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{{Short description|Single-winner cardinal voting system}}
{{Npov|date=October 2020}}
 
{{Electoral systems}}
'''Majority judgment''' ('''MJ''') is a single-winner [[voting system]] proposed in 20072010 by [[Michel Balinski]] and Rida Laraki.<ref name=":0" /><ref>{{cite book|authorfirst1= M. |last1=Balinski & |first2=R. |last2=Laraki|year=2010|title=Majority Judgment. |publisher=MIT |isbn=978-0-262-01513-4}}</ref><ref>{{Cite journal|last=de Swart|first=Harrie|date=2021-11-16|title=How to Choose a President, Mayor, Chair: Balinski and Laraki Unpacked|journal=The Mathematical Intelligencer|volume=44 |issue=2 |pages=99–107 |language=en|doi=10.1007/s00283-021-10124-3|s2cid=244289281 |issn=0343-6993|doi-access=free}}</ref> It is a kind of [[highest median voting rule|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 according to a series of grades. Balinski and Laraki suggest the options "Excellent, Very Good, Good, Acceptable, Poor, or Reject," but any scale can be used (e.g. the common [[letter grade]] scale). Voters can assign the same grade to multiple candidates.
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. 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 total. This is repeated until only one of the previously tied candidates is currently found to have the highest median-grade.<ref>Balinski and Laraki, ''Majority Judgment'', pp.5 & 14</ref>
 
As with all [[highest median voting rules]], the candidate with the highest [[median]] grade is declared winner. If more than one candidate has the same median grade, majority judgment breaks the tie by removing (one-by-one) any grades equal to the shared median grade from each tied candidate's column. This procedure is repeated until only one of the tied candidates is found to have the highest median grade.<ref>Balinski and Laraki, ''Majority Judgment'', pp.5 & 14</ref>
== Discussion ==
== Advantages and disadvantages ==
As it is a [[highest median voting rule|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. Only the published results of a election by the highest median would report exactly how all the voters had graded all the candidates.
(This same benefit is also offered by [[Proportional Representation#Evaluative Proportional Representation (EPR)|Evaluative Proportional Representation (EPR)]], an adaption of MJ to elect all the members of a legislature at the same time. With EPR, each voter can also guarantee that their vote will proportionately 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 vote|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 proportionately 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 ==
{{See also|Tactical voting#Majority judgment}}
 
Like most other [[cardinal voting]] rules, majority judgment satisfies the [[monotonicity criterion]], the [[later-no-help criterion]], and [[independence of irrelevant alternatives]].
Like other [[highest median voting rule|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|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 [[Ranked voting systems|preferences]] between the available candidates. It passes a rated ballot-analogue of the [[mutual majority criterion]]: 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.
 
Like any deterministic voting system (except [[Dictatorship mechanism|dictatorship]]), MJ allows for [[tactical voting]] in cases of more than three candidates, as a consequence of [[Gibbard's theorem]].
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.
 
Like any deterministic voting system without a dictator, MJ allows for [[tactical voting]] in case of three or more candidates. This is a consequence of [[Gibbard's theorem]].
Majority judgment voting fails the [[Condorcet criterion]],<ref group=note>{{efn|Strategically in the [[strong Nash equilibrium]], MJ passes the Condorcet criterion, just like [[score voting]].</ref>}} [[later-no-harm]],<ref group=note>{{efn|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.</ref> }}[[consistency criterion for voting systems|consistency]],<ref group=note>{{efn|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, [https://1007421605497013616-a-1802744773732722657-s-sites.googlegroups.com/site/ridalaraki/xfiles/BalinskiLarakiJudgeDontVotecahierderecherche2010-27.pdf "Judge, don't Vote"], November 2010</ref>}} the [[Condorcet loser criterion]],<ref group=note>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).</ref> and the [[participation criterion]].<ref, group=note>Itthe can[[Majority failfavorite the participationcriterion|majority criterion only when]],{{efn|MJ amongsatisfies othera conditions,weakened the new ballot rates bothversion of the candidatesmajority incriterion—if questiononly onone thecandidate samereceives sideperfect ofgrades thefrom winninga median, and the prior distributionmajority of ratingsall isvoters, morethis sharplycandidate peaked or irregular for one of thewill candidateswin.</ref>}} It also failsand the ranked or preferential [[mutual 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.
 
