Highest median voting rules are
cardinal voting
Cardinal voting refers to any electoral system which allows the voter to give each candidate an independent evaluation, typically a rating or grade. These are also referred to as "rated" (ratings ballot), "evaluative", "graded", or "absolute" ...
rules, where the winning candidate is a candidate with the highest median rating. As these employ ratings, each voter rates the different candidates on an ordered, numerical or verbal scale.
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 faithfully reflect the voter's opinion, that they satisfy the
independence of irrelevant alternatives
The independence of irrelevant alternatives (IIA), also known as binary independence or the independence axiom, is an axiom of decision theory and various social sciences. The term is used in different connotation in several contexts. Although it a ...
and, not being ranked ballots, do not fall within the scope of
Arrow's impossibility theorem
Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem in social choice theory that states that when voters have three or more distinct alternatives (options), no ranked voting electoral syste ...
.
Critics point out that highest median rules violate the
Condorcet criterion
An electoral system satisfies the Condorcet winner criterion () if it always chooses the Condorcet winner when one exists. The candidate who wins a majority of the vote in every head-to-head election against each of the other candidatesthat is, a ...
: a candidate can in principle be elected even if all voters but one prefer another candidate.
Definition and notation
Let
be the set of candidates,
the set of voters, and
an ordered finite set of ratings (e.g. the following ratings: "Very good", "Good", "Average", "Bad").
For any candidate
,
's median rating
is the median rating among the ratings that
received from voters. For example, if there are ten voters and if candidate
receives three ratings "Good", six ratings "Average", and one rating "Bad", its median rating
is "Average".
If, for any candidate
,
, then
obtained a higher median rating than all other candidates, and
is elected, regardless of which highest median rule was chosen.
When different candidates share the same median rating, a tie-breaking rule is required. This tie-breaking rule characterizes the highest median rule at use.
Tie-breaking rules often make use of two additional statistics about a candidate
's ratings:
* The ''share of proponents to
'', noted
, which is the share of voters attributing to
a rating greater than its median
. In the example above, the three ratings "Good" are above
's median "Average", so
.
* The ''share of opponents to
'', noted
, which is the share of voters attributing to
a rating lesser than its median
. In the example above, this correspond to the rating "Bad", so
.
Examples
* The typical judgment orders the candidates according to the largest difference between their share of proponents and opponents, i.e. according to the formula:
(the indices
are omitted for simplicity). In the above example, and identifying "Average" with the grade
, we have
.
* The
usual judgment
The usual judgment is a single-winner electoral system invented by Adrien Fabre, a French social scientist, in 2019. It is a highest median voting method, a system of cardinal voting in which the winner is decided by the median rather than the m ...
is the rule said to offer the best properties,
but it orders the candidates according to a slightly more complex formula:
; compared to the typical judgment, this leads to a more prominent score difference in small changes of number of proponents and opponents when the median share is low, as the equation approaches 0.5 both as either
or
approaches 0.5 (neither can actually be 0.5, as the median lies at the halfway point by definition).
* The central judgment orders the candidates according to the highest ratio between the shares of proponents and opponents, that is to say according to the formula:
(where
is an arbitrarily small number that simply allows the denominator to remain positive); compared to the typical judgment, this leads to a more prominent score difference in small changes of number of proponents and opponents when the median share is high, as the equation approaches 0.5 both as either
or
approaches 0.5. In fact, the usual judgment
is equivalent to the central judgment
.
* The
majority judgment
Majority judgment (MJ) is a single-winner voting system proposed in 2007 by Michel Balinski and Rida Laraki. It is a highest median rule, i.e., a cardinal voting system that elects the candidate with the highest median rating.
Unlike other vo ...
considers the candidate who is closest to having a rating other than its median and breaks the tie based on that rating. This is equivalent to ordering the candidates according to their score
,
defined by the following formula (the symbol
denotes the indicator function) :
.
* The
Bucklin rules are close to the highest median rules but have been developed for
ranked rules. They order the candidates according to the formula:
. In a ranked rule, this is equivalent to counting first choice votes first. If one candidate has a majority, that candidate wins. Otherwise the second choices are added to the first choices. If a candidate with a majority vote is found, the winner is the candidate with the most votes accumulated. Lower rankings are added as needed.
*
Approval voting
Approval voting is an electoral system in which voters can select many candidates instead of selecting only one candidate.
Description
Approval voting ballots show a list of the options of candidates running. Approval voting lets each voter i ...
corresponds to the degenerate case where there are only two possible ratings: approval and disapproval. In this particular case, all the tie-breaking rules are equivalent, and the
Condorcet criterion
An electoral system satisfies the Condorcet winner criterion () if it always chooses the Condorcet winner when one exists. The candidate who wins a majority of the vote in every head-to-head election against each of the other candidatesthat is, a ...
is satisfied.
See also
*
Cardinal voting
Cardinal voting refers to any electoral system which allows the voter to give each candidate an independent evaluation, typically a rating or grade. These are also referred to as "rated" (ratings ballot), "evaluative", "graded", or "absolute" ...
*
Majority judgment
Majority judgment (MJ) is a single-winner voting system proposed in 2007 by Michel Balinski and Rida Laraki. It is a highest median rule, i.e., a cardinal voting system that elects the candidate with the highest median rating.
Unlike other vo ...
*
Bucklin voting
Bucklin voting is a class of voting methods that can be used for single-member and multi-member districts. As in highest median rules like the majority judgment, the Bucklin winner will be one of the candidates with the highest median ranking o ...
*
Electoral System
An electoral system or voting system is a set of rules that determine how elections and Referendum, referendums are conducted and how their results are determined. Electoral systems are used in politics to elect governments, while non-political ...
*
Comparison of electoral systems
Electoral systems are the rules for conducting elections, a main component of which is the algorithm for determining the winner (or several winners) from the ballots cast. This article discusses methods and results of comparing different electora ...
References
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, journal=
European Journal of Political Economy
The ''European Journal of Political Economy'' is a quarterly peer-reviewed academic journal covering research on economic phenomena, including collective decision making, political behavior, and the role of institutions. The editors-in-chief are To ...
, volume=55, pages=14–28, doi=10.1016/j.ejpoleco.2017.09.006, issn=0176-2680, url=https://halshs.archives-ouvertes.fr/halshs-01618039/file/1729.pdf
External links
*
R package implementing different highest median rules, as well as
range voting
Score voting or range voting is an electoral system for single-seat elections, in which voters give each candidate a score, the scores are added (or averaged), and the candidate with the highest total is elected. It has been described by various ...
HighestMedianRules
Cardinal electoral systems
Electoral systems