Score voting

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Range voting, or ratings summation, or average voting, or cardinal ratings is a voting system used for single-seat elections. (It could also be used for multi-seat elections, but that is a poor idea since it would lead to massive proportionality failures, where, e.g. a 51% Whig electorate could elect a 100% Whig slate of winners.) It is also used on the web - 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.


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.

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. This idea is not recommended because with strategic voting it will typically lead to a vast number of tied winners.)


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 such that their most liked choice(s) got 4 points, and least liked choice(s) got 1 point.

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

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