From Academic Kids
Range voting, or average voting, or cardinal ratings is a voting system used for single or multiple-seat elections. It is also used on the web - for rating movies (Internet Movie Database), comments (Kuro5hin), and many other things.
For each candidate, each voter expresses the utility of that candidate's election to them in the form of a number. In "pure" range voting, each voter may give any candidate any real number, but as the potential for tactical voting is huge, most systems use some bounds. For example, each voter might give a real number between -1 and 1, or an integer between 1 and 10.
Range voting in which only two different votes may be submitted (0 and 1, for example) is equivalent to approval voting.
Range voting satisfies the monotonicity criterion.
Counting the votes
The scores for each candidate are summed, and the candidates with the highest sums are declared the winners.
Another method of counting is to find the median score of each candidate, and elect the candidate with the highest median score.
Imagine an election for the capital of Tennessee, a state in the United States that is over 500 miles (800 km) east-to-west, and only 110 miles (180 km) north-to-south. In this vote, the candidates for the capital are Memphis, Nashville, Chattanooga, and Knoxville. The population breakdown by metro area is as follows:
- Memphis: 826,330
- Nashville: 510,784
- Chattanooga: 285,536
- Knoxville: 335,749
If the voters cast their ballot based strictly on geographic proximity, the voters' sincere preferences might be as follows:
|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 each decided to grant from 1 to 10 points to each city such that their most liked choice got 10 points, and least liked choice got 1 point, with the intermediate choices getting 5 points and 2 points.
|Memphis||420 (42 * 10)||26||15||17||478|
|Nashville||210 (42 * 5)||260 (26 * 10)||30 (15 * 2)||34 (17 * 2)||534|
|Chattanooga||84 (42 * 2)||130 (26 * 5)||150 (15 * 10)||85 (17 * 5)||449|
|Knoxville||42||52 (26 * 2)||75 (15 * 5)||170 (17 * 10)||339|
Nashville wins. But Memphis would have won if the voters from Memphis had reduced the points they gave Nashville from 5 down to 1 and all other votes had remained the same; voters from Chattanooga or Knoxville could restore Nashville to first place over Memphis if they raised the points they gave Nashville from 2 up to 10.
In general, the optimal tactical voting strategy for range voting is to vote as if it were identical to approval voting, so that all candidates are given either the maximum score or the minimum score. For more detailed strategies, see approval voting.
Arrow's impossibility theorem
Arrow's impossibility theorem does not apply to range voting, in that it has the properties:
- unrestricted domain or universality: range voting produces a deterministic, complete societal preference order from every possible set of individual range votes.
- non-imposition or citizen sovereignty: every possible societal preference order can be achieved, for example if every voter gives points in that order.
- non-dictatorship: every voter is treated the same.
- positive association of social and individual values: if an individual modifies their range vote by increasing the points given to a certain option, then that option will either rise in the societal preference order or stay in the same place.
- independence of irrelevant alternatives: the relative ranking between elements of a subset of the options is unaffected by the points given to options not in that subset.
The reason that range voting is not treated as a counter-example to Arrow's theorem is that it is a cardinal voting system, while Arrow's theorem is restricted to the processing of ordinal preferences; two different sets of range votes may express the same ordinal preferences but lead to different overall rankings. By contrast, approval voting does not satisfy the preconditions for the theorem in that it fails to allow a full individual ranking of options if there are three or more options.