Majority Judgment
Encyclopedia
Majority Judgment is a single-winner voting system
proposed by Michel Balinski and Rida Laraki. Voters freely grade each candidate in one of several named ranks, for instance from "excellent" to "bad", and the candidate with the highest median
grade is the winner. If more than one candidate has the same median grade, a tiebreaker is used which sees the "closest-to-median" grade. Majority Judgment can be considered as a form of Bucklin voting
which allows equal ranks.
The median grade for each candidate is found, for instance by sorting their list of grades and finding the middle one. If the middle falls between two different grades, the lower of the two is used.
The candidate with the highest median grade wins. If several candidates share the highest median grade, all other candidates are eliminated. Then, one copy of that grade is removed from each remaining candidate's list of grades, and the new median is found, until there is an unambiguous winner. For instance, if candidate X's sorted ratings were {"Good", "Good", "Fair", "Poor"}, while candidate Y had {"Excellent", "Fair", "Fair", "Fair"}, the rounded medians would both be "Fair". After removing one "Fair" from each list, the new lists are, respectively, {"Good", "Good", "Poor"} and {"Excellent", "Fair", "Fair"}, so X would win with a recalculated median of "Good".
for rated ballots, the mutual majority criterion
, the monotonicity criterion
, and later-no-help. Assuming that ratings are given independently of other candidates, it satisfies the independence of clones criterion
and the independence of irrelevant alternatives criterion
- although this latter criterion is incompatible with the majority criterion if voters shift their judgments in order to express their preferences
between the available candidates.
Majority judgement fails reversal symmetry
, but this is only because of the rounding used. That is, a candidate whose ratings are {fair, fair} will beat a candidate whose ratings are {good, poor} in both directions, because the more variable rating will be rounded down. However, this failure only happens when a candidate's median is perfectly balanced between two scores, which is unlikely in proportion to the number of voters. The probability of this is negligible; it is only a constant factor higher than that of a perfect tie in a two way race, a situation in which any neutral system can fail to have reversal symmetry.
Majority Judgment voting fails the Condorcet criterion
, later-no-harm, consistency, the Condorcet loser criterion
, and the participation criterion
. It also fails the ranked or preferential majority criterion
, which is incompatible with the passed criterion independence of irrelevant alternatives
.
The median rating for Nashville and Chatanooga is "Good"; for Knoxville, "Fair"; and for Memphis, "Poor". Nashville and Chatanooga are tied, so "Good" ratings have to be removed from both, until their medians become different. After removing 16% "Good" ratings from the votes of each, the sorted ratings are now:
Chatanooga now has the same number of "Fair" ratings as "Good" and "Excellent" combined, so its median is rounded down to "Fair", while Nashville's median remains at "Good" and so Nashville, the capital in real life, wins.
If voters from Knoxville and Chattanooga were to rate Nashville as "Poor" and/or both sets of voters were to rate Chattanooga as "Excellent", in an attempt to make their preferred candidate Chatanooga win, the winner would still be Nashville.
, which has been independently reinvented many times, this aspect is probably the least original. However, voting theory has tended to focus more on ranked systems
, so this still distinguishes it from most voting system proposals. Second, it uses words, not numbers, to assign a commonly-understood meaning to each rating. Badinski and Laraki insist on the importance of the fact that ratings have a commonly-understood absolute meaning, and are not purely relative or strategic. Again, this aspect is unusual but not unheard-of throughout the history of voting. Finally, it uses the median to aggregate ratings. This method was explicitly proposed to assign budgets by Francis Galton
in 1907 and was implicitly used in Bucklin voting
, a ranked or mixed ranked/rated system used soon thereafter in Progressive era
reforms in the United States.
The full system of Majority Judgment was first proposed by Badinski and Laraki in 2007. That same year, they used it in an exit poll of French voters in the presidential election in the Bayrou area. 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 Bayrou, rather than the eventual runoff winner Sarkozy, would have won under most alternative rules, including Majority Judgment. They also note that:
It has since been used in judging wine competitions and in other political research polling in France and the US.
Voting system
A voting system or electoral system is a method by which voters make a choice between options, often in an election or on a policy referendum....
proposed by Michel Balinski and Rida Laraki. Voters freely grade each candidate in one of several named ranks, for instance from "excellent" to "bad", and the candidate with the highest median
Median
In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...
grade is the winner. If more than one candidate has the same median grade, a tiebreaker is used which sees the "closest-to-median" grade. Majority Judgment can be considered as a form of Bucklin voting
Bucklin voting
Bucklin voting is a class of voting systems that can be used for single-member and multi-member districts. It is named after its original promoter, James W. Bucklin of Grand Junction, Colorado, and is also known as the Grand Junction system...
which allows equal ranks.
