View source for Graduated majority judgment

{{Short description|Single-winner electoral system}}
{{more citations needed|date=January 2024}}
{{Electoral systems sidebar|expanded=Single-winner}}
'''Graduated majority judgment''' (GMJ), sometimes called the '''usual judgment'''<ref name=":0222222">{{Cite journal |last=Fabre |first=Adrien |year=2021 |title=Tie‐breaking the highest median: alternatives to the majority judgment |url=https://link.springer.com/article/10.1007%2Fs00355-020-01269-9 |journal=Social Choice and Welfare |volume=56 |pages=101–124 |doi=10.1007/s00355-020-01269-9 |s2cid=226196615 |via=Springer Link}}</ref> or '''continuous [[Bucklin voting]]''',<ref name=":022">{{Cite web |last=Smith |first=Warren D. |title=On Balinski & Laraki's "majority judgment" median-based range-like voting scheme |url=https://www.rangevoting.org/MedianVrange#GMJ |access-date=2024-01-02 |website=RangeVoting.org |publisher=Center for Range Voting |no-pp=y}}</ref> is a single-winner [[rated voting]] rule that selects the candidate with the [[Highest median voting rules|highest median score]].<ref name=":0222222" /> It was first suggested as an improvement on [[majority judgment]] by Andrew Jennings in 2010.<ref>{{Cite book |last=Jennings |first=Andrew |url=http://ajennings.net/dissertation.pdf |title=Monotonicity and Manipulability of Ordinal and Cardinal Social Choice Functions |date=2010 |publisher=Arizona State University |year=2010 |pages=25–30 |language=English}}</ref>

GMJ begins by counting all ballots for their first choice. If no candidate has a majority then later (second, third, etc.) preferences are gradually added in, continuing until one candidate reaches 50% approval. The first candidate to reach a majority of the vote is the winner.

== Highest medians ==
{{Main|Highest median voting rules}}
Votes should be cast using a [[Rated voting|cardinal]] (rated) ballot, which ask voters to give each candidate a separate grade, such as :
{| class="wikitable"
!
!Awful
!Bad
!Tolerable
!Neutral
!Acceptable
!Good
!Excellent
|-
!Candidate A
|
|
|
|
|
|
|X
|-
!Candidate B
|X
|
|
|
|
|
|
|-
!Candidate C
|
|
|
|
|
|X
|
|-
!Candidate D
|
|
|
|X
|
|
|
|}
When counting the votes, we calculate the share of each grade for each of the votes cast. This is the candidate's "merit profile":
{| class="wikitable"
!Candidate
!Awful
!Bad
!Tolerable
!Neutral
!Acceptable
!Good
!Excellent
|-
!A
|2%
|15%
|21%
|20%
|9%
|18%
|15%
|-
!B
|2%
|17%
|19%
|13%
|13%
|12%
|24%
|-
!C
|1%
|9%
|10%
|15%
|15%
|25%
|25%
|}
For each candidate, we determine the ''[[median]]'' or ''majority'' grade as the grade where a majority of voters would oppose giving the candidate a higher grade, but a majority would also oppose giving a lower grade.<ref>{{cite web |title=Le jugement majoritaire |url=https://lechoixcommun.fr/content/article/le-jugement-majoritaire.html |access-date=2021-02-08 |website=lechoixcommun.fr |language=fr |publication-date=}}</ref> This rule means that the absolute majority of the electors judge that a candidates merits ''at least'' its median grade, while half the electors judge that he deserves ''at most'' its median grade.<ref name=":122222">{{Cite web |last1=Leray |first1=Marjolaine |last2=Hogg |first2=Carol |date= |title=A little more democracy? Cartoons by Marjolaine Leray on the topic of Majority Judgment |url=https://www.lechoixcommun.fr/resource/BD-lechoixcommun-EN.pdf |access-date= |website=Le Choix commun}}</ref>

If only one candidate has the highest median grade, they are elected (as in all [[highest median voting rules]]). Otherwise, the election uses a tie-breaking procedure.

== Tie-breaking ==
Graduated majority judgment uses a simple [[Linear interpolation|line-drawing]] method to break ties.<ref name=":0222222" /> This rule is easier to explain than others such as [[majority judgment]], and also guarantees [[Continuous function|continuity]].

Graphically, we can represent this by drawing a plot showing [[Cumulative distribution function|the share of voters who assign an approval less than the given score]], then draw lines connecting the points on this graph. The place where this plot intersects 50% is each candidate's final score.