=== Participation failure ===
Unlike [[score voting]], majority judgment can have [[No show paradox|no-show paradoxes]],<ref>Felsenthal, Dan S. and Machover, Moshé, ''[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.324.1143&rep=rep1&type=pdf "The Majority Judgement voting procedure: a critical evaluation"]'', Homo oeconomicus, vol 25(3/4), pp. 319-334 (2008)</ref> situations where a candidate loses because they won "too many votes". In other words, adding votes that rank a candidate higher than their opponent can still cause this candidate to lose.
 
In their 2010 book, Balinski and Laraki demonstrate that the only join-consistent methods are point-summing methods, a slight generalization of [[score voting]] that includes [[positional voting]].<ref name=":2">{{Citation |last1=Balinski |first1=Michel |chapter=Majority Judgment |date=2011-01-28 |pages=295–301 |chapter-url=http://dx.doi.org/10.7551/mitpress/9780262015134.003.0001 |access-date=2024-02-08 |publisher=The MIT Press |last2=Laraki |first2=Rida|doi=10.7551/mitpress/9780262015134.003.0001 |isbn=978-0-262-01513-4 }}</ref> Specifically, their result shows the only methods satisfying the slightly stronger [[consistency criterion]] have:
 
<math>\sum_{\text{vote} \in \text{ballots}} f(\text{score}_\text{vote})</math>
 
Where <math>f</math> is a [[monotonic function]]. Moreover, any method satisfying both participation and either [[Continuous function|stepwise-continuity]] or the [[Archimedean property]]{{efn|Balinski and Laraki refer to this property as "respect for large electorates."}} is a point-summing method.<ref>{{Citation |last1=Balinski |first1=Michel |chapter=Majority Judgment |date=2011-01-28 |pages=300–301 |chapter-url=http://dx.doi.org/10.7551/mitpress/9780262015134.003.0001 |access-date=2024-02-08 |publisher=The MIT Press |last2=Laraki |first2=Rida|doi=10.7551/mitpress/9780262015134.003.0001 |isbn=978-0-262-01513-4 }}</ref>
=== Felsenthal and Macover ===
In 2008, Felsenthal and Macover's article <ref>Felsenthal, Dan S. and Machover, Moshé, ''[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.324.1143&rep=rep1&type=pdf "The Majority Judgement voting procedure: a critical evaluation"]'', Homo oeconomicus, vol 25(3/4), pp. 319-334 (2008)</ref> 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.
 
This result is closely related to and relies on the [[Von Neumann–Morgenstern utility theorem]] and [[Harsanyi's utilitarian theorem]], two critical results in [[social choice theory]] and [[decision theory]] used to characterize the conditions for [[Rational choice theory|rational choice]].
In their 2010 book, Balinski and Laraki respond by explaining (pp.&nbsp;285–295) that this theoretical possibility is inherent in any method which uses "medians" rather than any "point-summing" method to aggregate all citizens' votes. They go on to describe the following unlikely special circumstances that might allow such a "failure" to occur by chance when using MJ:
1) Initially, the total number of voters must be odd.
2) By these votes alone, candidate X would win and Y would lose (e.g. because X's median grade is "Very Good" and Y's is "Good").
3) Both of the potential additional grades to be given to X and Y must either happen to be higher or lower than each competitor's current median grade.
4) The grade immediately below the current median grade of the current winner(X) must be at least 2 grades lower, e.g. "Acceptable" rather than "Very Good". At the same time, the grade immediately below the current median grade of the current loser, but potential winner (Y), must be higher than the grade immediately below the current median grade of the current winner(X).
 
