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Simplify explanation and article by using a pairwise table
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(Simplify explanation and article by using a pairwise table)
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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.
 
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.
#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 theif candidatesperformed, using only the rankingranked information.
 
==Examples==
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{{Tenn voting example}}
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.
{| class="wikitable" border="1" style="empty-cells: show"
|-
!Voter from/<br />City Choice
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|}
 
Nashville is the score winner with 293 points. All other candidates are compared against it.
 
The following table shows preferences time 5 with score added.
{| class="wikitable" border="1" style="empty-cells: show"
 
{| class="wikitable"
|
|... over '''Memphis'''
|... over '''Nashville'''
|... over '''Chattanooga'''
|... over '''Knoxville'''
|-
|Prefer '''Memphis''' ...
!Voter from/<br />City Choice
| ---
!Memphis
|42x5 + 210
!Nashville
|42x5 + 210
!Chattanooga
|42x5 + 210
!Knoxville
!Relative
score
|-
|Prefer '''Nashville''' ...
|Memphis
|58x5 + 293
|210 (42 × +5)
| -130 (26 × -5)-
|68x5 + 293
| -75 (15 × -5)
|68x5 + 293
| -85 (17 × -5)
| -80
|-
|Prefer '''Chattanooga''' ...
|Nashville
|58x5 + 237
|0 (42 × 0)
|32x5 + 237
|0 (26 × 0)
| ---
|0 (15 × 0)
|83x5 + 237
|0 (17 × 0)
|0
|-
|Prefer '''Knoxville''' ...
|Chattanooga
|58x5 + 156
| -210 (42 × -5)
|32x5 + 156
| -130 (26 × -5)
|17x5 + 156
|75 (15 × +5)
|85 (17 × +5)---
|}
| -180
 
{| class="wikitable"
|
|... over '''Memphis'''
|... over '''Nashville'''
|... over '''Chattanooga'''
|... over '''Knoxville'''
|-
|Prefer '''Memphis''' ...
|Knoxville
| -210 (42 × -5)-
|bgcolor="#EE8080"|420
| -130 (26 × -5)
|bgcolor="#EE8080"|420
|75 (15 × +5)
|bgcolor="#EE8080"|420
|85 (17 × +5)
| -180
|Prefer '''Nashville''' ...
|bgcolor="#80EE80"|583
| ---
|bgcolor="#80EE80"|633
|bgcolor="#80EE80"|633
|-
|Prefer '''Chattanooga''' ...
|bgcolor="#80EE80"|527
|bgcolor="#EE8080"|397
| ---
|bgcolor="#80EE80"|652
|-
|Prefer '''Knoxville''' ...
|bgcolor="#80EE80"|446
|bgcolor="#EE8080"|316
|bgcolor="#EE8080"|241
| ---
|}
 
By being both the score and Condorcet winner the result is exaggerated in MARS voting, resulting is a clear victory for Nashville.
Adding absolute and relative scores together we get
 
===Cycle===
210 - 80 = '''30''' for Memphis
 
Suppose there are three candidates A, B, C and three groups of voters.
293 + 0 = '''293''' for Nashville
 
* 35 voters: A5, B5, C0
237 - 180 = '''57''' for Chattanooga
* 33 voters: A4, B5, C5
 
* 34 voters: A4, B0, C5
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===
 
{| class="wikitable" border="1" style="empty-cells: show"
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|}
 
Resulting in the following pairwise matrix.
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.
 
{| class="wikitable" border="1" style="empty-cells: show"
|
|... over '''A'''
|... over '''B'''
|... over '''C'''
|-
|Prefer '''A''' ...
!Voters
| ---
!35
|bgcolor="#EE8080"|34x5+443=613
!33
|bgcolor="#80EE80"|35x5+443=618
!34
!Relative
score
|-
|Prefer '''B''' ...
|A
|bgcolor="#EE8080"|33x5+340=505
|0 (35 × 0)
| ---
|0 (33 × 0)
|bgcolor="#80EE80"|35x5+340=515
|0 (34 × 0)
|0
|-
|Prefer '''C''' ...
|B
|bgcolor="#80EE80"|67x5+335=670
|0 (35 × 0)
|bgcolor="#EE8080"|34x5+335=505
|165 (33 × +5)
| -170 (34 × -5)-
| -5
|-
|C
| -175 (35 × -5)
|165 (33 × +5)
|170 (34 × +5)
|160
|}
 
There is a cycle A>B>C>A. In case of a cycle the score winner from within that cycle is elected, here A.
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.
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===Precinct summability===
DespiteLike severalmost roundsCondorcet of comparing candidatesmethods, 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. ScoresThe canresults simplyare be added up. For the preference wetabulated turnin a ballot into a preferencepairwise matrix andas addseen up those for all ballots. Forin the Tennessee exampleexamples above the summed up matrix is exemplified below.
 
{| class="wikitable"
|
|... 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.
 
{| class="wikitable"
|
|... 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.
 
{| class="wikitable"
|
|... 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 ==
Casimir
20

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