Majority judgment (MJ) is a single-winner
voting system
An electoral or voting system is a set of rules used to determine the results of an election. Electoral systems are used in politics to elect governments, while non-political elections may take place in business, nonprofit organizations and inf ...
proposed in 2010 by
Michel Balinski
Michel Louis Balinski (born Michał Ludwik Baliński; October 6, 1933 – February 4, 2019) was an American and French applied mathematician, economist, operations research analyst and political scientist. Educated in the United States, from 198 ...
and
Rida Laraki
Rida Laraki is a Moroccan researcher, professor, and engineer in the fields of game theory, social choice, theoretical economics, optimization, learning, and operations research at the French National Centre for Scientific Research.
Life
Born ...
.
It is a kind of
highest median rule, a
cardinal voting
Rated, evaluative, graded, or cardinal voting rules are a class of voting methods that allow voters to state how strongly they support a candidate, by giving each one a grade on a separate scale.
The distribution of ratings for each candidate� ...
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
Grading in education is the application of standardized measurements to evaluate different levels of student achievement in a course. Grades can be expressed as letters (usually A to F), as a range (for example, 1 to 6), percentages, or as num ...
scale). Voters can assign the same grade to multiple candidates.
As with all
highest median voting rules
The highest median voting rules are a class of graded voting rules where the candidate with the highest median rating is elected.
The various highest median rules differ in their treatment of ties, i.e., the method of ranking the candidates with ...
, the candidate with the highest
median
The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...
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.
Advantages and disadvantages
Like most other
cardinal voting
Rated, evaluative, graded, or cardinal voting rules are a class of voting methods that allow voters to state how strongly they support a candidate, by giving each one a grade on a separate scale.
The distribution of ratings for each candidate� ...
rules, majority judgment satisfies the
monotonicity criterion
Electoral system criteria
In social choice, the negative response, perversity, or additional support paradox is a pathological behavior of some voting rules where a candidate loses as a result of having too much support (or wins because of in ...
, the
later-no-help criterion, and
independence of irrelevant alternatives
Independence of irrelevant alternatives (IIA) is an axiom of decision theory which codifies the intuition that a choice between A and B (which are both related) should not depend on the quality of a third, unrelated outcome C. There are several dif ...
.
Like any deterministic voting system (except
dictatorship
A dictatorship is an autocratic form of government which is characterized by a leader, or a group of leaders, who hold governmental powers with few to no Limited government, limitations. Politics in a dictatorship are controlled by a dictator, ...
), MJ allows for
tactical voting
Strategic or tactical voting is voting in consideration of possible ballots cast by other voters in order to maximize one's satisfaction with the election's results.
Gibbard's theorem shows that no voting system has a single "always-best" stra ...
in cases of more than three candidates, as a consequence of
Gibbard's theorem
In the fields of mechanism design and social choice theory, Gibbard's theorem is a result proven by philosopher Allan Gibbard in 1973. It states that for any deterministic process of collective decision, at least one of the following three properti ...
.
Majority judgment voting fails the
Condorcet criterion
A Condorcet winner (, ) is a candidate who would receive the support of more than half of the electorate in a one-on-one race against any one of their opponents. Voting systems where a majority winner will always win are said to satisfy the Condo ...
,
later-no-harm,
consistency
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, the
Condorcet loser criterion
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.
A voting system complying wi ...
, the
participation criterion
The participation criterion is a voting system criterion that says candidates should never lose an election as a result of receiving too many votes in support. More formally, it says that adding more voters who prefer ''Alice'' to ''Bob'' should ...
, the
majority criterion, and the
mutual majority criterion
The mutual majority criterion is a criterion for evaluating electoral systems. It is also known as the majority criterion for solid coalitions and the generalized majority criterion. This criterion requires that whenever a majority of voters prefe ...
.
Participation failure
Unlike
score voting
Score voting, sometimes called range voting, is an electoral system for single-seat elections. Voters give each candidate a numerical score, and the candidate with the highest average score is elected. Score voting includes the well-known approva ...
, majority judgment can have
no-show paradoxes, 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
Score voting, sometimes called range voting, is an electoral system for single-seat elections. Voters give each candidate a numerical score, and the candidate with the highest average score is elected. Score voting includes the well-known approva ...
that includes
positional voting
Positional voting is a ranked voting electoral system in which the options or candidates receive points based on their rank position on each ballot and the one with the most points overall wins. The lower-ranked preference in any adjacent pair i ...
.
Specifically, their result shows the only methods satisfying the slightly stronger
consistency criterion
A voting system satisfies join-consistency (also called the reinforcement criterion) 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 ...
have:
Where
is a
monotonic function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of or ...
