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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 uses a highest median rule, i.e., a cardinal voting system that elects the candidate with the highest median rating.

Unlike other voting methods, MJ guarantees that the winner between three or more candidates will be the candidate who had received an absolute majority of the highest grades given by all the voters.

Voting process

Voters grade as many of the candidates' as they wish with regard to their suitability for office as either Excellent (ideal), Very Good, Good, Acceptable, Poor, or Reject. Multiple candidates may be given the same grade by a voter. Any candidate not explicitly graded by a voter is counted as having been rejected by the voter. Therefore, each candidate receives the same total number of grades, but a different distribution of them.

The candidate with the highest median grade is the winner. This median-grade can be found as follows: Place all the grades, high to low, top to bottom, in side-by-side columns, the name of each candidate at the top of each of these columns. The median-grade for each candidate is the grade located halfway down each column, i.e. in the middle if there is an odd number of voters, the lower middle if the number is even. If more than one candidate has the same highest median-grade, the MJ winner is discovered by removing (one-by-one) any grades equal in value to the shared median grade from each tied candidate's column. This is repeated until only one of the previously tied candidates is currently found to have the highest median grade.^[4] Equivalently, the candidates can be ranked according to a simple mathematical formula described on the page of highest median voting rules, or we can consider the median as being estimated with nearest-neighbor interpolation.

Discussion

As it is a highest median rule, MJ produces more informative results than the existing alternatives. It is true that if only one of two candidates is to be elected, and the winner has only a few votes more than the near winner, MJ and all the alternative voting methods would discover the same winner. However, unlike highest median rules, none of the alternative methods inform us whether the voters saw great merit in both, saw little merit in either, or saw merit in one but not the other. The published results of an MJ election reveal how many grades of each level were given to each candidate by all voters, including the median grade of each candidate.

This same educative benefit is also offered by evaluative proportional representation (EPR), an adaption of MJ to elect all the members of a legislature at the same time.^[5] With EPR, each voter can also guarantee that their vote will proportionally add to the voting power of the elected member of the legislature to whom they had given their highest grade, highest remaining grade, or proxy vote. No vote is "wasted". Each voter and each self-identifying minority or majority is represented proportionately. EPR offers voters an even smaller incentive to vote tactically than does MJ (see below). Unlike MJ, each EPR voter is assured that their vote will equally increase the voting power in the legislature of the winner they give their highest grade, highest remaining grade, or proxy vote.)

Satisfied and failed criteria

Like other highest median rules, majority judgment satisfies the majority criterion for rated ballots, the monotonicity criterion, and the later-no-help criterion. By assuming that grades are given independently of other candidates, it satisfies the independence of clones criterion and the independence of irrelevant alternatives criterion, but the latter criterion is less compatible with the majority criterion if voters instead use their grades only to express preferences between the available candidates. MJ may also pass a rated ballot-analogue of the mutual majority criterion, but this is hard to understand or seems irrelevant given that this criterion asserts that if a majority of voters prefer a set of candidates above all others, then someone in this set will win so long as the majority gives everyone in the set a perfect rating and everyone not in the set a less-than-perfect rating. This is because the median voter will be someone in the majority, and they will give everyone in the set a perfect rating, and everyone not in the set a less-than-perfect rating.

Majority judgment fails reversal symmetry. For example, a candidate whose grades are {Acceptable, Acceptable} will still beat a candidate whose ratings are {good, poor} in both directions.

Like any deterministic voting system without a dictator, MJ allows for tactical voting in cases of more than three candidates. This is a consequence of Gibbard's theorem.

Majority judgment voting fails the Condorcet criterion,^{[note 1]} later-no-harm,^{[note 2]} consistency,^{[note 3]} the Condorcet loser criterion, and the participation criterion. It also fails the majority criterion but satisfies a weakened version of it—if only one candidate receives perfect grades from a majority of all voters, this candidate will win.

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 to 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 the state
• Knoxville, with 17% of the voters
• Chattanooga, with 15% of the voters

The preferences of the voters would be divided like this:

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.

If voters were more strategic, those from Knoxville and Chattanooga might rate Nashville as "Poor" and Chattanooga as "Excellent", in an attempt to make their preferred candidate Chattanooga win. Also, Nashville voters might rate Knoxville as "poor" to distinguish it from Chattanooga. In spite of these attempts at strategy, the winner would still be Nashville. .

History

Voting theory has tended to focus more on ranked systems, so this still distinguishes MJ from most voting system proposals. Second, it uses words, not numbers, to assign a commonly understood meaning to each rating. Balinski and Laraki insist on the importance of the fact that ratings have a commonly understood absolute meaning. Firstly, MJ prompts voters to clarify in their own minds what qualities the office requires. These qualities are "absolute" in the sense that they are independent from any of the qualities any candidates might have or might not have in a future election. They 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^[11] 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. Also, hybrid mean/median systems, which throw away a certain predefined number of outliers on each side and then average the remaining scores, have long been used to judge contests such as Olympic figure skating; such systems, like majority judgment, are intended to limit the impact of biased or strategic judges.

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.^[12]

It has since been used in judging wine competitions and in other political research polling in France and in the US.^[13]

Variants

Varloot and Laraki^[14] 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
• Range voting
• Voting system
• 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