Despite this result, Balinski and Laraki claim that participation failures would be rare in practice for majority judgment.<ref name=":2" />
For example, if the additional grades given to X were "Excellent", and "Very Good" to Y, Y would become the winner instead. Y's new median grade would be "Good" (given the above examples). X's new median grade would be "Acceptable".
Balinski accepts that such paradoxes are not possible with "point-summing" counting methods, but are possible with MJ. At the same time, he argues that these are not very important. This is illustrated by the above example. The additional voter should not be very disappointed if, very rarely if ever, their "Very Good" candidate wins instead of their "Excellent" candidate. Consequently, such paradoxes are a very small price to pay for the clear advantages offered by MJ. Unlike MJ, they do not guarantee that the winner is supported by an absolute majority of all the voters, do not reveal all the grades given to all the candidates by all the voters, and do not remove, almost by "half", the opportunities and incentives to vote insincerely (tactical or strategically), and do not prompt voters as clearly--- firstly to consider the qualities required by the office being sought.
 
=== Claimed resistance to tactical voting ===
In arguing for majority judgment, Balinski and Laraki (the system's inventors) logically and mathematically prove that [[highest median voting rule|highest median rulerules]] are the most "strategy-resistant" of any system, that satisfies certain criteria considered desirable byin the authors. They showsense that MJthey providesminimize onlythe aboutshare "half"of the opportunitieselectorate andwith incentivesan incentive to vote tactically (dishonestly, strategically) when comparedbe with the alternative methodsdishonest.<ref>Balinski and Laraki, ''Majority Judgment'', pp.&nbsp;15,17,19,187-198, and 374</ref> However, some writers have disputed the significance of these results, as they do not apply in cases of imperfect information or collusion between voters.{{cn|date=March 2024}}
 
=== OutcomeMedian invoter political environmentsproperty ===
In 2010"left-right" environments, [[:fr:Jean-Françoismajority Laslier|J.judgment tends to favor the most homogeneous camp, instead of picking the middle-F.of-the-road, Laslier]]Condorcet showedwinner candidate.<ref>{{cite web |author=Jean-François Laslier |date=2010 |title=On choosing the alternative with the best median evaluation |url=https://halshs.archives-ouvertes.fr/hal-00397403/document |website=Public Choice|date=2010}}</ref> that in "left-right" environments, majorityMajority judgment tendstherefore to favorfails the most[[Median homogeneous camp, instead of picking the middle-of-the-road, Condorcet winner candidate. The reason is that, by definition, finding the highestvoter theorem|median is something like finding the best Rawlsian compromise ([[Minimax|maximinvoter criterion]]) when one allows disregarding almost half of the population.<ref>{{cite web |author=Jean-François Laslier |date=2018 |title=The strange "Majority Judgment" |url=https://halshs.archives-ouvertes.fr/hal-01965227 |website=Hal|date=2018}}</ref>
 
