# 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.

## Voting

Ballots are cast as score ballots (here we use a 0 to 5 rating). A pairwise table shows each match. The pairwise scores (M) are calculated as number of votes that prefer (P) the candidate over the competitor times the maximum rating (r), plus the total score (S) for the candidate: M(A,B) = P(A,B) x r + S(A). From within the Schwartz set the hightest scoring candidate is elected.

To resolve a tie, an automated runoff if performed, using only the ranked 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)
1. Memphis
2. Nashville
3. Chattanooga
4. Knoxville
1. Nashville
2. Chattanooga
3. Knoxville
4. Memphis
1. Chattanooga
2. Knoxville
3. Nashville
4. Memphis
1. Knoxville
2. Chattanooga
3. Nashville
4. Memphis

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.

The following table shows preferences time 5 with score added.

 ... over Memphis ... over Nashville ... over Chattanooga ... over Knoxville Prefer Memphis ... --- 42x5 + 210 42x5 + 210 42x5 + 210 Prefer Nashville ... 58x5 + 293 --- 68x5 + 293 68x5 + 293 Prefer Chattanooga ... 58x5 + 237 32x5 + 237 --- 83x5 + 237 Prefer Knoxville ... 58x5 + 156 32x5 + 156 17x5 + 156 ---
 ... over Memphis ... over Nashville ... over Chattanooga ... over Knoxville Prefer Memphis ... --- 420 420 420 Prefer Nashville ... 583 --- 633 633 Prefer Chattanooga ... 527 397 --- 652 Prefer Knoxville ... 446 316 241 ---

By being both the score and Condorcet winner the result is exaggerated in MARS voting, resulting is a clear victory for Nashville.

### Cycle

Suppose there are three candidates A, B, C and three groups of voters.

• 35 voters: A5, B5, C0
• 33 voters: A4, B5, C5
• 34 voters: A4, B0, C5
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

Resulting in the following pairwise matrix.

 ... over A ... over B ... over C Prefer A ... --- 34x5+443=613 35x5+443=618 Prefer B ... 33x5+340=505 --- 35x5+340=515 Prefer C ... 67x5+335=670 34x5+335=505 ---

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, 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

Like most Condorcet methods, 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. The results are tabulated in a pairwise matrix as seen in the examples above.

## Footnotes

Original proposal on the EndFPTP subreddit under the name "score better balance"

Implementation in Go by u/sxan