STAR voting

STAR voting is an electoral system for single-seat elections, though variations exist for multi-seat and multi-seat proportional representation elections. The name stands for "score then automatic runoff", referring to the fact that this system is a combination of score voting and runoff voting. 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 from 0-5. The two candidates with the highest total are selected as finalists, and then in the "automatic runoff", the finalist who is preferred on more ballots wins.

Usage
The concept was invented by Mark Frohnmayer and Clay Shentrup and was first proposed publicly in October 2014 by Center for Election Science co-founder Clay Shentrup. The runoff step was introduced in order to correct for strategic distortion in ordinary score voting, such as bullet voting and tactical maximization. Thus, STAR is intended to be a compromise between score voting and instant runoff voting. The movement to implement STAR voting was initially centered in Oregon, and in July 2018, supporters submitted over 16,000 signatures for a ballot initiative in Lane County. The initiative received 47.6% of the vote. Ballot initiatives are currently underway for Lane County, Eugene, and Troutdale, Oregon.

Method
STAR voting uses a ratings ballot; each voter scores each candidate from 0 to 5 stars. Candidates who are left blank receive a 0. No preference can be indicated by giving multiple candidates the same score. 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 scored higher on a greater number of ballots is the winner.

Ties
Tie votes in STAR Voting are rare, but as with any voting method they can occur, especially in elections without many voters. In most cases, ties in STAR voting can be broken by referring back to the ballots themselves for either the Scoring or Runoff round. Ties in the Scoring round are broken in favor of the candidate who was preferred by more voters. Ties in the Runoff round are broken in favor of the candidate who was scored higher. Ties which can not be broken as above are considered a true tie.

Example
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.

The finalists 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 first-past-the-post would elect Memphis, even though most citizens consider it the worst choice, because 42% is larger than any other single city. Instant-runoff voting would elect the 2nd-worst 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 two-round system would have a runoff between Memphis and Nashville where Nashville would win.

Variations
STAR Voting can be used for multi-winner elections as in STAR bloc voting or it can be used for proportional representation elections.

Bloc STAR Voting: Each voter scores all the candidates on a scale from 0-5. All the scores are added and the two highest scoring candidates advance to an automatic runoff. The finalist who was preferred by (scored higher by) more voters wins the first seat. The next two highest scoring candidates then runoff, with the finalist preferred by more voters winning the next seat. This process continues until all positions are filled.

Proportional STAR Voting: Each voter scores all the candidates on a scale from 0-5. The results are tabulated using a proportional STAR algorithm such as Allocated Score or Sequentially Spent STAR. In this context STAR stands for Score Then Automatic Runoffs, and refers to fact that an automatic runoff between all remaining candidates is performed until all seats have been filled. The Equal Vote Coalition has designated Allocated Score as the official recommendation for Proportional STAR Voting (STAR-PR.) following a two year research study on the recommendation of the Equal Vote 0-5 Star Proportional Research Committee.

Properties
STAR voting allows voters to express both preference order and degree of support.

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 all-strategic voters and perfect information, the Condorcet winner is a Nash equilibrium. 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 later-no-harm criterion, meaning that giving a positive rating to a less preferred candidate can cause a more-preferred candidate to lose.

Majority-related criteria
STAR passes the majority criterion, and thus is equivalent to choose-one FPTP voting whenever voters indicate all of their preferences, only in the one- or two-candidate 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 n-candidate 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 highest-scored 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 highest-scored 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
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.

Precinct summability
STAR voting can most simply be counted by first adding up the scores, and then doing a second round of tallying where the two highest-scoring candidates are compared on every ballot. It can be made precinct-summable (countable in one round) by using pairwise counting to determine who voters prefer in every possible automatic runoff.

Comprehensive 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 head-to-head matchup. Starting with the first 2 voters, they collectively gave White Chocolate 5*2=10 points and Dark Chocolate 2*2=4 points. The head-to-head 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 head-to-head 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: 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 head-to-head 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 write-in candidates; see below). (Technical side note: With this compactifying modification, there really are N^2 data values to capture, rather than the only N*(N-1) = 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: The two matrices can be combined to yield: 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 head-to-head 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. higher-scored 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 head-to-head matchup between them.

So, for example, if the above matrix was the final matrix, then we can find the result as follows: 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 head-to-head matchup.

(Note that precinct-summability becomes more complex when dealing with write-in candidates. See Pairwise counting 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 head-to-head matchup, and subtract a vote from each head-to-head matchup where the voter scored another candidate above or equal to the scored candidate. Since write-ins so rarely win, in practice, it is possible to simply record how many voters scored the write-in and what score they gave that candidate, and only do a second round of tallying in the rare case that the write-in makes it to the automatic runoff. If it is predicted that the write-in may garner significant support, then the election officials can be told to record this candidate's head-to-head matchups as well.)

Alternative examples
More complex example of the STAR result being calculated from precinct-summable data (example modified from https://star.vote/txzfc3c9/):

Legend: For, Against, No Preference (i.e. in the Green Party>Working Families Party cell, 97-60-132 means that 97 voters prefer Green>WFP, 60 WFP>Green, and 132 have no preference between the two). 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.