STAR voting
STAR voting^{[1]}^{[2]} is an electoral system for singleseat elections. The name stands for "score then automatic runoff", referring to the fact that this system is a combination of score voting to pick two frontrunners with the highest total scores, followed by a "virtual runoff" in which the frontrunner who is preferred on more ballots wins. It is a type of cardinal voting electoral system. It was previously known as score runoff voting (SRV).
Voters cast ballots as in score voting, rating each candidate on a numerical scale. The two candidates with the highest total or average are selected, and the pairwise winner between those two is the overall winner.
Contents
Usage[edit  edit source]
The concept was first proposed publicly in October 2014 by Center for Election Science cofounder Clay Shentrup.^{[3]} The runoff step was introduced in order to correct for strategic distortion in ordinary score voting,^{[4]} such as Bullet voting and tactical maximization.^{[5]} Thus, STAR is intended to be a compromise between score voting and instant runoff voting.^{[6]} The movement to implement STAR voting was centered in Oregon, and in July 2018, supporters submitted over 16000 signatures for a ballot initiative in Lane County. This is enough to qualify this proposal to be on the ballot, if the measure is deemed welldrafted.^{[7]}
Method[edit  edit source]
STAR voting uses a ratings ballot; that is, each voter rates each candidate with a number within a specified range, such as 0 to 5 stars.^{[8]} In the simplest system, all candidates must be rated. The scores for each candidate are then summed, and the two candidates with the highest sums go to the runoff. Of these two, the one that is higher on a greater number of ballots is the winner.
Example[edit  edit source]
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 stars 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  Total 

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 
The frontrunners are Nashville and Chattanooga. Of the two, Nashville is preferred by 68% (42+26) to 32% (15+17), so Nashville, the capital in real life, likewise wins in the example. In this particular case, there is no way for any single city of voters to get a better outcome through tactical voting. However, Chattanooga and Knoxville voters combined could vote strategically to make Chattanooga win; while Memphis voters could defend against that strategy and ensure Nashville still won by strategically giving Nashville a higher rating.
For comparison, note that traditional firstpastthepost would elect Memphis, even though most citizens consider it the worst choice, because 42% is larger than any other single city. Instantrunoff voting would elect the 2ndworst choice (Knoxville), because the central candidates would be eliminated early. In Approval voting, with each voter selecting their top two cities, Nashville would win because of the significant boost from Memphis residents. A tworound system would have a runoff between Memphis and Nashville where Nashville would win.
Properties[edit  edit source]
STAR voting allows voters to express preferences of varying strengths.
STAR voting satisfies the monotonicity criterion, i.e. raising your vote's score for a candidate can never hurt their chances of winning, and lowering it can never help their chances. It fails several generalized versions of monotonicity that Score voting passes.
In summary, STAR voting satisfies the monotonicity criterion, the resolvability criterion, and reversal symmetry. It does not satisfy the Condorcet criterion (i.e., is not a Condorcet method), although with allstrategic voters and perfect information, the Condorcet winner is a Nash equilibrium.^{[9]} It, like any voting method with a final runoff, passes the Condorcet loser criterion, so long as voters indicate their preference for candidates other than the Condorcet loser. It does not satisfy the laternoharm criterion, meaning that giving a positive rating to a less preferred candidate can cause a morepreferred candidate to lose.
STAR passes the majority criterion, and thus is equivalent to chooseone FPTP voting whenever voters indicate all of their preferences, only in the one or twocandidate cases (because a majority of voters will have scored their 1st choice higher than all other candidates, and with only at most two candidates in the race, the runoff must include all candidates, so majority's 1st choice will win the runoff), and passes the majority criterion for rated ballots in the ncandidate case if the majority gives no support to at least (n  2) of the candidates (because the majority's 1st choice will have over 50% approval, while the most approval the (n  2) candidates with the fewest points can get is strictly less than 50%, due to receiving support from at most 50%  1 of the voters. Thus, the majority's 1st choice will be one of the top two highestscored candidates and enter the runoff, and then win).
