# Technically Exists

Within social choice theory, there are two major approaches to evaluating voting methods: pass/fail analysis and probabilistic analysis. Pass/fail analysis primarily consists of defining mathematical criteria that seem desirable to have, then proving which methods pass them and which methods fail them. It also has other aspects like proving that certain sets of criteria cannot all be satisfied by the same voting method; this is where the famous Arrow’s impossibility theorem comes from. On the other hand, probabilistic analysis does away with this and instead assesses voting method performance using simulations and, where possible, data from real-world elections. This approach has its downsides, but overall I think it is by far the superior evaluation method.

As a simple example of the limitations of pass/fail analysis, I’d like to consider reversal symmetry. Reversal symmetry is passed if reversing the preferences of every voter will always result in a different winner when there are at least two candidates running. The idea behind this criterion is that voting methods should not consider the best candidate to also be the worst candidate. Single-choice plurality fails reversal symmetry, since it is possible for one candidate to have both a plurality of first-choice support and a plurality of last-choice “support”, which becomes first-choice support if preferences are inverted. On the other hand, approval voting passes reversal symmetry, since the candidate with the most approvals cannot also be the candidate with the least approvals.1

1. For simplicity, we ignore all elections in which ties occur, as is common in pass/fail analysis.

## The motivation behind SPSV, part 5

In the last post, we went over how RRV extends SPAV to work with 0-n rated ballots. I mentioned that SPSV also extends SPAV to work with such ballots, but that it does so using a different method. In this post, I want to start off by explaining how this method works and why I find it superior to RRV’s approach. After this, we can move on to the full explanation of the motivation behind SPSV and to comparisons with other proportional methods.

### Section A: The Kotze-Pereira transformation

The extension method that SPSV uses is called the Kotze-Pereira transformation, or KP transform for short. The basic idea behind the KP transform is that it’s possible to use approval ballots as building blocks for creating a score ballot. As an example, let’s use a 0-3 rated ballot that rates candidate A 0, candidate B 1, candidate C 2, and candidate D 3. We can construct this ballot out of 3 approval ballots by having the first ballot approve B, C, and D, the second approve C and D, and the third approve only D. That way, each candidate’s score equals the number of approvals they have.

This isn’t our only option though. We could also use 1 ballot that approves B and D, and 2 ballots that approve C and D. This is a problem because not only do we want to construct score ballots from approval ballots, we also want to decompose score ballots into approval ballots. Thus, we need a means of choosing a unique set of approval ballots for each score ballot.

## The motivation behind SPSV, part 4

In the last post, we considered how to improve upon the strange behavior that D’Hondt has when dealing with voters that have preferences which don’t match up with party lines. We went over how SPAV modifies the approach that D’Hondt used by reweighting individual ballots instead of a party’s full set of votes, and how this allows it to handle more complex voter groups in a sensible manner. This gave voters more freedom to express their preferences, but they still had to either fully support or not at all support each candidate.

Can SPAV be adapted to use 0-n rated ballots instead of approval ballots? As it turns out, this is fairly straightforward to do. Instead of using the formula 1/(1 + m), we can use 1/(1 + s/n), where s is the sum of all the scores given to candidates who have already been elected. This method is known as Reweighted Range Voting (RRV), where range voting is used as a synonym for score voting.

## The motivation behind SPSV, part 3

In the previous post we went over how going from closed party-list to D’Hondt allowed voters to have a say in which candidates are elected from each party while still maintaining proportionality. However, we were still using single-mark ballots like those employed under plurality voting, also known as first-past-the-post. This meant each voter could only support one candidate along with that candidate’s party.

This causes two problems, one with allocating seats to parties and one with assigning seats a party won to that party’s candidates. First of all, because voters can only support a single party, similar parties risk splitting the vote with each other. Because seats are assigned proportionally, this isn’t too much of a problem, but when D’Hondt has to decide how to round the level of support for each party to allow an integer number of seats to be assigned, it will tend to favor larger parties at the expense of smaller ones. Tweaking the formula can change what size of parties are favored, but it cannot remove the favoritism.

The second problem is that within a party, seats are assigned using what is essentially multi-winner plurality voting. This leads to a potentially much more severe vote-splitting problem than what occurs between parties. Thus, factions within a party are incentivized to run the minimum number of candidates that they predict will be able to win seats, which will often just be 1 or 2. If a party is uniform enough to not have factions, then internal vote-splitting will likely be far less of a concern than it is for other parties.