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Majority judgment (MJ) is a single-winner voting system proposed in 2010 by Michel Balinski and Rida Laraki.[1][2][3] It is a kind of highest median rule, a cardinal voting system that elects the candidate with the highest median rating.
Voting process
Voters grade as many of the candidates as they wish with regard to their suitability for office according to a series of grades. Balinski and Laraki suggest the options "Excellent, Very Good, Good, Acceptable, Poor, or Reject," but any scale can be used (e.g. the common letter grade scale). Voters can assign the same grade to multiple candidates.
As with all highest median voting rules, the candidate with the highest median grade is declared winner. If more than one candidate has the same median grade, majority judgment breaks the tie by removing (one-by-one) any grades equal to the shared median grade from each tied candidate's column. This procedure is repeated until only one of the tied candidates is found to have the highest median grade.[4]
Advantages and disadvantages
Like most other cardinal voting rules, majority judgment satisfies the monotonicity criterion, the later-no-help criterion, and independence of irrelevant alternatives.
Like any deterministic voting system (except dictatorship), MJ allows for tactical voting in cases of more than three candidates, as a consequence of Gibbard's theorem.
Majority judgment voting fails the Condorcet criterion,[a] later-no-harm,[b]consistency,[c] the Condorcet loser criterion, the participation criterion, the majority criterion,[d] and the mutual majority criterion.
Participation failure
Unlike score voting, majority judgment can have no-show paradoxes,[5] situations where a candidate loses because they won "too many votes". In other words, adding votes that rank a candidate higher than their opponent can still cause this candidate to lose.
In their 2010 book, Balinski and Laraki demonstrate that the only join-consistent methods are point-summing methods, a slight generalization of score voting that includes positional voting.[6] Specifically, their result shows the only methods satisfying the slightly stronger consistency criterion have:
Where is a monotonic function. Moreover, any method satisfying both participation and either stepwise-continuity or the Archimedean property[e] is a point-summing method.[7]
This result is closely related to and relies on the Von Neumann–Morgenstern utility theorem and Harsanyi's utilitarian theorem, two critical results in social choice theory and decision theory used to characterize the conditions for rational choice.
Despite this result, Balinski and Laraki claim that participation failures would be rare in practice for majority judgment.[6]
Claimed resistance to tactical voting
In arguing for majority judgment, Balinski and Laraki (the system's inventors) prove highest median rules are the most "strategy-resistant" system, in the sense that they minimize the share of the electorate with an incentive to be dishonest.[8] However, some writers have disputed the significance of these results, as they do not apply in cases of imperfect information or collusion between voters.[citation needed]
Median voter property
In "left-right" environments, majority judgment tends to favor the most homogeneous camp, instead of picking the middle-of-the-road, Condorcet winner candidate.[9] Majority judgment therefore fails the median voter criterion.[10]
Here is a numerical example. Suppose there were seven ratings named "Excellent," "Very good," "Good", "Mediocre," "Bad," "Very Bad," and "Awful." Suppose voters belong to seven groups ranging from "Far-left" to "Far-right," and each group runs a single candidate. Voters assign candidates from their own group a rating of "Excellent," then decrease the rating as candidates are politically further away from them.
Votes Candidate |
101 votes
Far-left |
101 votes
Left |
101 votes
Cen. left |
50 votes
Center |
99 votes
Cen. right |
99 votes
Right |
99 votes
Far-right |
Score |
---|---|---|---|---|---|---|---|---|
Far left | excel. | v. good | good | med. | bad | very bad | awful | med. |
Left | v. good | excel. | v. good | good | med. | bad | very bad | good |
Cen. left | good | v. good | excel. | v. good | good | med. | bad | good |
Center | med. | good | v. good | excel. | v. good | good | med. | good |
Cen. right | bad | med. | good | v. good | excel. | v. good | good | good |
Right | very bad | bad | med. | good | v. good | excel. | v. good | good |
Far right | awful | very bad | bad | med. | good | v. good | excel. | med. |
The tie-breaking procedure of majority judgment elects the Left candidate, as this candidate is the one with the non-median rating closest to the median, and this non-median rating is above the median rating. In so doing, the majority judgment elects the best compromise for voters on the left side of the political axis (as they are slightly more numerous than those on the right) instead of choosing a more consensual candidate such as the center-left or the center. The reason is that the tie-breaking is based on the rating closest to the median, regardless of the other ratings.
Note that other highest median rules such as graduated majority judgment will often make different tie-breaking decisions (and graduated majority judgment would elect the Center candidate). These methods, introduced more recently, maintain many desirable properties of majority judgment while avoiding the pitfalls of its tie-breaking procedure.[11]
Candidate |
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Left |
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Center left |
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Center |
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Center right |
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Right |
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Example application
Suppose that Tennessee is holding an election on the location of its capital. The population is concentrated around four major cities. All voters want the capital to be as close to them as possible. The options are:
- Memphis, the largest city, but far from the others (42% of voters)
- Nashville, near the center of the state (26% of voters)
- Chattanooga, somewhat east (15% of voters)
- Knoxville, far to the northeast (17% of voters)
The preferences of each region's voters are:
42% of voters Far-West |
26% of voters Center |
15% of voters Center-East |
17% of voters Far-East |
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Suppose there were four ratings named "Excellent", "Good", "Fair", and "Poor", and voters assigned their ratings to the four cities by giving their own city the rating "Excellent", the farthest city the rating "Poor" and the other cities "Good", "Fair", or "Poor" depending on whether they are less than a hundred, less than two hundred, or over two hundred miles away:
City Choice |
Memphis voters |
Nashville voters |
Chattanooga voters |
Knoxville voters |
Median rating[f] |
---|---|---|---|---|---|
Memphis | excellent | poor | poor | poor | poor+ |
Nashville | fair | excellent | fair | fair | fair+ |
Chattanooga | poor | fair | excellent | good | fair- |
Knoxville | poor | fair | good | excellent | fair- |
Then the sorted scores would be as follows:
City |
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Nashville |
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Knoxville |
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Chattanooga |
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Memphis |
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The median ratings for Nashville, Chattanooga, and Knoxville are all "Fair"; and for Memphis, "Poor". Since there is a tie between Nashville, Chattanooga, and Knoxville, "Fair" ratings are removed from all three, until their medians become different. After removing 16% "Fair" ratings from the votes of each, the sorted ratings are now: