MARS voting
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MARS voting is a single-winner electoral system that combines cardinal and ordinal information. The name stands for "Mixed Absolute Relative Score", as it combines score voting with relative preferences. It was created to address shortcomings in STAR voting. In particular cloning and edge cases of favorite betrayal.
Ballots are cast as score ballots (in this Article we use a 0 to 5 rating). The candidate with the highest score is found. Then, in an automatic runoff step, ballots are examined for preference relative to that candidate. Every candidate scored higher than the score winner gets the maximum rating added to their score, every candidate lower gets the maximum rating subtracted from their score. The candidate with the highest combined (absolute and relative) score wins.
Voting
Voters fill out a score ballot with a 0 to 5 range (blanks count as 0). These are evaluated in four steps.
- The candidate with the highest total score is found ("score winner").
- Ballots are evaluated again. Each candidate is compared against ("competitor") the score winner . When scored higher they receive 5 points. When scored lower they lose 5 points.
- For every candidate their scores of step 1. and the ratings of step 2. are added together. Note that for the score winner this will be score+0. The candidate with the highest combined sum wins.
- Repeat step 2. and 3. with the previous winner in place of the score winner, until the process terminates or a cycle is found. In a cycle the candidate with the highest score from within the cycle is elected.
To resolve a tie perform an automated runoff between the candidates, using only the ranking information.
Examples
Clear Winner
Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near the capital as possible.
The candidates for the capital are:
- Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
- Nashville, with 26% of the voters, near the center of Tennessee
- Knoxville, with 17% of the voters
- Chattanooga, with 15% of the voters
The preferences of the voters would be divided like this:
| 42% of voters (close to Memphis) |
26% of voters (close to Nashville) |
15% of voters (close to Chattanooga) |
17% of voters (close to Knoxville) |
|---|---|---|---|
|
|
|
|
Suppose that 100 voters each decided to grant from 0 to 5 points to each city such that their most liked choice got 5 stars, and least liked choice got 0 stars, with the intermediate choices getting an amount proportional to their relative distance.
| Voter from/ City Choice |
Memphis | Nashville | Chattanooga | Knoxville | Absolute
score |
|---|---|---|---|---|---|
| Memphis | 210 (42 × 5) | 0 (26 × 0) | 0 (15 × 0) | 0 (17 × 0) | 210 |
| Nashville | 84 (42 × 2) | 130 (26 × 5) | 45 (15 × 3) | 34 (17 × 2) | 293 |
| Chattanooga | 42 (42 × 1) | 52 (26 × 2) | 75 (15 × 5) | 68 (17 × 4) | 237 |
| Knoxville | 0 (42 × 0) | 26 (26 × 1) | 45 (15 × 3) | 85 (17 × 5) | 156 |
Nashville is the score winner with 293 points. All other candidates are compared against it.
| Voter from/ City Choice |
Memphis | Nashville | Chattanooga | Knoxville | Relative
score |
|---|---|---|---|---|---|
| Memphis | 210 (42 × +5) | -130 (26 × -5) | -75 (15 × -5) | -85 (17 × -5) | -80 |
| Nashville | 0 (42 × 0) | 0 (26 × 0) | 0 (15 × 0) | 0 (17 × 0) | 0 |
| Chattanooga | -210 (42 × -5) | -130 (26 × -5) | 75 (15 × +5) | 85 (17 × +5) | -180 |
| Knoxville | -210 (42 × -5) | -130 (26 × -5) | 75 (15 × +5) | 85 (17 × +5) | -180 |
Adding absolute and relative scores together we get
210 - 80 = 30 for Memphis
293 + 0 = 293 for Nashville
237 - 180 = 57 for Chattanooga
156 - 180 = -24 for Knoxville
Nashville wins the first round. Since it also is the score winner no further count is needed. By being both the score and Condorcet winner the result is exaggerated in MARS voting, resulting is a clear victory for Nashville.
Cycle
| Voters | 35 | 33 | 34 | Absolute
score |
|---|---|---|---|---|
| A | 175 (35 × 5) | 132 (33 × 4) | 136 (34 × 4) | 443 |
| B | 175 (35 × 5) | 165 (33 × 5) | 0 (34 × 0) | 340 |
| C | 0 (35 × 0) | 165 (33 × 5) | 170 (34 × 5) | 335 |
A is the score winner. When comparing B and C against A the summed scores are: A 443+0=443, B 340-5=335, C 335+160=495. Therefor C wins the first round.
| Voters | 35 | 33 | 34 | Relative
score |
|---|---|---|---|---|
| A | 0 (35 × 0) | 0 (33 × 0) | 0 (34 × 0) | 0 |
| B | 0 (35 × 0) | 165 (33 × +5) | -170 (34 × -5) | -5 |
| C | -175 (35 × -5) | 165 (33 × +5) | 170 (34 × +5) | 160 |
In the second round we compare A and B against the previous winner C. The summed scores are: A 443-160=283, B 340+5=345, C 335+0=335. B wins the second round.
In the third round we compare A and C against the previous winner B. The summed scores are: A 443+5=448, B 340+0=340, C 335-5=330. A wins the third round.
We discovered that there is a cycle A>B>C>A. In case of a cycle the score winner from within that cycle is elected, here A.
Properties
MARS voting reduces the incentive for strategic voting in the form of burying, min-max or bullet voting. Voter can make use of the full range of scores with only a small probability of having a less preferred candidate beat their favorite because of the vote.
It satisfies the following criteria: equal vote criterion ("Frohnmayer balance"), monotonicity, participation, clones, favorite betrayal, precinct summability, reversal symmetry.
MARS voting intentionally fails the Condorcet winner criterion in cases where the score winner outweighs the Condorcet winner. For the same reason it also fails the Condorcet looser criterion and majority winner, but less so then pure score (consider 51 voters: A0 B1, 49 voters: A5, B0, A wins). Further failed criteria are: Later-no-harm, IIA.
Ties
By using two types of information MARS voting can resolve top ties in most cases. The amount of true ties that can not be resolved is reduced to a small fraction.
Precinct summability
Despite several rounds of comparing candidates, MARS voting is precinct summable. Ballots need to be counted only once. All we need to know are the scores and for every pair of candidates how many voters prefer one over the other. Scores can simply be added up. For the preference we turn a ballot into a preference matrix and add up those for all ballots. For the Tennessee example above the summed up matrix is exemplified below.
| ... over Memphis | ... over Nashville | ... over Chattanooga | ... over Knoxville | |
| Prefer Memphis ... | 0 | 42 | 42 | 42 |
| Prefer Nashville ... | 58 | 0 | 68 | 68 |
| Prefer Chattanooga ... | 58 | 32 | 0 | 83 |
| Prefer Knoxville ... | 58 | 32 | 17 | 0 |
In order to compare against Nashville single out Nashville and subtract competing votes.
| ... over Nashville | |
| Prefer Memphis ... | 42-58=-16 |
| Prefer Nashville ... | 0 |
| Prefer Chattanooga ... | 32-68=-36 |
| Prefer Knoxville ... | 32-68=-36 |
The combined score then is the score plus the number of remaining votes times the maximum range.
| ... over Nashville | |
| Prefer Memphis ... | 210 + (-16 x 5) = 30 |
| Prefer Nashville ... | 293 + 0 = 293 |
| Prefer Chattanooga ... | 237 + (-36 x 5) = 57 |
| Prefer Knoxville ... | 156 + (-36 x 5) = -24 |
Footnotes
Original proposal on the EndFPTP subreddit under the name "score better balance"
Implementation in Go by u/sxan