Sequential Monroe (SMV) is a sequential Multi-Member System created by Parker Friedland that is built on Score voting. Each winner is that candidate with the highest possible sum-of-score in a Hare Quota of the remaining ballots. That Hare quota of ballots is then removed from subsequent rounds.
If you want to sequentially maximize the Monroe Function, it can be argued that this method is a better approach for doing so then Allocated Score. 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.
- For candidate X, sort the ballots in order of highest score given to candidate X to lowest score given to candidate X.
- Calculate the sum of score given to X on the first hare quota of ballot weight. Record this score as that candidate's hare quota score
- If Fractional Surplus Handling is used, any fractional ballots included in the Quota will have their contribution to that sum reweighted corresponding to their non-exhausted weight.
- Elect the candidate with the highest hare quota score
- If two candidates tie for having the highest score among their hare quota then elect whichever of the tied candidates has the highest sum of score among all the non-exhausted ballot weight.
- 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 given the same score at the cusp of the Hare Quota then Fractional Surplus Handling is applied to those voters
- Repeat this process until all the seats are filled.
Fractional Surplus Handling is used 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 too 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).
SMV passes a stronger property related to Proportionality for Solid Coalitions than most cardinal PR methods: a solid coalition comprising k Hare quotas can force the election of at least k of their preferred candidates by max-scoring them; this is because their preferred candidates will have the highest possible Monroe scores, since they have maximal support from their most-supporting Hare quota of voters. Most other cardinal PR methods further require that the solid coalition min-score all non-preferred candidates in order to receive this guarantee.