View source for Highest median voting rules

{{Electoral systems}}

The '''highest median voting rules''' are a class of [[cardinal voting|graded voting]] rules where the candidate with the highest [[median]] rating is elected.

The various highest median rules differ in their treatment of ties, i.e., the method of ranking the candidates with the same median rating.

Proponents of highest median rules argue that they provide the most faithful reflection of the voters' opinion. They note that as with other [[cardinal voting]] rules, highest medians are not subject to [[Arrow's impossibility theorem]].

However, critics note that highest median rules violate [[Participation criterion|participation]] and the [[Archimedean property]]; highest median rules can fail to elect a candidate almost-unanimously preferred over all other candidates.

== Example ==
As in [[score voting]], voters rate candidates along a common scale, e.g.:
{| class="wikitable" style="text-align: center"
!
!Excellent
!Very Good
!Good
!Fair
!Passable
!Inadequate
!Bad
|-
|Candidate A
|
|
|
|
|
|X
|
|-
|Candidate B
|
|X
|
|
|
|
|
|-
|Candidate C
|
|
|
|
|
|X
|
|-
|Candidate D
|
|
|
|X
|
|
|
|}
An elector can give the same appreciation to several different candidates. A candidate not evaluated automatically receives the mention "Bad".<ref>{{Cite web |title=Le jugement majoritaire |url=https://lechoixcommun.fr/content/article/le-jugement-majoritaire.html |access-date=2021-02-10 |website=lechoixcommun.fr |language=fr |archive-date=2021-02-04 |archive-url=https://web.archive.org/web/20210204213945/https://www.lechoixcommun.fr/content/article/le-jugement-majoritaire.html |url-status=dead }}</ref>

Then, for each candidate, we calculate what percentage of voters assigned them each grade, e.g.:
{| class="wikitable"
!Candidate
!Excellent
!Very Good
!Good
!Fair
!Passable
!Inadequate
!Bad
!TOTAL
|-
|A
|5%
|13%
|21%
|20%
|9%
|17%
|15%
|100%
|-
|B
|5%
|14%
|19%
|13%
|13%
|12%
|24%
|100%
|-
|C
|4%
|6%
|10%
|15%
|16%
|24%
|25%
|100%
|}
This is presented graphically in the form of a cumulative histogram whose total corresponds to 100% of the votes cast:
[[File:Highest_median_voting_rule_cumulative_histogram.svg|center|frame|An example of a cumulative histogram for a highest-median voting rule.]]
For each candidate, we then determine the majority (or median) grade (shown here in bold). This rule means that an absolute majority (more than 50%) of voters judge that a candidate deserves at least its majority grade, and that half or more (50% or more) of the electors judges that he deserves at the most its majority grade. Thus, the majority grade looks like a median.

If only one candidate has the highest median score, they are elected. Otherwise, highest median rules must invoke a tiebreaking procedure to choose between the candidates with the highest median grade.

== Tiebreaking procedures ==
When different candidates share the same median rating, a tie-breaking rule is required, analogous to interpolation. For discrete grading scales, the median is insensitive to changes in the data and highly sensitive to the choice of scale (as there are large "gaps" between ratings).

Most tie-breaking rules choose between tied candidates by comparing their relative shares of proponents (above-median grades) and opponents (below-median grades).<ref name="Fabre202">{{Cite journal |last=Fabre |first=Adrien |year=2020 |title=Tie-breaking the Highest Median: Alternatives to the Majority Judgment |url=https://github.com/bixiou/highest_median/raw/master/Tie-breaking%20Highest%20Median%20-%20Fabre%202019.pdf |journal=[[Social Choice and Welfare]] |volume=56 |pages=101–124 |doi=10.1007/s00355-020-01269-9 |issn=0176-1714 |s2cid=226196615}}</ref> The share of proponents and opponents are represented by <math>p</math> and <math>q</math> respectively, while their share of median grades is written as <math>m</math>.

* '''[[Bucklin voting|Bucklin's rule]]''' orders candidates by (one minus) the number of opponents.<ref>[https://books.google.com/books?id=RN5q_LuByUoC&q=Bucklin Collective decisions and voting: the potential for public choice], Nicolaus Tideman, 2006, p. 204</ref> '''Anti-Bucklin''' reverses this (choosing the candidate with the highest share of proponents).
* The '''[[majority judgment]]''' considers the candidate who is closest to having a rating other than its median and breaks the tie based on that rating.<ref name="Fabre202" />
* The '''typical judgment''' ranks candidates by the number of proponents minus the number of opponents,<ref name="Fabre202" /> i.e. <math>p-q</math>.
* The '''central judgment''' divides the typical judgment by the total number of proponents and opponents.
* '''Continuous Bucklin voting''' or '''[[Graduated majority judgment|Graduated Majority Judgment]] (GMJ)''', also called the '''usual judgment''',<ref name="Fabre202" /> ranks candidates by the share of their median grades needed to reach 50% support.
** This is equivalent to using a [[linear interpolation]] between the current score and the next-highest score.
** Compared to typical judgment, this leads to a more prominent score difference when the median share is low; in other words, candidates who are more "polarizing" receive more extreme evaluations.