Voting process
Voters are allowed rated ballots, on which they may assign a grade or judgment to each candidate. Badinski and Laraki suggest six grading levels, from "Excellent" to "Reject". Multiple candidates may be given the same grade if the voter desires.The median grade for each candidate is found, for instance by sorting their list of grades and finding the middle one. If the middle falls between two different grades, the lower of the two is used.
The candidate with the highest median grade wins. If several candidates share the highest median grade, all other candidates are eliminated. Then, one copy of that grade is removed from each remaining candidate's list of grades, and the new median is found, until there is an unambiguous winner. For instance, if candidate X's sorted ratings were {"Good", "Good", "Fair", "Poor"}, while candidate Y had {"Excellent", "Fair", "Fair", "Fair"}, the rounded medians would both be "Fair". After removing one "Fair" from each list, the new lists are, respectively, {"Good", "Good", "Poor"} and {"Excellent", "Fair", "Fair"}, so X would win with a recalculated median of "Good".
Satisfied and failed criteria
Majority Judgment voting satisfies the majority criterionMajority criterion
The majority criterion is a single-winner voting system criterion, used to compare such systems. The criterion states that "if one candidate is preferred by a majority of voters, then that candidate must win"....
for rated ballots, the mutual majority criterion
Mutual majority criterion
The mutual majority criterion is a criterion used to compare voting systems. It is also known as the majority criterion for solid coalitions and the generalized majority criterion...
, the monotonicity criterion
Monotonicity criterion
The monotonicity criterion is a voting system criterion used to analyze both single and multiple winner voting systems. A voting system is monotonic if it satisfies one of the definitions of the monotonicity criterion, given below.Douglas R...
, and later-no-help. Assuming that ratings are given independently of other candidates, it satisfies the independence of clones criterion
Independence of clones criterion
In voting systems theory, the independence of clones criterion measures an election method's robustness to strategic nomination. Nicolaus Tideman first formulated the criterion, which states that the addition of a candidate identical to one already present in an election will not cause the winner...
and the independence of irrelevant alternatives criterion
Independence of irrelevant alternatives
Independence of irrelevant alternatives is an axiom of decision theory and various social sciences.The word is used in different meanings in different contexts....
- although this latter criterion is incompatible with the majority criterion if voters shift their judgments in order to express their preferences
Preferential voting
Preferential voting is a type of ballot structure used in several electoral systems in which voters rank candidates in order of relative preference. For example, the voter may select their first choice as '1', their second preference a '2', and so on...
between the available candidates.
Majority judgement fails reversal symmetry
Reversal symmetry
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, the Kemeny-Young method, and the Schulze method...
, but this is only because of the rounding used. That is, a candidate whose ratings are {fair, fair} will beat a candidate whose ratings are {good, poor} in both directions, because the more variable rating will be rounded down. However, this failure only happens when a candidate's median is perfectly balanced between two scores, which is unlikely in proportion to the number of voters. The probability of this is negligible; it is only a constant factor higher than that of a perfect tie in a two way race, a situation in which any neutral system can fail to have reversal symmetry.
Majority Judgment voting fails the Condorcet criterion
Condorcet criterion
The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared with every other candidate, is preferred by more voters. Informally, the Condorcet winner is the person who would win a two-candidate election against each of the other candidates...
, later-no-harm, consistency, the Condorcet loser criterion
Condorcet method
A Condorcet method is any single-winner election method that meets the Condorcet criterion, which means the method always selects the Condorcet winner if such a candidate exists. The Condorcet winner is the candidate who would beat each of the other candidates in a run-off election.In modern...
, and the participation criterion
Participation criterion
The participation criterion is a voting system criterion. It is also known as the "no show paradox". It has been defined as follows:* In a deterministic framework, the participation criterion says that the addition of a ballot, where candidate A is strictly preferred to candidate B, to an existing...
. It also fails the ranked or preferential majority criterion
Majority criterion
The majority criterion is a single-winner voting system criterion, used to compare such systems. The criterion states that "if one candidate is preferred by a majority of voters, then that candidate must win"....