Here is a numerical example. Suppose there were seven ratings named "Excellent"," "Very good"," "Good", "Passable"Mediocre," "Inadequate"Bad," "MediocreVery Bad," and "Bad"Awful." SupposedSuppose voters belong to seven groups: Extremeranging Left,from Left,"Far-left" Centerto Left"Far-right, Center, Center Right, Right" and Extreme Right, and the size of the groups are respectively : 101 voters for each ofgroup theruns threea groupssingle on the left, 99 for each of the three groups on the right and 50 for the centrist groupcandidate. SupposeVoters there are sevenassign candidates, one from each group, and voters assigned their ratings to the seven candidates by giving the candidate closest to their own ideologicalgroup position thea rating of "Excellent", and" then decreasingdecrease the rating as candidates are politically further away from them:.
{| class="wikitable"
|-
!<br /><br />{{Diagonal split header|Candidate|Votes}}!! 101 votes
!<br /><br />Candidate !! 101 voters <br />Ext. left !! 101 voters <br /> Left !! 101 voters<br /> Cent. left!! 50 voters <br /> Center!!99 voters <br /> Cent. right !!99 voters<br /> Right!! 99 voters <br /> Ext. right !!Median
Far-left
! 101 votes
Left
! 101 votes
Cen. left
! 50 votes
Center
!99 votes
Cen. right
!99 votes
Right
! 99 votes
Far-right
!Score
|-
| ExtFar left|| bgcolor="darkgreen" | excellentexcel.|| bgcolor="forestgreen" | veryv. good || bgcolor="yellowgreen" | good || bgcolor="yellow"| passablemed. || bgcolor="orange" | inadequatebad|| bgcolor="orangered" |mediocrevery bad || bgcolor="red" | badawful || bgcolor="yellow" | passablemed.
|-
| Left|| bgcolor="forestgreen"| veryv. good || bgcolor="darkgreen" | excellentexcel. || bgcolor="forestgreen" | veryv. good || bgcolor="yellowgreen" | good || bgcolor="yellow"| passablemed.|| bgcolor="orange" | inadequatebad || bgcolor="orangered" | mediocrevery bad || bgcolor="yellowgreen" | good
|-
| CenterCen. left|| bgcolor="yellowgreen" | good || bgcolor="forestgreen"| veryv. good || bgcolor="darkgreen" | excellentexcel. || bgcolor="forestgreen" | veryv. good || bgcolor="yellowgreen" | good|| bgcolor="yellow"| passablemed. || bgcolor="orange" | inadequatebad || bgcolor="yellowgreen" | good
|-
| Center|| bgcolor="yellow"| passablemed. || bgcolor="yellowgreen" | good || bgcolor="forestgreen"| veryv. good || bgcolor="darkgreen" | excellentexcel. || bgcolor="forestgreen" | veryv. good || bgcolor="yellowgreen" | good|| bgcolor="yellow"| passablemed. || bgcolor="yellowgreen" | good
|-
| CenterCen. right|| bgcolor="orange" | inadequatebad|| bgcolor="yellow" | passablemed. || bgcolor="yellowgreen" | good || bgcolor="forestgreen"| veryv. good || bgcolor="darkgreen" | excellentexcel. || bgcolor="forestgreen" | veryv. good || bgcolor="yellowgreen" | good || bgcolor="yellowgreen"| good
|-
| Right|| bgcolor="orangered"| mediocrevery bad|| bgcolor="orange" | inadequatebad|| bgcolor="yellow" | passablemed. || bgcolor="yellowgreen" | good || bgcolor="forestgreen"| veryv. good || bgcolor="darkgreen" | excellentexcel. || bgcolor="forestgreen" | veryv. good || bgcolor="yellowgreen" | good
|-
| ExtFar right|| bgcolor="red" | badawful || bgcolor="orangered" | mediocrevery bad|| bgcolor="orange" | inadequatebad|| bgcolor="yellow" | passablemed. || bgcolor="yellowyellowgreen" |good || bgcolor="forestgreen"| veryv. good || bgcolor="darkgreen" | excellentexcel. || bgcolor="yellow" | passablemed.
|}
 
The tie-breaking procedure of Majoritymajority Judgmentjudgment elects the Left candidate, as this candidate is the one with the non-median rating closest to the median, and this non-median rating is above the median rating. In so doing, the majority judgment elects the best compromise for voters on the left side of the political axis (as they are slightly more numerous than those on the right) instead of choosing a more consensual candidate such as the centrecenter-left or the center. The reason is that the tie-breaking is based on the rating closest to the median, regardless of the other ratings.
 