=== Example ===<!--The spreadsheet calculating these election results can be found here:
https://docs.google.com/spreadsheets/d/1wHxF0PPs4hqKuQjt0HRE5uy-vHYK-oxhh3ohEZjqKhw/edit?usp=sharing-->
[[File:Highest-median-tiebreak-example.svg|thumb|511x511px|Example of an election where each choice (or candidate) A-F wins according to one of the tie-breaking rules: typical, central, [[Graduated majority judgment|graduated majority]], majority, Bucklin, and anti-Bucklin''.''<ref name="Fabre202" />]]
The example in the following table shows a six-way tied rating, where each alternative wins under one of the rules mentioned above. (All scores apart from Bucklin/anti-Bucklin are scaled to fall in <math display="inline">\left[-\frac{1}{2}, \frac{1}{2} \right]</math>to allow for interpreting them as interpolations between the next-highest and next-lowest scores.)
{| class="wikitable sortable"
|+
!Candidate
!Against
!For
!Diff
!Central
![[Majority judgment|Nearest]]
![[Graduated majority judgment|GMJ]]
|-
!A
|15%
|30%
|'''15%'''
|17%
|30%
|14%
|-
!B
|4%
|11%
|7%
|'''23%'''
|11%
|4%
|-
!C
|27%
|40%
|13%
|10%
|40%
|'''20%'''
|-
!D
|43%
|45%
|2%
|1%
|'''45%'''
|8%
|-
!E
|'''3%'''
|0%
| -3%
| -50%
| -3%
| -2%
|-
!F
|49%
|'''46%'''
| -3%
| -2%
| -49%
| -30%
|-
!'''Formula'''
!<math>p</math>
!<math>q</math>
!<math>p-q</math>
!<math>\frac{p-q}{2(p+q)}</math>
!<math>\min(p, -q)</math>
!<math>\frac{p -q}{2 m}</math>
|}

== Advantages and Disadvantages ==

=== Advantages ===

==== Common to cardinal voting methods ====
[[Cardinal voting|Cardinal voting systems]] allow voters to provide much more information than [[Ordinal ballots|ranked-choice ballots]] (so long as there are enough categories); in addition to allowing voters to specify which of two candidates they prefer, cardinal ballots allow them to express how ''strongly'' they prefer such candidates.<ref name=":2222">{{Cite journal |last=Balinski |first=Michel |date=2019 |title=Réponse à des critiques du jugement majoritaire |url=https://www.cairn.info/revue-economique-2019-4-page-589.htm |journal=Revue Économique |volume=70 |issue=4 |pages=589–610 |doi=10.3917/reco.704.0589 |s2cid=199348869 |via=CAIRN}}</ref> Voters can choose between a wide variety of options for rating candidates, allowing for nuanced judgments of quality.<ref name=":2222" /><ref>{{Cite journal |last1=Balinski |first1=Michel |last2=Laraki |first2=Rida |date=2012 |title=Jugement majoritaire versus vote majoritaire |url=https://www.cairn.info/revue-francaise-d-economie-2012-4-page-11.htm |journal=Revue Française d'Économie |volume=27 |pages=33 |via=CAIRN}}</ref>

Because highest median methods ask voters to [[Cardinal voting|evaluate candidates]] rather than rank them, they escape [[Arrow's impossibility theorem]], and satisfy both unanimity and [[independence of irrelevant alternatives]].<ref name=":1222">{{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 |archive-url=https://web.archive.org/web/20230214040420/https://lechoixcommun.fr/resource/BD-lechoixcommun-EN.pdf |archive-date=2023-02-14 |access-date= |website=Le Choix commun |url-status=dead }}</ref> However, highest medians fail the slightly stronger near-unanimity criterion (see [[#Disadvantages]]).

Several candidates belonging to a similar political faction can participate in the election without helping or hurting each other, as highest median methods satisfy [[Independence of irrelevant alternatives|independence from irrelevant alternatives]]:<ref name=":1222" /> Adding candidates does not change the ranking of previous candidates. In other words, if a group ranks A higher than B when choosing between A and B, they should not rank that B higher than A when choosing between A, B, and C.

==== Unique to highest medians ====
The most commonly-cited advantage of highest median rules over their mean-based counterparts is they minimize the number of voters who have an incentive to be dishonest.<ref name=":2222" /> Voters with weak preferences in particular will not have much incentive to give candidates very high or very low scores. On the other hand, all voters in a [[score voting]] system have an incentive to exaggerate, which in theory would lead to ''de facto'' [[approval voting]] for a large share of the electorate most voters will only give the highest or lowest score to every candidate).

=== Disadvantages ===
==== Participation failure ====
Highest median rules violate the [[participation criterion]]; in other words, a candidate may lose because they have "too many supporters."