. Moreover, any method satisfying both participation and either
stepwise-continuity or the
Archimedean property
In abstract algebra and mathematical analysis, analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, Italy, Syracuse, is a property held by some algebraic structures, such as ordered or normed g ...
is a point-summing method.
This result is closely related to and relies on the
Von Neumann–Morgenstern utility theorem
In decision theory, the von Neumann–Morgenstern (VNM) utility theorem demonstrates that rational choice under uncertainty involves making decisions that take the form of maximizing the expected value of some cardinal utility function. The theo ...
and
Harsanyi's utilitarian theorem, two critical results in
social choice theory
Social choice theory is a branch of welfare economics that extends the Decision theory, theory of rational choice to collective decision-making. Social choice studies the behavior of different mathematical procedures (social welfare function, soc ...
and
decision theory
Decision theory or the theory of rational choice is a branch of probability theory, probability, economics, and analytic philosophy that uses expected utility and probabilities, probability to model how individuals would behave Rationality, ratio ...
used to characterize the conditions for
rational choice
Rational choice modeling refers to the use of decision theory (the theory of rational choice) as a set of guidelines to help understand economic and social behavior. The theory tries to approximate, predict, or mathematically model human behav ...
.
Despite this result, Balinski and Laraki claim that participation failures would be rare in practice for majority judgment.
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. However, some writers have disputed the significance of these results, as they do not apply in cases of imperfect information or collusion between voters.
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. Majority judgment therefore fails the
median voter criterion.
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.
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.
Example application
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:
Then the sorted scores would be as follows:
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:
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.
Real-world examples
The somewhat-related
median voting rule method was first explicitly proposed to assign budgets by
Francis Galton
Sir Francis Galton (; 16 February 1822 – 17 January 1911) was an English polymath and the originator of eugenics during the Victorian era; his ideas later became the basis of behavioural genetics.
Galton produced over 340 papers and b ...
in 1907. 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
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 ...
, a system used by
Progressive Era
The Progressive Era (1890s–1920s) was a period in the United States characterized by multiple social and political reform efforts. Reformers during this era, known as progressivism in the United States, Progressives, sought to address iss ...
reformers in the United States.
The full system of majority judgment was first proposed by Balinski and Laraki in 2007.
[Balinski M. and R. Laraki (2007) ]
A theory of measuring, electing and ranking
�. Proceedings of the National Academy of Sciences USA, vol. 104, no. 21, 8720-8725. 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
François René Jean Lucien Bayrou (; born 25 May 1951) is a French politician who has served as Prime Minister of France since December 2024. He has presided over the European Democratic Party (EDP) since 2004 and the Democratic Movement (France ...
, rather than the eventual runoff winner,
Nicolas Sarkozy
Nicolas Paul Stéphane Sarközy de Nagy-Bocsa ( ; ; born 28 January 1955) is a French politician who served as President of France from 2007 to 2012. In 2021, he was found guilty of having tried to bribe a judge in 2014 to obtain information ...
, or two other candidates (
Ségolène Royal
Ségolène Royal (; born Marie-Ségolène Royal; 22 September 1953) is a French politician who took part in the 2007 French presidential election, losing to Nicolas Sarkozy in the second round. She was the first woman in France's history to r ...
or
Jean-Marie Le Pen
Jean Louis Marie Le Pen (20 June 1928 – 7 January 2025), commonly known as Jean-Marie Le Pen (), was a French politician, lawyer and activist. He founded the far-right National Front (now National Rally) party and served as the party's presi ...
) 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.
It has since been used in judging wine competitions and in other political research polling in France and in the US.
[Balinski M. and R. Laraki (2010) «Judge: Don't vote». Cahier du Laboratoire d’Econométrie de l’Ecole Polytechnique 2010-27.]
Variants
Varloot and Laraki
present a variant of majority judgement, called majority judgement with uncertainty (MJU), which allows voters to express uncertainty about each candidate's merits.
See also
*
Usual judgment
*
Approval voting
Approval voting is a single-winner rated voting system where voters can approve of all the candidates as they like instead of Plurality voting, choosing one. The method is designed to eliminate vote-splitting while keeping election administration ...
*
Range voting
Score voting, sometimes called range voting, is an electoral system for single-seat elections. Voters give each candidate a numerical score, and the candidate with the highest average score is elected. Score voting includes the well-known approva ...
*
Voting system
An electoral or voting system is a set of rules used to determine the results of an election. Electoral systems are used in politics to elect governments, while non-political elections may take place in business, nonprofit organizations and inf ...
*
List of democracy and elections-related topics
Notes
References
Further reading
*Balinski, Michel, and Laraki, Rida (2010). ''Majority Judgment: Measuring, Ranking, and Electing'', MIT Press
{{voting methods
Single-winner electoral systems
Cardinal electoral systems
Monotonic electoral systems