Note that other [[highest median voting rules|highest median rules]] that take into account the ratings on either side of the median, such as the [[typicalgraduated judgment]] or the [[usualmajority judgment]], wouldwill notoften electmake thedifferent Lefttie-breaking candidatedecisions as(and in the case of the[[graduated majority judgment, but]] would elect the Center candidate. These other rules would in this case respect the [[Condorcet criterion]]). These methods, introduced more recently, thusmaintain verify themany desirable properties of the majority judgment while avoiding the pitfalls of its maintie-breaking pitfallsprocedure.<ref name="Fabre20">{{Cite journal |first=Adrien |last=Fabre |title=Tie-breaking the Highest Median: Alternatives to the Majority Judgment |journal=[[Social Choice and Welfare]]|date=2020 |volume=56 |pages=101–124 |url=https://github.com/bixiou/highest_median/raw/master/Tie-breaking%20Highest%20Median%20-%20Fabre%202019.pdf |doi=10.1007/s00355-020-01269-9 |s2cid=253851085 }}</ref>
{|
| align="right" |Candidate&nbsp;&nbsp;&nbsp;
|-
| align=right | Candidate&nbsp;&nbsp;&nbsp;
|
{| cellpaddingcellspacing="0 width=650" border="0" cellspacingwidth="650" cellpadding="0"
| width="49%" |&nbsp;
|-
| width=49"2%" | &nbsp;↓
| width=2"49% textalign=center" | ↓Median
| width=49% | Median
|}
|-
| align="right" | Left
|
{| cellpaddingcellspacing="0 width=650" border="0" cellspacingwidth="650" cellpadding="0"
| width="101" bgcolor="darkgreen" |&nbsp;
|-
| width="202" bgcolor="forestGreen" |
| bgcolor=darkgreen width=101 | &nbsp;
| width="50" bgcolor="yellowGreen" |
| bgcolor=forestGreen width=202 |
| width="99" bgcolor="yellow" |
| bgcolor=yellowGreen width=50 |
| bgcolor=yellow width="99" bgcolor="orange" |
| bgcolor=orange width="99" bgcolor="orangered" |
| bgcolor=orangered width=99 |
|}
|-
| align="right" | CentreCenter left
|
{| cellpaddingcellspacing="0 width=650" border="0" cellspacingwidth="650" cellpadding="0"
| width="101" bgcolor="darkgreen" |&nbsp;
|-
| width="151" bgcolor="forestGreen" |
| bgcolor=darkgreen width=101 | &nbsp;
| width="200" bgcolor="YellowGreen" |
| bgcolor=forestGreen width=151 |
| width="99" bgcolor="yellow" |
| bgcolor=YellowGreen width=200 |
| bgcolor=yellow width="99" bgcolor="orange" |
| bgcolor=orange width=99 |
|}
|-
| align="right" | CentreCenter
|
{| cellpaddingcellspacing="0 width=650" border="0" cellspacingwidth="650" cellpadding="0"
| width="50" bgcolor="darkgreen" |&nbsp;
|-
| width="200" bgcolor="forestGreen" |
| bgcolor=darkgreen width=50 | &nbsp;
| bgcolor=forestGreen width="200" bgcolor="YellowGreen" |
| bgcolor=YellowGreen width="200" bgcolor="yellow" |
| bgcolor=yellow width=200 |
|}
|-
| align="right" | CentreCenter right
|
{| cellpaddingcellspacing="0 width=650" border="0" cellspacingwidth="650" cellpadding="0"
| width="99" bgcolor="darkgreen" |&nbsp;
|-
| width="149" bgcolor="forestGreen" |
| bgcolor=darkgreen width=99 | &nbsp;
| width="200" bgcolor="YellowGreen" |
| bgcolor=forestGreen width=149 |
| width="101" bgcolor="yellow" |
| bgcolor=YellowGreen width=200 |
| bgcolor=yellow width="101" bgcolor="orange" |
| bgcolor=orange width=101 |
|}
|-
| align="right" | Right
|
{| cellpaddingcellspacing="0 width=650" border="0" cellspacingwidth="650" cellpadding="0"
| width="99" bgcolor="darkgreen" |&nbsp;
|-
| width="198" bgcolor="forestGreen" |
| bgcolor=darkgreen width=99 | &nbsp;
| width="50" bgcolor="yellowGreen" |
| bgcolor=forestGreen width=198 |
| width="101" bgcolor="yellow" |
| bgcolor=yellowGreen width=50 |
| bgcolor=yellow width="101" bgcolor="orange" |
| bgcolor=orange width="101" bgcolor="orangered" |
| bgcolor=orangered width=101 |
|}
|-
| &nbsp;
| &nbsp;
|-
| &nbsp;
|
{| cellpaddingcellspacing="1" border="0" cellspacingcellpadding="1"
|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
|-
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;bgcolor="darkgreen" |&nbsp;
| bgcolor=darkgreen | &nbsp;Excellent&nbsp;&nbsp;
| &nbsp;Excellent&nbsp;bgcolor="forestGreen" |&nbsp;
|&nbsp;Very good&nbsp;&nbsp;
| bgcolor=forestGreen | &nbsp;
| &nbsp;Verybgcolor="YellowGreen" good&nbsp;|&nbsp;
| bgcolor=YellowGreen | &nbsp;Good&nbsp;&nbsp;
| &nbsp;Good&nbsp;bgcolor="Yellow" |&nbsp;
| bgcolor=Yellow | &nbsp;Passable&nbsp;&nbsp;
| &nbsp;Passable&nbsp;bgcolor="Orange" |&nbsp;
| bgcolor=Orange | &nbsp;Inadequate&nbsp;&nbsp;
| &nbsp;Inadequate&nbsp;bgcolor="Orangered" |&nbsp;
| bgcolor=Orangered | &nbsp;Mediocre&nbsp;&nbsp;
| &nbsp;Mediocre&nbsp;&nbsp;
|}
|}
Line 161 ⟶ 162:
{| class="wikitable"
|-
!<br /><br />City Choice !! Memphis<br />voters !! Nashville<br />voters !! Chattanooga<br />voters !! Knoxville<br />voters !! Median<br />rating<ref group=note>{{efn|A "+" or "-" is added depending on whether the median would rise or fall if median ratings were removed, as in the tiebreakingtie-breaking procedure.</ref>}}
|-
| Memphis || bgcolor="green"|excellent || bgcolor="orangered"| poor || bgcolor="orangered"| poor || bgcolor="orangered"| poor || bgcolor="orangered"| poor+
Line 295 ⟶ 296:
|}
|}
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.
 
