# Vote unitarity

Vote Unitarity is a three part criteria relating to the concept of 'One person, One vote' and to the concept of an Equally Weighted Vote as applied to multi-winner, sequential, proportional election methods.

### Vote Unitarity requires:

1.) An Equal Vote: The weight and worth of each vote must be equal. Each voter must have the ability to cast an equally weighted vote to that of any other voter, including the ability to support as many candidates as they like, and to support candidates equally. An equally weighted vote is the legal definition of one person, one vote.

2.) Proportionate Spending: The total cost to elect a candidate voters support at each level will be the same for every voter and for every candidate elected. For any voter, the cost in vote weight to elect a candidate shall always equal the amount of support given to that candidate on a voter's ballot, minus that voter's share of any surplus votes received by that candidate. The highest possible cost to elect a voter's favorite is equal to the full weight of their vote.

3.) Unitary Transformations:  A voter's total vote weight cannot be created or destroyed but may be spent to elect winners who represent them through the rounds of tabulation, winner selection, and reweighting. The amount of remaining vote weight available to be used by any voter at any stage in the tabulation must always be their initial vote weight minus the amount they have spent. A voter who has achieved full representation by electing their favorite will have no vote weight left in play in subsequent rounds unless their favorite candidate received a surplus of votes and didn't need the full weight of their supporters votes to be elected.

## Rationale and Critique

When Single Transferable Vote allocates entire 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.

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 stands to reason that it should be possible to find a balanced middle ground, ensuring that every voter has an equally weighted vote and that power of the vote is able to be used efficiently to help voters achieve representation which is as fair and equitable as possible, while maintaining other desirable features.

This rational became the foundation of a new class of Unitary Proportional voting methods which were invented to satisfy Vote Unitarity and Hare Proportionality, including Sequentially Spent Score Voting.

The principle of proportionate spending can also be critiqued in that it increases strategic incentives. Note that "proportionate spending" only refers to how a ballot is reweighted after a candidate is elected. This leaves open the question of how much power a vote has to elect a candidate in the first place. Thus, there are two possibilities.

1. If a rating of (for example) 3 out of 5 has only 3/5 the power (or less) to elect a candidate in the first place, then the strategic incentive is to give higher ratings to your favorite among the most borderline-viable candidates, in order to get that candidate elected.
2. Following the same example, if a rating of 3 out of 5 has more than 3/5 the power to elect a candidate in the first place, the strategic incentive is to give lower ratings, so that the amount of your ballot that is used up when a candidate is elected is less than the voting power you exerted in electing them.

This leads some voting theorists, such as Jameson Quinn, to consider vote unitarity an actively harmful criterion, as opposed to otherwise-similar methods which tend to allocate votes more fully.

## Compliant Voting Methods

Sequentially Spent Score (SSS) is a sequential, multi-winner proportional, cardinal voting method. Voters score candidates, generally from 0-5, using Score voting ballots. Each round's winner is the candidate who has the highest sum total score. When tabulating the ballots, each voter begins with 5 stars to spend in order to gain representation and voters spend those stars when a candidate they supported is elected. If a voter scored a candidate 3 stars, that voter could only spend up to three stars to help elect that candidate.-- Voters cannot influence subsequent rounds more than the stars they have remaining.

Sequentially Spent STAR voting is a variation of Sequentially Spent Score in which a runoff is added to the final seat up for election so that the two highest scoring candidates are finalists and the finalist preferred by more voters wins the seat. This variation is helps to incentivize voters to show their full preference order.

## Discussion of Common Failure Modes

### Failure to allow voters to cast an Equal Vote:

Most voting methods and implementations ensure that the initial weight of a vote is equal between voters, but in order to ensure that it's possible to cast an equally powerful vote, there can be no caps on the number of candidates who can be ranked or rated, and there can be no limit to the number of candidates who can be ranked or rated at any level.

