1. Voting process

2. Satisfied and failed criteria
     Claimed resistance to tactical voting    Outcome in political environments    Effect of the range of appreciations used  

3. Example application

4. History

5. See also

6. Notes

7. References

8. Further reading

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.

Voting process

Voters are allowed rated ballots, on which they may assign a grade or judgment to each candidate. Balinski and Laraki suggest six grading levels, from "Excellent" to "Reject", as used in some French schools. Multiple candidates may be given the same grade if the voter wants to.

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". To help communicate how this tiebreaker works, a plus sign or minus sign can be added to the median of each candidate, depending on whether the median would eventually rise or fall if the tiebreaker were applied to them.

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 ratings are given independently of other candidates, it satisfies the independence of clones criterion and the independence of irrelevant alternatives criterion,^[1] but the latter criterion is incompatible with the majority criterion if voters shift their judgments in order to express their preferences between the available candidates.

Majority judgment 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,^[2] later-no-harm,^[3] consistency,^[4] the Condorcet loser criterion,^[5] and the participation criterion.^[6] It also fails the ranked or preferential majority criterion, which is incompatible with the passed criterion independence of irrelevant alternatives.

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 that satisfies certain criteria considered desirable by the authors. While the definition of "strategy-resistant" used for the proof is not generally shared, their overall case is bolstered by a study they did on simulated elections in which, of the systems that they studied, majority judgment was the system that had its results swayed by strategic voters the least often.

Outcome in political environments

In 2010, J.-F. Laslier showed^[7] 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 best median is equivalent to finding the best rawlsian compromise (maximin criterion) when one allows to disregard up to half of the population.^[8]

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 their own candidate the rating "Excellent", and then decreasing the rating as candidates are politically further away from them:

The tie-breaking procedure of the Majority Judgment imposes to elect the Left candidate whose tally is the closest to "very good". In this example, the Majority Judgment elects the Left candidate despite the fact that a large majority (448 against 202) would prefer a more moderate candidate. Precisely, the sorted scores would be as follows:

The total population is 650, and 303 voters value Left as Very good or more and 297 voters value Right as Very good or more, so the tie-break elects Left.

Effect of the range of appreciations used

An apparent limitation ^[9] of this system can be illustrated by the following example :

To generalise, if voters give grades that are on the same side of the final median for all candidates, their votes have no impact on the result. This incentivizes using a wider range of grades to distinguish between candidates.

Example application

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, so their medians are rounded down to "Poor", 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

This system has several salient features, none of which is original in itself. First, it is a rated system. Since such systems include approval voting, 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. Balinski 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^[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.^[12] 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.^[13]

It has since been used in judging wine competitions and in other political research polling in France and in the US.^[14]

See also

• 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

• John Sloat
• John Sloboda
• John Sloman
• John Sloman's Badlands
• John Slough
• John S. Macdonald
• John S. MacDonald
• John Smalbroke
• John Small
• John Smallberries
• John Small (Canadian MP)
• John Small (Canadian politician)
• John Small (cricket)
• John Small (disambiguation)
• John Small (Hambledon cricketer)
• John Small MacDonald
• John Small, Sr
• John Small (Upper Canada politician)
• John Smallwood
• Johns-Manville Corporation
• John S. McCain (DDG 56)
• John S. McCain Sr.
• John S. McGroarty
• John S. McKiernan
• John Smeaton Academy