STLR voting

STLR voting (pronounced 'Stellar Voting') is an electoral system for single-seat elections, though it can be extended to a Multi-member system with a sequential elimination method. The name stands for "score then levelled runoff", and can be thought of as a utilitarian version of STAR voting. It is a type of cardinal voting electoral system and while the name is a clear reference to STAR voting it is actually a compromise between STAR voting and Score voting.

Voters cast ballots as in Score voting, rating each candidate from 0 to a maximum value, $$MAX$$. The two candidates with the highest total are selected as finalists, and then in the "levelled runoff" a winner is chosen. In the runoff the scores are levelled such that the higher of the two on each ballot is $$MAX$$. The ratio between the two finalists on each ballot stay constant while the ballots are levelled between one-another to ensure equal influence.

Method

 * 1) STLR voting uses a ratings ballot; each voter scores each candidate from 0 to $$MAX $$
 * A $$MAX$$ is typically either 5 or 9
 * Candidates who are left blank receive a 0.
 * 1) The scores for each candidate are then summed
 * 2) The two candidates with the highest sums go to the runoff.
 * 3) On each ballot, the scores of these two are levelled by multiplying both with the same number such that the higher scored candidate's score is equal to $$MAX$$
 * Explicitly, if the scores for the candidates are $$a$$ and $$b$$ then they are levelled to $$ a_{new} = a \frac{MAX}{max(a,b)}$$ and $$ b_{new} = b \frac{MAX}{max(a,b)}$$
 * 1) The levelled scores for each candidate are then summed and the winner is the one with the higher sum

Invention and Motivation
The concept was invented by Equal Vote Coalition Director Keith Edmonds and was first proposed publicly in July 2020.

The originally proposed runoff method was not levelling but the normalization from Instant Runoff Normalized Ratings (IRNR). Levelling chosen to be better shortly after. The version with IRNR normalization was previously and independently invented by user lucasvb on Reddit. The idea came from an attempt to solve the issue of arbitrary scale in Score voting. STAR voting solves this with the majoritarian runoff but as a result makes the system a majoritarian system.

Scale in Score Voting
When analyzing the ballots from the French studies and primary election for the Independent Party of Oregon   a well known criticism of score voting arose. This is that some voters tend to be too honest and to not score any candidate at the $$MAX$$ and $$MIN$$ values. In a score election the amount of influence a voter has is the difference between the maximum and minimum score they give a candidate. The standard response is that voters are told how the system works and it is up to them to decide how to use the scores.

A related issue is that of deciding who gets a score of 0. Only the worst or just those unfavoured? What is typically advised is for voters to give their the maximum value to their favourite and zero to any candidate they do not support.

The final issue is then to decide how to scale the scores of semi-supported candidates. Should a candidate who is half as supported as the favourite get half the score? Is this linear scaling what all voters do or do some use a different way?

Problem of being Majoritarian
STAR is intended to be a compromise between score voting and instant runoff voting. . It solves the above issue of scale because as long as a preference is shown between the two candidates in the runoff, then their influence is maximized. It is a majoritarian system as shown in the example

Red 51%: A[5] B[4] C[0] D[0] Blue 49%: A[0] B[4] C[0] D[5]

It elects A when both Score and Approval voting elect B. B is a compromise and the Utilitarian winner. STAR argues that it is better because it recovers the way that people would have voted given the top two utilitarian winners.

One can still solve the scale problem in score voting without resorting to majoritarianism with a different normalization before the runoff. STAR puts the favoured candidate to $$MAX$$ but also puts the other candidate in the runoff to 0. In order to preserve utility the amount of utility that this candidate is worth to the voter needs to be preserved instead of being zeroed. Some assumption needs to be made and the assumption for STLR voting is that that relative utility should be constant. This is the same as assuming the scale of linearity as in the previous section.

In summary, the favoured candidate is levelled up to $$MAX$$ so that all voters have the same maximum endorsement of the preferred candidate in the runoff. The less preferred candidate is given the same scaling so that the relative utility on the ballot is preserved. This means that the Utilitarian winner will be chosen in a runoff where each voter has equal influence.

Examples
An example was already given above showing that STLR voting is indeed Utilitarian. There are other situations where this helps vs either STAR or score.

Losing favourite
Suppose a voter in a [0,5] system gave candidates A, B and C the following scores; A:5, B:1, C:0. Clearly A is their favourite and they have expressed this. However, if A is unviable and the top two are B and C they have ruined their influence by giving B only a score of 1. Proponents of score voting argue that an informed voter should know the polling data and vote strategically as A:5, B:5, C:0. This is wrong for two reasons. Firstly, incentive to vote dishonestly should never be encouraged. Secondly, polls can be wrong or manipulated. If B was supported by the media then they could have their standing inflated so that people would vote for them as a lesser evil. In both STLR and STAR voting. The runoff between B and C converts the ballot to B:5, C:0 without any need for the voter to vote strategically.

Preserving Utility
Suppose a voter in a [0,5] system gave candidates A, B, C, D and E the following scores; A:5, B:4, C:2, D:1, E:0. If the runoff come to A and B STLR voting leaves the scores as is since one is already at $$MAX$$. However, STAR would convert them to A:5, B:0 which greatly exaggerates the voters preference between A and B.

If the runoff comes to C and D then STLR voting levels it to C:5, D:5*1/2=2.5. The impact on the final sum is 5-2.5 = 2.5. The impact from the score method would be 2-1=1 so significantly smaller. The impact in the STAR method is always 5 so significantly larger.

The voter themselves would always want their impact maximized but they also do not want tyranny of the majority. STLR finds the middle ground. No matter who is in the runoff the voter has their score levelled in a fair manner.

Simple Variant
Some find the concepts of multiplication in the levelling procedure difficult to understand. Because of this, a simplified variant of STLR voting exists. In this method the runoff is done with putting only the best of the 2 candidates to $$MAX$$. The other candidate keeps the score he already has to avoid all references to levelling or normalization. This pushes the method slightly more in the majoritarian direction but not so far as STAR.

Sequential elimination System
STAR is basically Cardinal Baldwin, but only doing the last round instead of all rounds. It is ONLY applying Baldwins method in the last step to normalize without losing monotonicity. STLR does a similar thing but with levelling as the normalization

A similar sequential elimination method can be made by eliminating the worst candidate then applying the levelling to all remaining candidates. A Multi-member system can be made from this by stopping while there are still multiple candidates remaining.

This puts this new system in a class of systems with Cardinal Baldwin and IRNR distinguished by their normalization mechanisms.