Most ranked proportional methods, including Single Transferable Vote, and some proportional Condorcet methods do not allow voters to rank candidates equally, and some cap the number of candidates which can be ranked. These ballot limitations prevent a vote cast from carrying equal weight when compared with other possible votes cast, and put voters at a measurable disadvantage if they have more candidates on their side.

Cumulative Voting, where voters can distribute a limited number of votes between their candidates, is another example of a method which does not allow voters to cast an equally weighted vote because of the limit on the number and degree that voters can support their candidates.

In voting methods which don't ensure an Equal Vote, similar candidates will dilute each others' votes, resulting in vote-splitting, higher frequency of the Spoiler Effect, and less accurate and representative outcomes the more candidates are in the race.

### Failure to ensure Proportionate Spending:

In Allocated voting methods the principle of Proportionate Spending is violated when a voter's full vote weight is spent on a candidate, even if they didn't fully support that candidate.

For example, in Allocated Score, somebody who only gave a score of 1 out of 5 to the winner could lose all future influence in some scenarios. This can be thought of as 'overcharging' the voter, because from their perspective if they are not fully represented, they should still have a chance to earn more representation in subsequent rounds of the election's tabulation.

Single Transferable Vote has the same issue of overcharging a voter for an outcome that is not ideal for them due to the fact that a ranked ballot only allows a voter to show their preference order, not their level of support. For example, if a voter liked three candidates equally but was forced to rank them, then support for their 3rd choice was actually just as strong as their support for their first choice. When a 3rd choice is actually just as good as a favorite, it is fair for a vote to use their full vote weight to elect that candidate, but if their 3rd choice was a lesser-evil candidate who the voter didn't really like, it would be unfair for them to use up their full vote. Because proportionate spending cannot be ensured, and because the cost to elect a less preferred candidate may be unfairly high, STV does not pass vote unitarity.

Reweighted Range Voting, which reduces influence fractionally, has the inverse problem; where the cost to elect a candidate may be unfairly low and may also be inconsistent between voters. For example, a voter who elected a candidate they gave a max score to in the first round would have their ballot weight reduced by only ½. Then, in the next round, the cost in vote weight to elect a candidate they also gave a max score to would only be 1/4. This violates the Proportionate Spending principle since the cost to elect candidates who a voter likes equally is not consistent though the rounds of tabulation.

For 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%.

In contrast, with Vote Unitary-based reweighting like Sequentially Spent Score, they'd have their ballot weight reduced to 1-1/(90%*5) = 77%. Note that in Sequentially Spent Score the reweighting cost to elect a candidate remains the same no matter which round a candidate is elected in, and the only variables as to how much the cost is to elect a candidate are the amount of support given by each voter and the number of voters who supported that candidate.

Cumulative Voting does not ensure proportionate spending because it doesn't use surplus handling. Thus, supporters of candidates who win in a landslide will end up paying more vote weight than was needed to elect their candidate, and this creates biases in much the same way a gerrymandered district might.

Bloc voting methods violate Proportionate Spending even more than any proportional methods since a voter can fully influence the election of multiple candidates independently without any reweighting. This is why Bloc Voting is majoritarian and Bloc systems are not proportional representation.

### Failure to ensure Unitary Transformations:

Reweighted Range Voting doesn't ensure unitary transformations because of the fractional nature of the system. In RRV, a voter can only ever spend a fraction of their vote weight, which means that they have less than a full vote's weight at their disposal in any given round.

## 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.

## Definitions

Vote Weight = The weight and worth of a person's vote. The amount of power that a voter has to elect candidates in each round. Ballots are fully weighted at the beginning of tabulation, and ballots may be reweighted to reflect that some or all of a vote's weight has been spent to win representation.

## Priceability

Vote Unitarity is related to the more recent concept of priceability[2] A committee is priceable if there is a price such that voter spending can be arranged in such a way that each committee member gets a total spending of exactly the price, and voters do not have enough money left to buy additional candidates. The intuition behind this condition is that it encodes that each voter has (approximately) equal influence on the committee (since each voter starts out with an equal budget), and this ensures proportionality.