

Kevin
votingmethods.net

> So thanks to the feedback I got on my previous post,
> I made an attempt at modifying Ranked Pairs
> in hopes to create a voting system satisfying
> Avoids Favorite Betrayal (AFB), Majority-Beat Condorcet critiera,
> Monotonicity, Mutual Majority, and 1 of Later-No-Help/Harm.
> 
> But first I will do some definitions for clarification.
> A candidate Alice Plurality-Beats another candidate Bob
> if the number of votes strictly prefering Alice to Bob
> is greater then the number of votes strictly prefering Bob to Alice,
> or v(Alice > Bob) > v(Bob > Alice).
> We futher define that Alice Majority-Beats Bob at the minimal majority
> threshold if the number of votes strictly prefering Alice to Bob
> is strictly greater then half the total number of votes.
> I will use minimal majority of half as the majority threshold
> unless otherwise specified.
> If a voting method always without exception inferes complete
> preference orders from the ballots,
> and the total number of voter are predetermined and fixed
> (spoiled ballots and noshows still counts to the total),
> then Alice Majority-Beating Bob at the minimal majority threshold
> conicides with v(Alice > Bob) > v(Alice = Bob) + v(Alice < Bob).
> The classical Condorcet winner/loser candidate and criteria
> as well as Smith and ISDA implicitly assumes Plurality-Beats,
> but I will prefix the cirteria with PB and MB to avoid ambiguity.
> A result of PB vs MB is that the PB-Smith set is a subset of the MB-Smith set,
> and PB-Smith and PB-Condorcet criteras imply their MB counterparts,
> but it is possible to be a PB-Condorcet winner without being a MB-Condorcet winner
> as well as a MB-Smith set member without being a PB-Smith set winner.
> 
> The incompatability of PB-Condorcet cirteria with AFB and LN-Help/Harm
> does not apply to the MB-Condorcet criteria, so I have been trying to create
> a system satisfying AFB, MB-Condorcet critera, Monotonicity and 2 out of the
> following 3 Mutual Majority, LN-Help, LN-Harm in hopes of getting a useful
> baseline comparison.
> 
> Assuming the above assumptions for Majority-Beating coinciding
> with v(A>B) > v(A=B)+v(A<B) holds, I propose the following system:
> 1) Go through each pairwise matchup and check if it is a Majority-Beat,
> if so add it to the list in the form of
> Winner Majority-Beats Loser with Score
> where Score is the number of votes where the pairwise winner
> is ranked greater or equal to the pairwise loser,
> or Score = v(Winner > Loser) + v(Winner = Loser).
> This means pairwise matchups where neither Majority-Beats the other
> won't end up on the list.
> 2) Sort the list according to the highest score first, lowest last.
> 3) Go through the sorted list in order and lock in the pairwise matchup,
> and marking the pairwise looser of each entry as defeated,
> unless it would result in a cycle,
> in which case we skip the entry.
> 4) If there is only 1 candidate which have not been marked defeated
> after locking in the matchups, elect that candidate as the winner.
> If there are more then 1, do tie-breaking among them.
> There will always be at least 1 candidate which have not benn marked as defeated.
> 
> Compared to Rank Pairs, step (2), (3), and (4) are the same if I have correctly
> understood whats on electowiki correctly.
> https://electowiki.org/wiki/Ranked_pairs
> https://electowiki.org/wiki/Maximize_Affirmed_Majoritues
> https://electowiki.org/wiki/River
> Step (1) differ in two ways, only pairwise matchups resulting in Majority-Beats
> are considered, and how we score each matchup.
> What the tie breaker actually is was unclear,
> the links for random voter hierarchy went to the MAM page,
> which description appears to be specific to MAM.
> I will assume the tie-breaker used won't cause any confilict
> with criteria compatability.
> 
> So far for criteria compatability I have only managed to prove
> MB-ISDA which implies MB-Smith which implies Mutal Majority.
> MB-Condorcet winner criteria, and MB-Condorcet loser criteria.
> Every member of the MB-Smith set will have all their pairwise
> matchups against non-members of the MB-Smith set locked in,
> without risk of marking MB-Smith set members defeated,
> resulting in MB-ISDA satisfaction.
> Monotonicity should carry over from regular Ranked Pairs,
> but I am unsure how to fully prove it.
> I also believe it satisfy AFB and one of LN-Help/Harm based
> on my understand of ICA and MMPO, but no proof so far.
> I futher suspect that if set (2) is changed to sort the score in reverse,
> that is lowest first to highest last, we get the other of LN-Help/Harm.
> 