In general, STAR voting passes a weak form of the mutual majority criterion. This is because whenever at least half of the voters give every candidate in a set of candidates the maximal score and at least all but one candidate not in the set the minimal score, and all candidates not in the set a lower score than any candidates in the set (i.e. a less than maximal score), then they guarantee that at least one of the candidates in the set at least ties to enter the runoff, or ties or wins overall (because every candidate in the set has at least 50% approval, and all but one of the candidates not in the set has at most 50% approval, with only one candidate not in the set possibly having over 50% approval. Therefore, at worst, all candidates in the set are tied with some candidates not in the set to take the second spot in the runoff; if at least one candidate in the set enters the runoff, one of the candidates in the set is guaranteed to at least tie in the runoff, since at least half of the voters scored them higher than the other candidate, unless both candidates in the runoff are candidates from the set, in which case, one of the candidates from the set is guaranteed to win). If a majority does this, they guarantee one of the candidates in the set enters the runoff and wins, rather than only tie to enter the runoff or tie or win overall (because the candidates in the set will have over 50% approval, with the most approval any of all but one of the candidates not in the set possibly getting being less than 50% approval, therefore at least one of the candidates in the set will be among the two highestscored candidates, and then one of the candidates in the set will win the runoff because either a majority scored them higher than the other candidate, or both candidates in the runoff are candidates from the set).
Complete ranking[edit  edit source]
A STAR voting ranking of candidates can be done by using the Bloc STAR voting procedure: find the STAR winner, put them in 1st place, then remove them from the election, and repeat, putting each consecutive STAR winner in a lower rank than all previous STAR winners. Optionally, if two candidates tie in the automatic runoff during this procedure, they can both be put as tied for the same rank, and then both are removed from the election. Note that while STAR voting can never put someone ranked 3rd or worse by Score voting as 1st i.e. its winner (when run on the same ballots; this is because only the two candidates ranked highest by Score voting can enter the STAR automatic runoff and thus even be eligible to win), it can put the candidate Score ranked 1st (i.e. the Score winner) as its last place candidate using this procedure, since the Score winner may be a Condorcet loser i.e. a candidate who would lose an automatic runoff against any other candidate.
A simpler way of finding the STAR ranking is simply to put the STAR winner 1st, and then rank all of the other candidates below the STAR winner according to their Score voting ranking.
Modifications[edit  edit source]
A modification to STAR that takes degree of preference more into account would be to make each voter's vote in the runoff only as strong as the highest score they gave to any candidate. In other words, a voter who gave their favorite a 3/5 (3 out of 5) would have only 3/5ths of a vote in the runoff, rather than a full vote. This modification allows voters to express lessthanfull support for any candidate in both the score round and the runoff.
Example:
2 A:1 D:5
3 A:5
6 B:4
Scores are A 17 B 24 D 10, with B pairwise beating A 6 to 5. So the Score and STAR winner is B. But taking into account that those who preferred B over A all have their favorite a maximum of a 4 out of 5, if we weight their runoff votes at 80%, then now B loses the runoff 4.8 to 5. So A wins under modified STAR.^{[10]}
IRNR is related to STAR in the same way that IRV is related to Toptwo runoff.
Given that STAR is an automatic form of Score voting + Runoff, one can also create an automatic "Approval voting + Runoff" method by allowing voters to rank or score candidates, and then indicate an approval threshold for a particular rank or score such that they'd approve all candidates at that same rank or score or a higher rank or score, and then use the ranks or scores to figure out which of the two most approved candidates is preferred by a majority.
