Score voting

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Score voting, also known as range voting, ratings summation, average rating, cardinal ratings is a voting system used for single-seat elections. (It can also be used for multi-seat elections, but not with proportional results.) 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.


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.

Counting the Votes

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.

(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.)


Tennessee's four cities are spread throughout the state

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.


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.

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


  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". 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.
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