# Score voting

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Score voting, also known as range voting, ratings summation and average rating, is a type of cardinal voting systems 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], though this has often also been used to mean Score voting.

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.

## Criteria

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 criterion, Plurality Criterion for rated ballots, and independence of clones.

### Majority-related criteria

Range voting does not comply with the Condorcet criterion because it allows for the difference between 'rankings' to matter (though see the rated pairwise preference ballot for an alternative way to do this). 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.

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, so therefore Score voting passes a weak form of the mutual majority criterion.

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 normalization, which in itself involves 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.

In general, unlike almost any other voting method, the only strategies that are possible in Score voting are to either score a candidate higher to increase their chances of winning, or score a candidate lower to decrease their chances of winning. There is never incentive for "order-reversal" (i.e. voting as if candidate X is better than candidate Y if you prefer Y to X). Because of this, Score can be thought of as a "perfectly monotonic" voting method.

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

#### Computing the result using pairwise counting

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.

#### Limits on strength of transitive pairwise preference

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 transitive 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."

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.

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. When the Score winner is the Condorcet winner, and all voters expressed all of their ranked preferences with the scores, then this means that each voter could exaggerate their preference in each head-to-head matchup from the weak pairwise preference they expressed in Score to a maximal pairwise preference and obtain the same result if the head-to-head matchups are used to find the Condorcet winner in both circumstances. The same holds for when the Score ranking is equivalent to the Condorcet ranking.

#### Connection to pairwise counting

When using the Rated pairwise preference ballot#Rated or ranked preference implementation with Pairwise counting#Negative vote-counting approach, the score for a candidate can be interpreted as a partial ballot marking that candidate i.e. a voter with rated preferences giving a candidate a 3/5 would be considered to give them 0.6 votes in every matchup, whereas a voter who has ranked preferences would instead give 1 vote.

## Notes

It is often argued that Score can do a much better job of electing consensus candidates than many ranked methods; one commonly discussed implication of this is that Score may neuter the effectiveness of gerrymandering (see Gerrymandering#Cardinal methods), and thus could be even better than Proportional representation in the sense of also giving voters good geographical single-winner district representation.

Score voting can be simulated with Approval ballots if every voter votes probabilistically according to their utility value for each candidate i.e. a voter who thinks a candidate is a 6 out of 10 would use a dice or other randomizing device to approve that candidate with only 60% probability. With this approach, the Approval voting winner will probabilistically be the Score voting winner so long as there are many voters. In some sense, cardinal utility is tied to randomness in that it is often considered a better measure than ordinal utility when analyzing decision-making under uncertainty. See also the KP transform.

### Two-candidate case

Score is probably the only major deterministic voting method which can fail the majority criterion in the two-candidate case. It is, however, practically impossible to happen in practice since the scale for score voting is set by the candidates who run. With only two candidates each voter will vote one candidate at each extreme of the scale resulting in a reduction to approval voting. It is then also very unlikely that any voter would approve of both or neither candidate when voting. Oftentimes, voting reform proposals for governmental elections implement an alternative voting method when 3+ candidates run, but allow or mandate using FPTP when only two candidates run, ostensibly for simplicity and because it will elect the pairwise winner anyways. Some who prefer Score argue that FPTP is fine for two-candidate elections since score will reduce to FPTP for rational voters.

Despite this, if enough voters vote in a way contrary both to how the system is intended and their honest interest there can be results similar to failing Independence of irrelevant alternatives. When there is a Condorcet cycle involving the Score winner when 3+ candidates are running, then if all candidates except the Score winner and the candidate they pairwise lose to drop out of the election, then the use of FPTP will lead to the Score winner now losing even if every voter was honest and had the same preferences. Such a situation is virtually impossible since if a winner drops out after an election another will be held. This can be averted by the voters if they use a probabilistic strategy to decide how to vote in the FPTP election however, since they can simulate their margin of strength of preference between the two candidates by choosing to either vote for their preferred candidate between the two or not vote. See the Utility page for an example.

## 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.