Precinct summability[edit  edit source]
STAR voting can most simply be counted by first adding up the scores, and then doing a second round of tallying where the two highestscoring candidates are compared on every ballot. It can be made precinctsummable (countable in one round) by using pairwise counting to determine who voters prefer in every possible automatic runoff. Example:
Suppose there are 3 precincts, with 5 voters in each of them, and there are four "candidates": White Chocolate, Dark Chocolate, Milk Chocolate, and Chocolate with Almonds. In Precinct 1, we have the following votes:
2 voters vote White:5 Dark:2 (i.e. they give White a score of 5 points, Dark 2 points)
3 voters vote Dark:4 Milk:3 Almonds:3 White:1
To sum this, we add up both the scores and the preferences in each headtohead matchup. Starting with the first 2 voters, they collectively gave White Chocolate 5*2=10 points and Dark Chocolate 2*2=4 points. The headtohead matchups are computed by looking at which candidate a voter scored higher between every pair of candidates, so for example, between White and Dark, these 2 voters score White a 5 and Dark a 2, so they prefer (scored higher) White; thus, we add 2 votes to White>Dark (2 votes that help White beat Dark in the headtohead matchup between the two). Note that with 4 candidates in the election, the preferences in (4*3*0.5)=6 matchups must be known: White vs Dark, White vs Milk, White vs Almonds, Dark vs Milk, Dark vs Almonds, and Milk vs Almonds. In essence, we figure out what the result would be no matter which pair of candidates advance to the automatic runoff, so that we don't have to do a second round of tallying once we know who the pair are. For the first two voters, they score White above all other candidates, so that's 2 votes for White>(each of Dark, Almond, and Milk). Then, they score Dark above all candidates except White, so that's 2 votes for Dark>(Milk and Almonds). Finally, they show no preference between Almonds and White (they didn't score either), so no preference is recorded in the Almonds vs White matchup. We can represent these 2 voter's preferences like so:
White  Dark  Milk  Almond  

White  10 points  2 (+2 Win)  2 (+2 Win)  2 (+2 Win) 
Dark  0 (2 Loss)  4 points  2 (+2 Win)  2 (+2 Win) 
Milk  0 (2 Loss)  0 (2 Loss)  0 points  0 (Tie) 
Almond  0 (2 Loss)  0 (2 Loss)  0 (Tie)  0 points 
This should be read as "candidate on the left is preferred over candidate on the top by the number of voters in this cell". The margin in each matchup is recorded by taking the number of votes for one candidate and subtracting the number of votes for the other in their headtohead matchup. Note that, for compactness, the scores for each candidate can be held as a data value in the cell comparing them to themselves i.e. the White>White cell shows White Chocolate's score. If desired, it is possible to record as a separate data value in that cell the number of voters who scored that candidate (i.e. to calculate the average score for the candidate, or even to handle writein candidates; see below). (Technical side note: With this compactifying modification, there really are N^2 data values to capture, rather than the only N*(N1) = N^2  N = almost N^2 values that are needed for Condorcet).
For the 3 other voters in the precinct, they scored Dark higher than all others, so that's 3 votes for Dark>(Milk, Almonds, and White); they also scored Milk and Almonds above all others except each other and except Dark, so that's 3 votes for (each of Milk and Almonds)>White. So, their runoff matrix is:
White  Dark  Milk  Almond  

White  3 points  0 (3 Loss)  0 (3 Loss)  0 (3 Loss) 
Dark  3 (+3 Win)  12 points  3 (+3 Win)  3 (+3 Win) 
Milk  3 (+3 Win)  0 (3 Loss)  9 points  0 (Tie) 
Almond  3 (+3 Win)  0 (3 Loss)  0 (Tie)  9 points 
The two matrices can be combined to yield:
White  Dark  Milk  Almond  

White  13 points  2 (1 Loss)  2 (1 Loss)  2 (1 Loss) 
Dark  3 (+1 Win)  16 points  5 (+5 Win)  5 (+5 Win) 
Milk  3 (+1 Win)  0 (5 Loss)  9 points  0 (Tie) 
Almond  3 (+1 Win)  0 (5 Loss)  0 (Tie)  9 points 
As can be seen above, the number of points for each candidate can be added up (i.e. 10 + 3 = 13 points for White) and the number of votes for each candidate in each headtohead matchup can be added up (i.e. 3 + 2 = 5 votes for Dark>Milk), allowing us to see all of the final relevant preference information when everything is added up. All of the precincts' matrices can be combined in this manner until a final matrix is reached. If the matrix is organized such that it is in descending order of points (i.e. higherscored candidates go in higher rows), then the winner can be determined simply by, so long as there are no scorewise ties (ties on points) between the top two candidates and anyone else, looking at the top two rows and looking for the result in the headtohead matchup between them.
