Median Ratings

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Median Ratings is a voting system used for single-seat elections. It is like Cardinal Ratings except that the candidate with the highest median score, rather than the highest total score, is elected.

Example

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 on Wikipedia, the state's largest city, with 42% of the voters, but located far from the other cities
  • Nashville on Wikipedia, with 26% of the voters, near the center of Tennessee
  • Knoxville on Wikipedia, with 17% of the voters
  • Chattanooga on Wikipedia, 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
"4" votes 42 26 15 17
"3" votes 0 42 43 15
"2" votes 0 32 42 26
"1" votes 58 0 0 42
Median 1 3 3 2

There is a tie between Nashville and Chattanooga.

Advantages

The main advantages of using the median instead of the sum or arithmetic mean of the individual ratings are:

  • The result is determined by majorities: when a majority specifies a ranking at least a, then the median is also at least a, and when a majority specifies a ranking at most a, then the median is also at most a.
  • The resulting method possesses many more and more natural-looking group strategy equilibria than Cardinal Ratings in the case of a sincere Condorcet Winner.

Disadvantages

The example illustrates one of the major flaws of Median Ratings. Because the median, by definition, is either the value of a single rating or the average of two ratings, there are very few possible values for a median score, and this makes ties much more likely than in Cardinal Ratings.

Given that a big range like 0-999 is probably an awkward solution, because nobody senses so sophisticated differences between candidates, a shorter range like 0-9 and a resolving method for ties might work better. Ties could be solved by using the average. Or by using the interquartile mean. One could also select just enough ratings above and under the median that the average of those gives a clear frontrunner. But that would fail independence of irrelevant alternatives. Yet another solution would be to take the average of the rating above and under each median in a tie. If there is still a tie, we take the average of the rating above the rating above and the rating under the rating under each median that is still in a tie. And if there is still a tie, we take the rating above the rating above the rating above an the rating under the rating under the rating under each median that is still in a tie etc.

Another problem with Median Ratings is its failure of the Blank Ballot Criterion. For example, with the ballots

  • A=1, B=4
  • A=2, B=4
  • A=8, B=4
  • A=9, B=4

A (median rating=5) beats B (median rating=4), but if the ballot (A=0, B=0) is added, then A's median rating drops to 2, causing B to win.

See also