Majority Judgment

Majority Judgment is a single-winner voting system proposed by Michel Balinski and Rida Laraki. 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 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.

It 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, and reversal symmetry.

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:

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:

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" 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. . 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.