=== Example ===
Consider the same election as before, but relabeling the verbal grades as numbers on a scale from 0 to 6:
[[File:GMJ-tiebreaker.svg|thumb|472x472px|A plot showing how the overall score of a candidate can be calculated using graduated majority judgment.]]
{| class="wikitable sortable mw-collapsible mw-collapsed"
!{{Diagonal split header 2|Candidate|Grade}}
!0
!1
!2
!3
!4
!5
!6
|-
| rowspan="2" |A
([[Cumulative distribution function|Running total]])
|2%
|15%
|21%
|20%
|9%
|18%
|15%
|-
|2%
|17%
|'''39%'''
|'''58%'''
|68%
|85%
|100%
|-
| rowspan="2" |B
([[Cumulative distribution function|Running total]])
|2%
|17%
|19%
|13%
|13%
|12%
|24%
|-
|2%
|19%
|'''38%'''
|'''51%'''
|64%
|76%
|100%
|-
| rowspan="2" |C
([[Cumulative distribution function|Running total]])
|1%
|9%
|10%
|15%
|15%
|25%
|25%
|-
|1%
|10%
|20%
|35%
|'''50%'''
|75%
|100%
|}
Candidates A and B both cross the 50% threshold between 2 or 3, so we must invoke the tiebreaking procedure. When we do, we find that the median grades for candidates A, B, and C are 3.4, 3.1, and 2.0 respectively. Thus, Candidate A is declared the winner.

=== Race analogy ===
The tiebreaking rule can be explained using an analogy where every candidate is in a race. Each candidate takes 1 minute to run from one grade to the next, and they run at a constant speed when moving from one grade to the next. The winner is the first candidate to cross the finish line at 50% of the vote.

=== Mathematical formula ===
Say the median grade of a candidate <math>c</math> is <math>\alpha _c</math> (when there is a tie, we define the median as halfway between the neighboring grades). Let <math>p_c</math> (the share of ''proponents'') refer to the share of electors giving <math>c</math> a score strictly better than the median grade. The share of ''opponents'' of <math>c</math>, written <math>q_c</math>, is the share of grades falling below the median. Then the complete score for GMJ is given by the following formula:<ref name=":0222222" /><math display="block">n_c = \alpha_c + \frac{1}{2} \frac{p_c-q_c}{1-p_c-q_c}</math>

=== Additional tie-breaking ===
In the unusual case of a tie where the formula above does not determine a single winner (if several candidates have exactly the same score), ties can be broken by binning together the 3 grades closest to the median, then repeating the tie-breaking procedure.<ref name=":0222222" /> In the example above, we would combine all "Good," "Fair," and "Passable" grades into a new "Passable to Good" grade, then apply the same tie-breaking formula as before. This process can be repeated multiple times (binning more and more grades) until a winner is found.

== Properties and advantages ==
=== Advantages and disadvantages common to highest-median rules ===
{{Main|Highest median voting rules}}
As an electoral system, the graduated majority judgment shares most of its advantages with other [[Highest median voting rules|highest-median voting rules]] such as [[majority judgment]], including its resistance to tactical voting. It also shares most of its disadvantages (for example, it fails the [[participation criterion]], and can fail the [[Majority favorite criterion|majority criterion]] arbitrarily badly).

=== Specific advantages of graduated majority judgment ===
The tie-breaking formula of the graduated majority judgment presents specific advantages over the other [[Highest median voting rules|highest-median voting rules]].

==== Continuity ====
The function defined by the graduated majority judgment tie-breaking formula is a [[continuous function]] (as well as being almost-everywhere [[Differentiable function|differentiable]]), whereas the functions of majority judgment and typical judgment are [[Continuous function|discontinuous]].<ref name=":0222222" /> In other words, a small change in the number of votes for each candidate is unlikely to change the winner of the election, because small changes in vote shares result in only small changes in the overall rating.

This property makes the graduated majority judgment a more robust voting method in the face of accusations of fraud or demands of a recount of all votes. As small differences of votes are less likely to change the outcome of the election, candidates are less likely to contest results.<ref name=":0222222" />

== See also ==
{{Portal|Mathematics}}

* [[Bucklin voting]], a similar voting rule
* [[Majority judgment]]
* [[Highest median voting rules]]
* [[Score voting|Range voting]]
== References ==
<references />
{{Voting systems}}
[[Category:Single-winner electoral systems]]