# Majority judgment

Majority judgment (MJ) is a single-winner voting system proposed by Michel Balinski and Rida Laraki.[1] 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.

## Satisfied and failed criteria

Majority judgment voting satisfies the majority criterion for rated ballots, the mutual majority criterion, 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.

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

Majority judgment voting fails the Condorcet criterion,[note 1] later-no-harm,[note 2] consistency,[note 3] the Condorcet loser criterion,[note 4] and the participation criterion.[note 5] It also fails the ranked or preferential majority criterion, which is incompatible with the passed criterion independence of irrelevant alternatives. However, the importance of these failures are diminished by Balinski's response to the following article.

### Felsenthal and Macover’s Discuss of MJ

In 2008, Felsenthal and Macover's article [3] usefully discussed MJ as presented by Balinski and his associates in 2007. However, the last part of their discussion claims that MJ is "afflicted" most seriously by the fact that it can fail the tests of "participant-consistency". For example, the "no-show objection" refers to the paradox that a candidate who is given a higher grade than is need to win can lose as a result.

### Claimed resistance to tactical voting

In arguing for majority judgment, Balinski and Laraki (the system's inventors) logically and mathematically prove that MJ is the most "strategy-resistant" of any system that satisfies certain criteria considered desirable by the authors. They show that MJ provides only about "half" the opportunities and incentives to vote tactically (dishonestly, strategically) when compared with the alternative methods.[4]

### Outcome in political environments

In 2010, J.-F. Laslier showed [5] that in "left-right" environments, majority judgment tends to favor the most homogeneous camp, instead of picking the middle-of-the-road, Condorcet winner candidate. The reason is that, by definition, finding the highest median is something like finding the best rawlsian compromise (maximin criterion) when one allows disregarding almost half of the population.[6]

Here is a numerical example. Suppose there were seven ratings named "Excellent", "Very good", "Good", "Passable", "Inadequate", "Mediocre" and "Bad". Supposed voters belong to seven groups: Extreme Left, Left, Center Left, Center, Center Right, Right and Extreme Right, and the size of the groups are respectively : 101 voters for each of the three groups on the left, 99 for each of the three groups on the right and 50 for the centrist group. Suppose there are seven candidates, one from each group, and voters assigned their ratings to the seven candidates by giving the candidate closest to their own ideological position the rating "Excellent", and then decreasing the rating as candidates are politically further away from them:

Candidate
101 Ext. left
voters
101 Left
voters
101 Cent. left
voters
50 Center
voters
99 Cent. right
voters
99 Right
voters
99 Ext. right
voters
Median
rating
Centerpassablegoodvery goodexcellentvery goodgoodpassablegood

The tie-breaking procedure of Majority Judgment elects the Left candidate by an absolute majority, even though the Center Left candidate received a larger absolute majority. This is because the majority of the Center Left candidate is composed proportionately of fewer Excellents and Very Goods than the majority of the Left candidate (see above).

The above analysis may suggest that the winner should instead be the candidate who has receive the largest absolute majority with the highest median-grade. Accordingly, Bosworth, et al. suggest that a modification of MJ would guarantee this preferred result. They call this modification MJ+. MJ+ replaces MJ's existing tie breaking procedure by instead electing the tied candidate who has received the largest majority of grades equal to or higher than the highest median-grade.[7] MJ+ would elect the Center Left candidate in the above example.

Candidate
 ↓ Median point
Left
Center left
Center
Center right
Right

 Excellent Very good good passable or less

## 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:

42% of voters
(close to Memphis)
26% of voters
(close to Nashville)
15% of voters
(close to Chattanooga)
17% of voters
(close to Knoxville)
1. Memphis
2. Nashville
3. Chattanooga
4. Knoxville
1. Nashville
2. Chattanooga
3. Knoxville
4. Memphis
1. Chattanooga
2. Knoxville
3. Nashville
4. Memphis
1. Knoxville
2. Chattanooga
3. Nashville
4. Memphis

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:

City Choice
Memphis
voters
Nashville
voters
Chattanooga
voters
Knoxville
voters
Median
rating[note 6]
Memphisexcellentpoorpoorpoorpoor+
Nashvillefairexcellentfairfairfair+
Chattanoogapoorfairexcellentgoodfair-
Knoxvillepoorfairgoodexcellentfair-

Then the sorted scores would be as follows:

City
 ↓ Median point
Nashville
Knoxville
Chattanooga
Memphis

 Excellent Good Fair Poor

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:

City
 ↓ Median point
Nashville
Knoxville
Chattanooga

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[8] 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.[9] 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.[10]

It has since been used in judging wine competitions and in other political research polling in France and in the US.[11]

## Notes

1. Strategically in the strong Nash equilibrium, MJ passes the Condorcet criterion.
2. MJ provides a weaker guarantee similar to LNH: rating another candidate at or below your preferred winner's median rating (as opposed to one's own rating for the winner) cannot harm the winner.
3. Majority judgment's inventors argue that meaning should be assigned to the absolute rating that the system assigns to a candidate; that if one electorate rates candidate X as "excellent" and Y as "good", while another one ranks X as "acceptable" and Y as "poor", these two electorates do not in fact agree. Therefore, they define a criterion they call "rating consistency", which majority judgment passes. Balinski and Laraki, "Judge, don't Vote", November 2010
4. Nevertheless, it passes a slightly weakened version, the majority condorcet loser criterion, in which all defeats are by an absolute majority (if there are not equal rankings).
5. It can fail the participation criterion only 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 sharply peaked or irregular for one of the candidates.
6. A "+" or "-" is added depending on whether the median would rise or fall if median ratings were removed, as in the tiebreaking procedure.

## References

1. M. Balinski & R. Laraki (2010). Majority Judgment. MIT. ISBN 978-0-262-01513-4.
2. Balinski and Laraki, Majority Judgment, pp.5 & 14
3. Felsenthal, Dan S. and Machover, Moshé, "The Majority Judgement voting procedure: a critical evaluation", Homo oeconomicus, vol 25(3/4), pp. 319-334 (2008)
4. Balinski and Laraki, Majority Judgment, pp. 15,17,19,187-198, and 374
5. Jean-François Laslier (2010). "On choosing the alternative with the best median evaluation". Public Choice.
6. Jean-François Laslier (2018). "The strange "Majority Judgment"". Hal.
7. Bosworth, Stephen; Corr, Ander & Leonard, Stevan (July 8, 2019). "Legislatures Elected by Evaluative Proportional Representation (EPR): an Algorithm; Endnote 8". Journal of Political Risk. 7 (8). Retrieved August 19, 2019.
8. Francis Galton, "One vote, one value," Letter to the editor, Nature vol. 75, Feb. 28, 1907, p. 414.
9. 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.
10. Balinski M. and R. Laraki (2007) «Election by Majority Judgment: Experimental Evidence». Cahier du Laboratoire d’Econométrie de l’Ecole Polytechnique 2007-28. Chapter in the book: «In Situ and Laboratory Experiments on Electoral Law Reform: French Presidential Elections», Edited by Bernard Dolez, Bernard Grofman and Annie Laurent. Springer, to appear in 2011.
11. Balinski M. and R. Laraki (2010) «Judge: Don't vote». Cahier du Laboratoire d’Econométrie de l’Ecole Polytechnique 2010-27.
• Balinski, Michel, and Laraki, Rida (2010). Majority Judgment: Measuring, Ranking, and Electing, MIT Press