[EM] AFB Ranked Pairs Attempt

[Gustav Thorzen]

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.

Hopefully of interest
Gustav