Score voting
Score voting, also known as range voting, ratings summation and average rating, is a type of Cardinal voting system used for singleseat 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 realworld systems in which the voter may give to each candidate any number of points within some specified range, such as 05 or 010. "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.
Contents
Voting[edit  edit source]
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 Noncompulsory support criterion, MonoAddTop, 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[edit  edit source]
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.)
Example[edit  edit source]
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) 





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[edit  edit source]
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[edit  edit source]
Supporters of range voting include Warren Smith, Clay Shentrup, Jan Kok, Keith Edmonds and Steve Gruber.
Connection to Condorcet methods[edit  edit source]
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 "morepreferred candidate is 1 vote i.e. 5 points better than lesspreferred candidates." Both Score and Condorcet elect the candidate who can get more points/votes than any other opponent in oneonone comparisons, though in Condorcet such a candidate may not always exist.
Score's satisfaction of the abovementioned 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 beatsall winner exists, and forcing Condorcet methods to satisfy that property ensures a beatsortiesall winner will exist.
References[edit  edit source]
 ↑ Hillinger, Claude (20050501). "The Case for Utilitarian Voting". Open Access LMU. Munich. Retrieved 20180515.
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.
 ↑ "Should you be using a more expressive voting system?". VoteUp app. Retrieved 20180515.
Score Voting  it’s just like range voting except the scores are discrete instead of spanning a continuous range.
 ↑ "Good criteria support range voting". RangeVoting.org. Retrieved 20180515.
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 scoresum, is elected.
 ↑ Smith, Warren D. (December 2000). "Range Voting" (PDF).
The “range voting” system is as follows. In a ccandidate 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 5candidate election. The votevectors are summed to get a cvector x and the winner is the i such that xᵢ is maximum.
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