In the example below, notice how adding the two ballots labeled "+" causes A (the initial winner) to lose to B:
{| class="wikitable sortable"
|+
!
!
!+
!+
!New Median
!Old Median
!
!
|-
!A
|9
|9
|9
|''6''
|''5''
|3
|0
|-
!B
|9
|7
|7
|''7''
|''4''
|2
|0
|-
!C
|9
|0
|0
|''4''
|''3''
|2
|0
|}
It can be proven that [[score voting]] (i.e. choosing highest mean instead of highest median) is the unique voting system satisfying the participation criterion, [[Archimedean property]], and [[Independence of spoilers|independence of irrelevant alternatives]], as a corollary of the [[Von Neumann–Morgenstern utility theorem|VNM utility theorem]].<ref>{{Cite book |last1=Balinski |first1=Michel |url=https://books.google.com/books?id=acrxCwAAQBAJ |title=Majority Judgment: Measuring, Ranking, and Electing |last2=Laraki |first2=Rida |date=2011 |publisher=The MIT Press |isbn=978-0-262-01513-4 |edition=1 |pages=285–287}}</ref>

==== Archimedean property ====

Highest median rules violate the [[Archimedean property]]; informally, the Archimedean property says that if "99.999...%" of voters prefer Alice to Bob, Alice should defeat Bob. As shown below, it is possible for [[Alice and Bob|Alice]] to defeat [[Alice and Bob|Bob]] in an election, even if only one voter thinks Bob is better than Alice, and a very large number of voters (up to 100%) give Alice a higher rating:

{| class="wikitable defaultright"
|+Ballots (Bolded medians)
!# ballots
!Alice
!Bob
!Charlie
|-
|[[Arbitrarily large|Many]]
|100/100
|52/100
|0/100
|-
|{{Math|1}}
|'''50/100'''
|'''51/100'''
|'''1/100'''
|-
|[[Arbitrarily large|Many]]
|49/100
|0/100
|100/100
|}
In this election, Bob has the highest median score (51) and defeats Alice, even though every voter except one (perhaps Bob himself) thinks Alice is a better candidate. This is true no matter how many voters there are. As a result, even a single voter's weak preferences can override the strong preferences of the rest of the electorate.

The above example restricted to candidates Alice and Bob also serves as an example of highest median rules failing the [[Majority favorite criterion|majority criterion]], although highest medians can pass the majority criterion with normalized ballots (i.e. ballots scaled to use the whole 0-100 range). However, normalization cannot recover the Archimedean criterion.

==== Feasibility ====
A poll of French voters found a majority would be opposed to implementing [[majority judgment]], but a majority would support conducting elections by [[score voting]].<ref>{{Cite web |title=RangeVoting.org - What voters want |url=https://www.rangevoting.org/WhatVotersWant.html |access-date=2023-12-30 |website=www.rangevoting.org}}</ref><ref>https://www.rangevoting.org/Sondageopinionway2012FR.pdf {{Bare URL PDF|date=August 2024}}</ref>{{Better source needed|reason=The current source is insufficiently reliable ([[WP:NOTRS]]).|date=May 2024}}

== Related rules ==
* [[Cardinal voting]] systems are similar to highest median methods, but determine winners using a statistic other than the median; the most common of these is [[score voting]], which uses the mean.
* [[Approval voting]] corresponds to the [[Degeneracy (mathematics)|degenerate case]] where there are only two possible ratings: approval and disapproval. In this case, all tie-breaking rules are equivalent.<ref>{{Cite journal |last1=Brams |first1=Steven |last2=Fishburn |first2=Peter |year=1978 |title=Approval Voting |journal=American Political Science Review |volume=72 |issue=3 |pages=831–847 |doi=10.2307/1955105 |jstor=1955105 |s2cid=251092061}}</ref>

== See also ==

* [[Cardinal voting]]
* [[Majority judgment]]
* [[Bucklin voting]]
* [[Electoral System]]
* [[Comparison of electoral systems]]

== Further reading ==

* {{Cite journal |last1=Baujard |first1=Antoinette |last2=Gavrel |first2=Frédéric |last3=Igersheim |first3=Herrade |last4=Laslier |first4=Jean-François |last5=Lebon |first5=Isabelle |date=September 2017 |title=How voters use grade scales in evaluative voting |url=https://halshs.archives-ouvertes.fr/halshs-01618039/file/1729.pdf |journal=[[European Journal of Political Economy]] |volume=55 |pages=14–28 |doi=10.1016/j.ejpoleco.2017.09.006 |issn=0176-2680}}

* [[R (language)|R]] package implementing different highest median rules, as well as [[range voting]]: [https://cran.r-project.org/web/packages/HighestMedianRules/index.html HighestMedianRules]

== References ==
<references responsive="1"></references>{{Voting systems}}
[[Category:Cardinal electoral systems]]
[[Category:Electoral systems]]