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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.
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]
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, [lower-alpha 1] later-no-harm, [lower-alpha 2] consistency, [lower-alpha 3] the Condorcet loser criterion, the participation criterion, the majority criterion, [lower-alpha 4] and the mutual majority criterion.
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 [lower-alpha 5] 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]
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 ]
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 |
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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 | ||||||||||||||
Center left | ||||||||||||||
Center | ||||||||||||||
Center right | ||||||||||||||
Right | ||||||||||||||
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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:
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 [lower-alpha 6] |
---|---|---|---|---|---|
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 | ||||||||||
Knoxville | ||||||||||
Chattanooga | ||||||||||
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:
City |
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Nashville | |||||||
Knoxville | |||||||
Chattanooga |
Chattanooga and Knoxville now have the same number of "Poor" ratings as "Fair", "Good" and "Excellent" combined. As a result of subtracting one "Fair" from each of the tied cities, one-by-one until only one of these cities has the highest median-grade, the new and deciding median-grades of these originally tied cities are as follows: "Poor" for both Chattanooga and Knoxville, while Nashville's median remains at "Fair". So Nashville, the capital in real life, wins.
The somewhat-related median voting rule method was first explicitly proposed to assign budgets by Francis Galton in 1907. [12] Hybrid mean/median systems based on the trimmed mean have long been used to assign scores in contests such as Olympic figure skating, where they are intended to limit the impact of biased or strategic judges.
The first highest median rule to be developed was Bucklin voting, a system used by Progressive era reformers in the United States.
The full system of majority judgment was first proposed by Balinski and Laraki in 2007. [1] That same year, they used it in an exit poll of French voters in the presidential election. Although this regional poll was not intended to be representative of the national result, it agreed with other local or national experiments in showing that François Bayrou, rather than the eventual runoff winner, Nicolas Sarkozy, or two other candidates (Ségolène Royal or Jean-Marie Le Pen) would have won under most alternative rules, including majority judgment. They also note:
Everyone with some knowledge of French politics who was shown the results with the names of Sarkozy, Royal, Bayrou and Le Pen hidden invariably identified them: the grades contain meaningful information. [13]
It has since been used in judging wine competitions and in other political research polling in France and in the US. [14]
Varloot and Laraki [15] present a variant of majority judgement, called majority judgement with uncertainty (MJU), which allows voters to express uncertainty about each candidate's merits.
Score voting or range voting is an electoral system for single-seat elections, in which voters give each candidate a score, the scores are added, and the candidate with the highest total is elected. It has been described by various other names including evaluative voting, utilitarian voting, interval measure voting, point-sum voting, ratings summation, 0-99 voting, and average voting. It is a type of cardinal voting electoral system that aims to approximate the utilitarian social choice rule.
Strategic or tactical voting is a situation where a voter casts a ballot in a way other than to obtain a more desirable outcome. For example, in a plurality or instant-runoff voting, a voter can gain a better outcome by supporting a less-preferred but more broadly popular candidate.
A Condorcet method is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, whenever there is such a candidate. A candidate with this property, the pairwise champion or beats-all winner, is formally called the Condorcet winner. The head-to-head elections need not be done separately; a voter's choice within any given pair can be determined from the ranking.
Copeland's method, also called Llull's method or round-robin voting, is a ranked-choice voting system based on scoring pairwise wins and losses.
Bucklin voting is a class of voting methods that can be used for single-member and multi-member districts. As in highest median rules like the majority judgment, the Bucklin winner will be one of the candidates with the highest median ranking or rating. It is named after its original promoter, the Georgist politician James W. Bucklin of Grand Junction, Colorado, and is also known as the Grand Junction system.
Ranked pairs, sometimes called the Tideman method, is a tournament-style system of ranked-choice voting first proposed by Nicolaus Tideman in 1987.
In an election, a candidate is called a Condorcet, beats-all, or majority-rule winner if more than half of voters would support them in any one-on-one matchup with another candidate. Such a candidate is also called an undefeated, or tournament champion, by analogy with round-robin tournaments. Voting systems where a majority-rule winner will always win the election are said to satisfy the Condorcetcriterion. Condorcet voting methods extend majority rule to elections with more than one candidate.
The participation criterion, also called vote or population monotonicity, is a voting system criterion that says that a candidate should never lose an election because they have "too much support." It says that adding voters who support A over B should not cause A to lose the election to B.
The majority criterion is a voting system criterion. The criterion states that "if only one candidate is ranked first by a majority of voters, then that candidate must win."
A voting system satisfies join-consistency if combining two sets of votes, both electing A over B, always results in a combined electorate that ranks A over B. It is a stronger form of the participation criterion, which only requires join-consistency when one of the sets of votes unanimously prefers A over B.
In single-winner voting system theory, the Condorcet loser criterion (CLC) is a measure for differentiating voting systems. It implies the majority loser criterion but does not imply the Condorcet winner criterion.
Anti-plurality voting describes an electoral system in which each voter votes against a single candidate, and the candidate with the fewest votes against wins. Anti-plurality voting is an example of a positional voting method.
Reversal symmetry is a voting system criterion which requires that if candidate A is the unique winner, and each voter's individual preferences are inverted, then A must not be elected. Methods that satisfy reversal symmetry include Borda count, ranked pairs, Kemeny–Young method, and Schulze method. Methods that fail include Bucklin voting, instant-runoff voting and Condorcet methods that fail the Condorcet loser criterion such as Minimax.
The Kemeny–Young method is an electoral system that uses ranked ballots and pairwise comparison counts to identify the most popular choices in an election. It is a Condorcet method because if there is a Condorcet winner, it will always be ranked as the most popular choice.
The later-no-harm criterion is a voting system criterion first formulated by Douglas Woodall. Woodall defined the criterion by saying that "[a]dding a later preference to a ballot should not harm any candidate already listed." For example, a ranked voting method in which a voter adding a 3rd preference could reduce the likelihood of their 1st preference being selected, fails later-no-harm.
Instant-runoff voting (IRV), also known as plurality with elimination or plurality loser, is a ranked-choice voting system that modifies plurality by repeatedly eliminating the last-place winner until only one candidate is left. In the United Kingdom, it is generally called the alternative vote (AV). In the United States, IRV is often referred to as ranked-choice voting (RCV), by way of conflation with ranked voting systems in general.
A major branch of social choice theory is devoted to the comparison of electoral systems, otherwise known as social choice functions. Viewed from the perspective of political science, electoral systems are rules for conducting elections and determining winners from the ballots cast. From the perspective of economics, mathematics, and philosophy, a social choice function is a mathematical function that determines how a society should make choices, given a collection of individual preferences.
STAR voting is an electoral system for single-seat elections. The name stands for "Score then Automatic Runoff", referring to the fact that this system is a combination of score voting, to pick two finalists with the highest total scores, followed by an "automatic runoff" in which the finalist who is preferred on more ballots wins. It is a type of cardinal voting electoral system.
The highest median voting rules are a class of graded voting rules where the candidate with the highest median rating is elected.
Graduated majority judgment (GMJ), sometimes called the usual judgment or continuous Bucklin voting, is a single-winner electoral system. It was invented independently three times in the early 21st century. It was first suggested as an improvement on majority judgment by Andrew Jennings in 2010, then by Jameson Quinn, and later independently by the French social scientist Adrien Fabre in 2019. In 2024, the latter coined the name "median judgment" for the rule, arguing it was the best highest median voting rule.
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