Score voting

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Score voting, also known as range voting, ratings summation and average rating, is a type of Cardinal voting system used for single-seat elections. This is the familiar "Points System", used for rating movies (Internet Movie Database), comments (Kuro5hin), and many other things - and something very similar to it is used in the Olympics to award gold medals to gymnasts.

"Score voting" typically refers to real-world systems in which the voter may give to each candidate any number of points within some specified range, such as 0-5 or 0-10. "Range voting" is the more theoretical mathematical model of Score, in which voters express a real number from 0 to 1.[1][2][3][4]

Approval voting is equivalent to Score voting with only 0 or 1 (approve or abstain) as scores.

Voting

Range voting uses a ratings ballot; that is, each voter rates each candidate with a number. In "pure numerical voting," each voter may give any candidate any real number (i.e. not restricted to any finite range), but as the potential for tactical voting would then be huge, most systems use upper and lower bounds. For example, each voter might give a real number between -1 and 1, or between 0 and 99; in the latter case little is lost by also demanding that the scores be integers.

Range voting in which only two different votes may be submitted (0 and 1, for example) is equivalent to approval voting.

In range (or approval) voting with blanks, the voter is allowed to leave some scores blank to denote ignorance about those candidates.

Range voting satisfies the monotonicity criterion, the participation criterion, the Consistency Criterion, the summability criterion, the Favorite Betrayal criterion, Independence of irrelevant alternatives, the Non-compulsory support criterion, Mono-Add-Top, Pareto, Plurality Criterion, and independence of clones.

Range voting does not comply with the Condorcet criterion because it allows for the difference between 'rankings' to matter. E.g. 51 people might rate A at 100, and B at 90, while 49 people rate A at 0, and B at 100. Condorcet would consider this 51 people voting A>B, and 49 voting B>A, and A would win. Range voting would see this as A having support of 5100/100 = 51%, and B support of (51*90+49*100)/100 = 94.9%. Score voting advocates say that in this case, the Condorcet winner is not the socially ideal winner.

The scores for each candidate are summed, and the candidate with the highest sum is declared the winner. In range voting with blanks the candidate with the highest average score (where only nonblank scores are incorporated into the average) is the winner.

One way of averaging scores is to use the Simple Majority Minimum Denominator - the average for candidates scored by at least a majority of voters is computed normally, while for all other candidates, their averages are calculated as if they had been scored by exactly a simple majority of voters.

(Another method of counting is to find the median score of each candidate, and elect the candidate with the highest median score - see Median Ratings. Because strategic voting will typically lead to a vast number of candidates with the same median, a secondary measure to resolve ties is needed.)

Example

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 voters were told to grant 1 to 4 points to each city, giving their most favorite 4 points, second favorite 3 points, third favorite 2 points, and least favorite 1 point. For simplicity, let’s say we had 42 voters from Memphis, 26 from Nashville, 15 from Chattanooga, and 17 from Knoxville. The votes would be as follows.

City Memphis Nashville Chattanooga Knoxville Total
Memphis 42 * 4 = 168 26 * 1 = 26 15 * 1 = 26 17 * 1 = 26 226
Nashville 42 * 3 = 126 26 * 4 = 104 15 * 2 = 30 17 * 2 = 34 294
Chattanooga 42 * 2 = 84 26 * 3 = 78 15 * 4 = 60 17 * 3 = 51 273
Knoxville 42 * 1 = 42 26 * 2 = 52 15 * 3 = 45 17 * 4 = 68 207

Nashville wins.

Strategy

In general, the optimal strategy for range voting is to vote it identically to approval voting, so that all candidates are given either the maximum score or the minimum score. For more detailed strategies, see approval voting.

In both Approval and Score, the best strategy will always involve giving maximal support to your 1st choice(s) and no support to your least favorite(s) (how you score the other candidates will depend more on the situation). This is because this maximizes the chances the candidates you prefer most win and minimizes the chances the candidates you want least win. Because of this, Score and Approval always pass the majority criterion in the two-candidate case, and the mutual majority criterion (indeed, even the Smith criterion and Condorcet criterion) when voters' preferences are dichotomous (i.e. they view all candidates as either good or bad, implying they are all either one of the voter's 1st choices or one of their last choices) for any number of candidates, if all voters are strategic.

If half of the voters give every candidate in a set of candidates the maximal score and all other candidates the minimal score, they can guarantee that one of the candidates in that set will tie or win (because every candidate in that set will have at least 50% approval, while all other candidates, receiving support from at most half of the voters, will have at most 50% approval). When a majority of voters do this, they can guarantee one of the candidates in the set will win, rather than only tie or win, therefore Score voting passes a weak form of the mutual majority criterion.

Range voting has an advantage over approval voting if voters are actually expressing their personal feelings rather than doing everything they can to cause their most favored outcomes.

Supporters

Supporters of range voting include Warren Smith, Clay Shentrup, Jan Kok, Keith Edmonds and Steve Gruber.

Variants and similar systems

Score voting is in the class of Cardinal voting systems which contains many other methods.

Other single winner cardinal systems which take the same ballot as Score exist. STAR voting introduces an extra instant-runoff step, in which the majority preferred out of the top two rated candidates is chosen. Median Ratings utilizes the median rating as opposed to the mean rating to select the winner. Majority Judgment can also be thought of score voting using the median instead of the mean, except ratings are replaced by words and has a specific tie-breaking rule. Reciprocal Score Voting re-weights scores based on reciprocity between factions to encourage cooperation and discourage exaggeration.

Multi-winner cardinal systems include many systems. The most popular for achieving high levels of Proportional representation are Reweighted Range Voting, Single distributed vote, Sequentially Spent Score and Sequential Monroe voting.

Connection to Condorcet methods

Usually, Score voting is computed by adding the scores on each voter's ballot to find the candidate with the most points, who wins. But one can also do it (in a theoretical, and more difficult manner) by, for each pair of candidates, subtracting the score of the lower-scored candidate from the higher-scored candidate, and putting this in a pairwise counting table. The candidate who gets more points in their matchups against all other candidates wins. Example:

2: A:5 B:4 C:1 1: C:5 B:3 A:2

The scores are A 12 B 11 C 7.

The pairwise matrix:

Score voting matchups
A B C
A --- 2 (+1 Win) 8 (+5 Win)
B 1 (-1 Loss) --- 6 (+4 Win)
C 3 (-5 Loss) 2 (-4 Loss) ---

A gets more points than B or C (2 voters gave A 1 more point than B, with 1 voter giving B 1 more point than A, so A>B. 2 voters gave A 4 points more than C, with 1 voter giving C 3 points more than A, so A>C), so A wins. B is 2nd place because B beats C, and C loses all of their matchups, so they're in last place. Note that the Score voting order of finish can be constructed using a Condorcet ranking from this matrix. Some information is captured with this counting approach that is not normally captured. Note that to handle write-ins, one would have to ensure that a voter who gave an on-ballot candidate a score of, say, 3, would be counted as scoring that candidate 3 points higher than any candidates the voter didn't personally write in, which could complicate things.

Score can be thought of as a Condorcet method where a voter may only put up to 1 vote (i.e. the maximum number of points allowed) in between any pair of candidates in a beatpath. That is, a strategic voter whose preference is A>B>C can maximally contribute to A getting more points than B or to B getting more points than C, but not both. A rated ballot A:5 B:4 C:0 with max score of 5 is treated as "A is 1 point better than B, B is 4 points better than C, and A is 5 points better than C", whereas in Condorcet all three pairwise comparisons are treated as "more-preferred candidate is 1 vote i.e. 5 points better than less-preferred candidates." Both Score and Condorcet elect the candidate who can get more points/votes than any other opponent in one-on-one comparisons, though in Condorcet such a candidate may not always exist. See Self-referential Smith-efficient Condorcet method.

Score's satisfaction of the above-mentioned property (max of 1 vote of differentiation in a beatpath) is one of the reasons it nominally passes Independence of Irrelevant Alternatives where Condorcet methods don't, as the only time those methods fail it is when no beats-all winner exists, and forcing Condorcet methods to satisfy that property ensures a beats-or-ties-all winner will exist.

References

1. Hillinger, Claude (2005-05-01). "The Case for Utilitarian Voting". Open Access LMU. Munich. Retrieved 2018-05-15. Specific UV rules that have been proposed are approval voting, allowing the scores 0, 1; range voting, allowing all numbers in an interval as scores; evaluative voting, allowing the scores -1, 0, 1.
2. "Should you be using a more expressive voting system?". VoteUp app. Retrieved 2018-05-15. Score Voting -- it’s just like range voting except the scores are discrete instead of spanning a continuous range.
3. "Good criteria support range voting". RangeVoting.org. Retrieved 2018-05-15. Definition 1: For us "Range voting" shall mean the following voting method. Each voter provides as her vote, a set of real number scores, each in [0,1], one for each candidate. The candidate with greatest score-sum, is elected.
4. Smith, Warren D. (December 2000). "Range Voting" (PDF). The “range voting” system is as follows. In a c-candidate election, you select a vector of c real numbers, each of absolute value ≤1, as your vote. E.g. you could vote (+1, −1, +.3, −.9, +1) in a 5-candidate election. The vote-vectors are summed to get a c-vector x and the winner is the i such that xᵢ is maximum.