Sequential Monroe voting

Sequential Monroe is sequential Multi-Member System built on Score voting ballots. Each winner is that candidate who which highest sum of score in a Hare Quota of ballots. That Hare quota of ballots is then removed from subsequent rounds.

Discussion

If you want to sequentially maximize the Monroe Function, it can be argued Allocated Score Voting is the best approach for doing so. The reason is that if in the first round you just pick the candidate with the highest score, that candidate might be scored high on average because everybody is giving that candidate a moderate score, but there could be another candidate that has a max score specifically among the first hare quota of voters, and electing that candidate instead is more likely to maximize the Monroe function because that That candidate's hare quota will be full of 5's instead of 3's.

While the Monroe's function isn't a bad measure of Proportional Representation it is at least highly nonstandard. If you are going to use the function as a measure of how efficient a voting method is, then it is worth noting that the function does produce some logical contradictions: adding ballots that approve of all candidates being able to change which outcome Monroe's function deems the best. This property may not seem too important, but some of it's ramifications include failing the consistency criterion (i.e. the best result in multiple districts not being the best result in those districts combined). Minor logical contradictions are not that bad in voting methods as long as they don't impact the results too much and all sequential algorithms will have these anyways, but if you want a measure of the quality of an election result that you can use to text voting methods, then you want to measure the quality of an election result that avoids logical contradictions.

Definition

1. For candidate X, sort the ballots in order of highest score given to candidate X to lowest score given to candidate X.
2. Calculate the sum of score given to X on the first hare quota of those ballots. Record this score as that candidate's hare quota score
3. Elect the candidate with the highest hare quota score
4. Set the ballot weight to zero for the voters that contribute to that candidate's hare quota score.

• If there are several voters who have the same effective score (score x ballot weight) at the cusp of the Hare Quota then Fractional Surplus Handling is applied to those voters

1. Repeat this process until all the seats are filled.

As a bonus, there is no need for a conformation loop because if the voters in X's highest scores hare quota gave Y a higher score on average, then Y's hare quota score must be at least equal to X's since Y's hare quota score is equal to the average score among the hare quota of ballots that does the best job of maximizing that average score and X's quota is a possible contender for "the best job of maximizing that average score and X's quota".

Fractional Surplus Handling to break ties: when calculating which ballots belong to a candidate's quota, if for a particular score including voters that gave that candidate that score in the quota would make the quota to large and excluding it would make it to small, exhaust a portion of those vote's weights such that the total weight of the exhausted ballots still equals the hare quota. The reason why Fractional Surplus Handling is preferred is that it preserves the Independence of irrelevant alternatives and Monotonicity criteria that Monroe's method passes).