Vote unitarity
Vote Unitarity is the concept of the Equally Weighted Vote or One-Person-One-Vote as applied to multi-winner election methods. Vote Unitarity ensures that each person should have one vote and that vote should not change in power during the rounds of tabulation in any system. More mathematically, it is the condition that the time evolution of the vote according to the tabulation procedure is mathematically represented only by Unitary Transformations. This means that ballot weight can be split between winners but never created or destroyed during the voting systems calculation of winners.
Rationale
When Single Transferable Vote allocates voters to winners it can violate vote unitarity by over removing influence in some cases. This occurs in all allocation systems; for example in Allocated Score somebody who only gave a score of 1 to the winner could lose all future influence. Reweighted Range Voting on the other hand only reduces influence fractionally so a voter who got a candidate they gave max score in the first round would only have their ballot weight reduced to 1/2. This violates the principle of one person one vote since this voter would essentially be allowed to vote with half weight in later rounds after "winning". Proponents of Single Transferable Vote would use this argument for its superior fairness over Reweighted Range Voting and the Reweighted Range Voting use the opposite argument. Since Reweighted Range Voting and Single Transferable Vote are very popular systems which violate Vote Unitarity in opposite ways it should be possible to find a balanced middle ground, ensuring that every voter has an equally weighted vote, while maintaining other desirable features.
On an even further extreme, "Choose One" Plurality Bloc voting when treated as a sequential method often violates Vote Unitarity even more than Reweighted Range Voting since a voter can fully influence the election of multiple candidates independently without any reweighing. Cumulative Voting attempts to mitigate this by giving voters the same number of "votes" beforehand with the understanding that it is up to them to chose how to distribute their vote's weight on their ballot. As a result, Cumulative Voting elections have higher Proportional Representation than standard Bloc Systems. Thiele methods such as Reweighted Range Voting violate Vote Unitarity less than Bloc elections because they at least reduce ballot weight to some degree. In addition they do this reweigting in such a way to satisfy the Hare Quota Criterion.
Example
In a 5-winner election, if there is a candidate that 90% of voters maximally support, and that the other 10% of voters don't support, supposing this candidate is the first one elected, with Thiele reweighting as in RRV, the 90% of voters supporting that candidate will have their ballot weight reduced from 100% to 50%, whereas with Vote unitary-based reweighting like Sequentially Spent Score, they'd have their ballot weight reduced to 1-1/(90%*5) = 77%. Note that Thiele's reweighting stays the same no matter how seats are to be filled, whereas Vote Unitarity takes this into account. Vote Unitarity takes into account the popularity of a candidate when deciding how much ballot weight should be spent.
Relation to Similar Concepts
Each voter gets one vote/ballot
This is the most literal interpretation, and it’s passed by pretty much every serious system. It is assumed to be the starting state for Vote Unitarity. Preserving this concept throughout tabulation is equivalent to Vote Unitarity.
Each vote/ballot has the same weight
The weight of each voters ballot is given the same initial weight. This is the interpretation that the U.S. Supreme Court holds states to. It’s failed by single-winner methods which do not ensure an Equally Weighted Vote and are thus vulnerable to Vote Splitting, and it is failed in unequally-populated districts, and in the Electoral College. This concept is independent from Vote Unitarity. If a voter's weight is initially unequal, Vote Unitarity will maintain that inequality.
The Test of Balance
The Test of Balance is defined as the following "Any way I vote, you should be able to vote in an equal and opposite fashion. Our votes should be able to cancel each other’s out."
Vote Unitarity is not incompatible with this but the concept of a utilitarian multi-winner score system is. These systems do not aim to cancel out the will of opposing groups and leave them with nothing. They aim to find an compromise for all conflicting voters. Vote Unitarity helps to ensure fairness in the compromise.
Compliance
Vote Unitarity can be turned into a criterion in specific ways for specific classes of systems.
Single member systems
In single member systems this property is defined by the Equal Vote Criterion.
Multi-member systems
In sequential multi-member systems this concept become especially relevant due to the different rounds of tabulation. Specifically, a voter whose favorite has been elected should not have influence over subsequent rounds. On the other side, a voter who has not been fully satisfied should still have some level of influence. This means that systems which allocate votes such as Single Transferable Vote and Sequential Monroe violate Vote Unitarity if they allocate the whole vote weight to a candidate the voter did not express maximal endorsement for. In ordinal systems it is not possible to know how much influence should be lost at each round since only relative endorsement is given. In cardinal voting systems the influence of each voter in each round goes down proportionally in relation to the amount of representation they have won in previous rounds.
Partisan systems
The versions of party-list proportional representation which comply with vote unitarity are those which follow a largest remainder method like the Hamilton method. This is because it apportions evenly.
History
Keith Edmonds saw a unification of Proportional Representation and the concept of one person one vote which was maintained throughout winner the winner selection method. He coined the term "vote unitarity" for the second concept[1] and designed a score reweighting system which satisfied both Hare Quota Criterion and Vote Unitarity. As such it would preserve the amount of score used through sequential rounds while attributing representation in a partitioned way. It would assign Hare Quotas of score to winners which allowed for a voters influence to be spread over multiple winners. The final system was originally proposed in a late stage of the W: 2018 British Columbia electoral reform referendum but was not selected for the referendum ballot. This system, Sequentially Spent Score, was the first sequential Multi-Winner Cardinal Voting System built on Score Voting ballots to satisfy Vote Unitarity. Variants were soon found.