, which is incompatible with the passed criterion independence of irrelevant alternatives
Independence of irrelevant alternatives
Independence of irrelevant alternatives is an axiom of decision theory and various social sciences.The word is used in different meanings in different contexts....
.
Claimed resistance to tactical voting
In arguing for majority judgment, Balinski and Laraki (the system's inventors) mathematically proved that this system was the most "strategy-resistant" of any system which satisfies certain criteria they consider desirable. While the definition of "strategy-resistant" which they used for this proof is not generally shared, their overall case is bolstered by a study they did on simulated elections, in which, of the systems they studied, majority judgment least-often had its results swayed by strategic voters.Example application
If there were four ratings named "Excellent", "Good", "Fair", and "Poor", and each voter assigned four different ratings to the four cities, then the sorted scores would be as follows:| City |
| |||||||||
| Nashville | ||||||||||
| Chattanooga | ||||||||||
| Knoxville | ||||||||||
| Memphis | ||||||||||
|
The median rating for Nashville and Chatanooga is "Good"; for Knoxville, "Fair"; and for Memphis, "Poor". Nashville and Chatanooga are tied, so "Good" ratings have to be removed from both, until their medians become different. After removing 16% "Good" ratings from the votes of each, the sorted ratings are now:
| City |
| |||||
| Nashville | ||||||
| Chattanooga | ||||||
|
Chatanooga now has the same number of "Fair" ratings as "Good" and "Excellent" combined, so its median is rounded down to "Fair", while Nashville's median remains at "Good" and so Nashville, the capital in real life, wins.
If voters from Knoxville and Chattanooga were to rate Nashville as "Poor" and/or both sets of voters were to rate Chattanooga as "Excellent", in an attempt to make their preferred candidate Chatanooga win, the winner would still be Nashville.
History
This system has several salient features, none of which is original in itself. First, it is a rated system. Since such systems include approval votingApproval voting
Approval voting is a single-winner voting system used for elections. Each voter may vote for as many of the candidates as the voter wishes. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most one vote.The...
, which has been independently reinvented many times, this aspect is probably the least original. However, voting theory has tended to focus more on ranked systems
Preferential voting
Preferential voting is a type of ballot structure used in several electoral systems in which voters rank candidates in order of relative preference. For example, the voter may select their first choice as '1', their second preference a '2', and so on...
, so this still distinguishes it from most voting system proposals. Second, it uses words, not numbers, to assign a commonly-understood meaning to each rating. Badinski and Laraki insist on the importance of the fact that ratings have a commonly-understood absolute meaning, and are not purely relative or strategic. Again, this aspect is unusual but not unheard-of throughout the history of voting. Finally, it uses the median to aggregate ratings. This method was explicitly proposed to assign budgets by Francis Galton
Francis Galton
Sir Francis Galton /ˈfrɑːnsɪs ˈgɔːltn̩/ FRS , cousin of Douglas Strutt Galton, half-cousin of Charles Darwin, was an English Victorian polymath: anthropologist, eugenicist, tropical explorer, geographer, inventor, meteorologist, proto-geneticist, psychometrician, and statistician...
in 1907 and was implicitly used in Bucklin voting
Bucklin voting
Bucklin voting is a class of voting systems that can be used for single-member and multi-member districts. It is named after its original promoter, James W. Bucklin of Grand Junction, Colorado, and is also known as the Grand Junction system...
, a ranked or mixed ranked/rated system used soon thereafter in Progressive era
Progressive Era
The Progressive Era in the United States was a period of social activism and political reform that flourished from the 1890s to the 1920s. One main goal of the Progressive movement was purification of government, as Progressives tried to eliminate corruption by exposing and undercutting political...
reforms in the United States.
The full system of Majority Judgment was first proposed by Badinski and Laraki in 2007. That same year, they used it in an exit poll of French voters in the presidential election in the Bayrou area. 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 Bayrou, rather than the eventual runoff winner Sarkozy, would have won under most alternative rules, including Majority Judgment. They also note that:
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 the US.
See also
- Approval votingApproval votingApproval voting is a single-winner voting system used for elections. Each voter may vote for as many of the candidates as the voter wishes. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most one vote.The...
- Range votingRange votingRange voting is a voting system for one-seat elections under which voters score each candidate, the scores are added up, and the candidate with the highest score wins.A form of range voting was apparently used in...
- Voting systemVoting systemA voting system or electoral system is a method by which voters make a choice between options, often in an election or on a policy referendum....
- List of democracy and elections-related topics