Majority Judgment
Majority Judgment is a singlewinner voting system proposed by Michel Balinski and Rida Laraki.^{[1]}^{[2]} Voters freely score each candidate in one of several named qualities, for instance from "excellent" to "bad". Each quality is associated with a numeric score and the candidate with the highest median score is the winner. If more than one candidate has the same median grade, a tiebreaker is used which sees how "broad" that median grade is. Majority Judgment can be considered as a form of Bucklin voting which allows equal ranks.
Voting process
Voters are allowed rated ballots, on which they may assign a grade or judgement to each candidate. Badinski and Laraki suggest six grading levels, from "Excellent" to "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 higest 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 an unambiguous winner is found.
Satisfied and failed criteria
Majority Judgment voting satisfies the majority criterion for rated ballots, and also a weak form of the mutual majority criterion (a majority giving only and all of their preferred candidates perfect grades will win), the monotonicity criterion, and laternohelp. Assuming that ratings are given independently of other candidates, it satisfies the independence of clones criterion and the independence of irrelevant alternatives criterion^{[3]}  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.
It fails the Condorcet criterion,^{[nb 1]} laternoharm,^{[nb 2]} consistency, the Condorcet loser criterion,^{[nb 3]} and the participation criterion.^{[nb 4]} It also fails the ranked or preferential majority criterion, which is incompatible with the passed criterion independence of irrelevant alternatives, and reversal symmetry.
Example application
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, the state's largest city, with 42% of the voters, but located far from the other cities
 Nashville, with 26% of the voters, near the center of Tennessee
 Knoxville, with 17% of the voters
 Chattanooga, 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) 





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 Chattanooga is "Good"; for Knoxville, "Fair"; and for Memphis, "Poor". Nashville and Chattanooga 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 



Chattanooga 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"^{[nb 5]} 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 Chattanooga win, the winner would still be Nashville.
Variants
Variants of majority judgment have been described. Fabre considers three: the typical judgment, usual judgment, and central judgment.^{[4]}. He argues that all of these are less sensitive to noise than the majority judgment, with the usual judgment being the most robust, though the calculation that determines the winner is more complex.
See also
Notes
 ↑ Strategically in the strong Nash equilibrium, MJ passes the Condorcet criterion.
 ↑ MJ provides a weaker guarantee similar to LNH: rating another candidate at or below your preferred winner's median rating (as opposed to your own rating for the winner) cannot harm the winner.
 ↑ Nevertheless, it passes a slightly weakened version, the majority condorcet loser criterion, in which all defeats are by an absolute majority (for instance, if there aren't equal rankings).
 ↑ It can only fail the participation criterion when, among other conditions, the new ballot rates both of the candidates in question on the same side of the winning median, and the prior distribution of ratings is more sharplypeaked or irregular for one of the candidates.
 ↑ After removal, Chattanooga has 42% of the initial electorate at "Fair", 27% "Good", and 15% "Excellent", while Nashville has 32% "Fair", 26% "Good", and 26% "Excellent"
References
 ↑ M. Balinski & R. Laraki (2010). Majority Judgment. MIT Press. ISBN 9780262015134.
 ↑ de Swart, Harrie (20211116). "How to Choose a President, Mayor, Chair: Balinski and Laraki Unpacked". The Mathematical Intelligencer. doi:10.1007/s00283021101243. ISSN 03436993.
 ↑ Badinski and Laraki, Majority Judgment, p. 217
 ↑ Fabre, Adrien (2020). "Tiebreaking the Highest Median: Alternatives to the Majority Judgment" (PDF). Social Choice and Welfare. 56: 101–124. doi:10.1007/s00355020012699. ISSN 01761714.