== Real-world examples ==
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.
.
 
The somewhat-related [[median voting rule]] method was first explicitly proposed to assign budgets by [[Francis Galton]] in 1907.<ref>Francis Galton, "One vote, one value," Letter to the editor, ''Nature'' vol. 75, Feb. 28, 1907, p. 414.</ref> Hybrid mean/median systems based on the [[trimmed mean]] have long been used to assign scores in contests such as [[Olympic figure skating]], where they are intended to limit the impact of biased or strategic judges.
== History ==
 
The first [[Highest median voting rules|highest median rule]] to be developed was [[Bucklin voting]], a system used by [[Progressive era]] reformers in the United States.
Voting theory has tended to focus more on [[Ranked voting systems|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<ref>Francis Galton, "One vote, one value," Letter to the editor, ''Nature'' vol. 75, Feb. 28, 1907, p. 414.</ref> 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 Majoritymajority judgment was first proposed by Balinski and Laraki in 2007.<ref name=":0">Balinski M. and R. Laraki (2007) «[https://www.pnas.org/content/pnas/104/21/8720.full.pdf A theory of measuring, electing and ranking]». Proceedings of the National Academy of Sciences USA, vol. 104, no. 21, 8720-8725.</ref> 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:
 
<blockquote>
Line 311:
 
It has since been used in judging wine competitions and in other political research polling in France and in the US.<ref>Balinski M. and R. Laraki (2010) «Judge: Don't vote». Cahier du Laboratoire d’Econométrie de l’Ecole Polytechnique 2010-27.</ref>
 
== Variants ==
Varloot and Laraki<ref name=":1">{{Cite book |last1=Varloot |first1=Estelle Marine |last2=Laraki |first2=Rida |chapter=Level-strategyproof Belief Aggregation Mechanisms |date=2022-07-13 |title=Proceedings of the 23rd ACM Conference on Economics and Computation |chapter-url=https://doi.org/10.1145/3490486.3538309 |series=EC '22 |location=New York, NY, USA |publisher=Association for Computing Machinery |pages=335–369 |doi=10.1145/3490486.3538309 |isbn=978-1-4503-9150-4|arxiv=2108.04705 }}</ref> 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==
* [[ApprovalUsual votingjudgment]]
*[[Approval voting]]
* [[Range voting]]
* [[Voting system]]
Line 319 ⟶ 323:
 
== Notes ==
{{notelist}}
{{Reflist|group=note}}
 
== References ==
Retrieved from "https://en.wikipedia.org/wiki/Majority_judgment"