So, for example, if the above matrix was the final matrix, then we can find the result as follows:
Dark  Milk  Almond  

Dark  16 points  3 (+1 Win)  5 (+5 Win)  5 (+5 Win) 
2 (1 Loss)  13 points  2 (1 Loss)  2 (1 Loss)  
Milk  0 (5 Loss)  3 (+1 Win)  9 points  0 (Tie) 
Almond  0 (5 Loss)  3 (+1 Win)  0 (Tie)  9 points 
As can be seen after sorting the candidates by points, Dark and White would go to the automatic runoff because they have more points than any other candidates, and then Dark would win, because Dark has 1 more vote than (is preferred/scored higher by 1 more voter than) White in their headtohead matchup.
(Note that precinctsummability becomes more complex when dealing with writein candidates. See Pairwise counting#Notes for ideas on how to deal with this; essentially, the most comprehensive way is to count the number of voters who score a candidate, filling in that number of votes preferring that candidate above each other candidate in a headtohead matchup, and subtract a vote from each headtohead matchup where the voter scored another candidate above or equal to the scored candidate. Since writeins so rarely win, in practice, it is possible to simply record how many voters scored the writein and what score they gave that candidate, and only do a second round of tallying in the rare case that the writein makes it to the automatic runoff. If it is predicted that the writein may garner significant support, then the election officials can be told to record this candidate's headtohead matchups as well.)
More complex example of the STAR result being calculated from precinctsummable data (example modified from https://star.vote/txzfc3c9/):
Legend: For, Against, No Preference (i.e. in the Green Party>Working Families Party cell, 9760132 means that 97 voters prefer Green>WFP, 60 WFP>Green, and 132 have no preference between the two).
Democratic Party  I don’t like party politics  Green Party  Working Families Party  Libertarian Party  

Democratic Party  >  642 points  100    104    85 (4 Loss)  107    88    94  114    57    118  133    59    97  
I don’t like party politics  >  104    100    85 (+4 Win)  615 points  109    82    98  97    67    125  101    69    119  
Green Party  >  88    107    94  82    109    98  523 points  97    60    132  111    78    100  
Working Families Party  >  57    114    118  67    97    125  60    97    132  450 points  86    74    129  
Libertarian Party  >  59    133    97  69    101    119  78    111    100  74    86    129  449 points 
For simplicity, the candidates have been sorted by scores, with their scores in bold in their own pairwise comparison cell. The top two candidates are not tied scorewise with anyone else, so they both are in the automatic runoff. Between the two, "I don't like party politics" is pairwise preferred (has 4 more votes in the matchup), so it wins.
See also[edit  edit source]
 Majority judgment — similar voting method, based on medians instead of averages
Notes[edit  edit source]
 ↑ "STAR voting  front page". starvoting.us. Retrieved 2018710.
STAR voting.
Check date values in:accessdate=
(help)  ↑ "Revolutionary New Voting Method Bolstered By over 16,000 Voters in Oregon County". The Independent Voter Network. 20150709. Retrieved 2016710. Check date values in:
accessdate=
(help)  ↑ "Google Groups". groups.google.com. Retrieved 20170405.
 ↑ "Score Runoff Voting: The New Voting Method that Could Save Our Democratic Process". IVN.us. 20161208. Retrieved 20170405.
 ↑ "Strategic SRV?  Equal Vote Coalition". Equal Vote Coalition. Retrieved 20170405.
 ↑ "Equal Vote Coalition". Equal Vote Coalition. Retrieved 20170405.
 ↑ "Revolutionary New Voting Method Bolstered By over 16,000 Voters in Oregon County". The Independent Voter Network. 20150709. Retrieved 2016710. Check date values in:
accessdate=
(help)  ↑ "Rating Scale Research". RangeVoting.org. Retrieved 20161211.
The evidence surveyed here currently suggests that the "best" scale for human voters should have 10 levels
 ↑ Laslier, J.F. (2006) "Strategic approval voting in a large electorate," IDEP Working Papers No. 405 (Marseille, France: Institut D'Economie Publique)
 ↑ https://forum.electionscience.org/t/starbutwithweakvotesintherunoffwhenvotersdontgiveanycandidatethetopscore/615/5
External links[edit  edit source]
 STAR voting
 The Equal Vote Coalition, an advocacy organization for STAR voting and other reforms.
 ★.✓ An online platform for carrying out STAR voting elections.
 Leading Election Systems: Pros and Cons
 Comparing Voting Systems: A Report Card