### Passing Monotonicity

These changes make the method significantly less “Monroe”, despite the original method being a constricted version of Monroe. The modified method also no longer requires “cheating steps” to ensure that it actually passes IIB and ULC. It is still optimization based.

To score a winner set, first, each ballot is split between each candidate in the winner set. The Monroe score for this division is the “base score”. Notice that it is just the

average of the unweighted scores for each candidate in the winner set.

Next: ballots (or portions of ballots) are switched between candidates. For a switch to be allowed, the following conditions must be met:

- Each candidate gains as many ballots as it gives.
- Each ballot (or portion of a ballot) that is switched gets reassigned to a candidate that it strictly prefers to its old designation.
- Notice that because of this rule, irrelevant ballots cannot be switched.
- Also notice that ULCs cannot trade any of their ballots.

The score for each switch is half the improvement of the ballot portion with the smallest gain from reassignment times the total weight of the switched ballots. The ballots must be switched in a way that maximizes the sum of the switch scores, while following the rules for switching. The total score of the winner set is the base score plus the switch scores.

The tiebreaker preferrably passes ULC factions. The base score inclusion is sufficient to pass Pareto.

It’s not at all obvious that this method is proportional. I have proven that in 2-party scenarios where the smaller party has exactly 1 Droop quota of support, it ties with the larger party for the final seat, for any number of total seats. While this is not sufficient to show that it generally passes PR, it suggests that it is possible.