Social:Highest median voting rules

Highest median voting rules are cardinal voting rules, where the winning candidate is a candidate with the highest median rating. As these employ ratings, each voter rates the different candidates on a 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 provide the most faithful reflection of the voters' opinion. They cite theorems showing that minimizes the share of voters with an incentive to vote strategically. They note that as with other cardinal voting rules, highest medians are not subject to Arrow's impossibility theorem, and so can satisfy both independence of irrelevant alternatives and Pareto efficiency.

However, critics note that highest median rules violate participation and fail the majority criterion. Highest median methods can sometimes fail to elect a candidate who is almost-unanimously preferred over all other candidates.

Example

As in score voting, voters rate candidates along a common scale, e.g.:

	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".

Then, for each candidate, we calculate what percentage of voters assigned them each grade, e.g.:

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:

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). For example, the following election results in a tied median grade, despite the second candidate being a strict improvement:

Most tie-breaking rules choose between tied candidates by comparing their relative shares of proponents (above-median grades) and opponents (below-median grades). The share of proponents and opponents are represented by [math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math] respectively, while their share of median grades is written as [math]\displaystyle{ m }[/math].

• Bucklin's rule orders candidates by Bucklin's rule orders candidates by (one minus) the number of opponents.
  • This results in a simple counting procedure, and allows for candidates to be ranked using an ordinal ballot, rather than requiring ratings. Bucklin's rule counts first choice votes first, then checks if any candidate has a majority. If not, we add second-, third-, etc. votes until one candidate has a majority of votes. If more than one candidate reaches a majority, ties are broken by choosing the candidate with the most votes.
  • Anti-Bucklin reverses Bucklin's rule (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.
• The typical judgment ranks candidates by the number of proponents minus the number of opponents, i.e. [math]\displaystyle{ p-q }[/math].
• The central judgment divides the typical judgment by the total number of proponents and opponents. Compared to the 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 moderate evaluations. (This method is rarely-used in practice.)
• Continuous Bucklin voting or Graduated Majority Judgment (GMJ), also called the usual judgment, uses the shares of proponents and opponents to linearly interpolate between the next-highest score and the next-lowest 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

Example of vote outcome where each choice (or candidate) A-F wins according to one of the four tie-breaking rules studied: respectively typical judgment, central judgment, usual judgment, majority judgment, Bucklin voting, and anti-Bucklin voting.

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]\displaystyle{ \left[-\frac{1}{2}, \frac{1}{2} \right] }[/math]to allow for interpreting them as interpolations between the next-highest and next-lowest scores.)

Candidate	Opponents (Bucklin)	Median	Proponents (Anti-Bucklin)	Typical Judgment	Central Judgment	Majority Judgment	Continuous (GMJ)
A	15%	55%	30%	15%	17%	30%	14%
B	4%	85%	11%	7%	23%	11%	4%
C	27%	33%	40%	13%	10%	40%	20%
D	43%	12%	45%	2%	1%	45%	8%
E	3%	97%	0%	-3%	-50%	-3%	-2%
F	49%	5%	46%	-3%	-2%	-49%	-30%
Formula	[math]\displaystyle{ p }[/math]	[math]\displaystyle{ m }[/math]	[math]\displaystyle{ q }[/math]	[math]\displaystyle{ p-q }[/math]	[math]\displaystyle{ \frac{p-q}{2(p+q)} }[/math]	[math]\displaystyle{ \begin{cases} p & p \gt q \\ -q & p \leq q \end{cases} }[/math]	[math]\displaystyle{ \frac{p -q}{2 m} }[/math]

Advantages and Disadvantages

Highest median rules have several unique advantages and disadvantages.

Advantages

Because higest median methods ask voters to evaluate the candidates rather than rank them, they escape Arrow's impossibility theorem, and satisfy both unanimity and independence of irrelevant alternatives. However, these methods 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 to irrelevant alternatives: Adding a losing candidate does not change the results of the election. In other words, if a group believes A is better than B given the choice, the group should not also believe that B is better than A when choosing between A, B, and C.

The most commonly-cited advantage of highest median rules over their mean-based counterparts is that they arguably minimize the amount of strategic voting. Highest median methods minimize the proportion of voters who have an incentive to be dishonest: Only voters who assign precisely the median vote will have an incentive to switch their votes. In other words, extremists will not have an 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 generally leads to de facto approval voting for a large share of the electorate (where most voters will only give the highest or lowest score to every candidate).

Unlike approval voting, which allows only two answers, voters can choose between a wide variety of options for rating candidates, allowing for more nuanced judgments.

The "merit profile" drawn from the results gives very detailed information on the popularity of each candidate or option across the whole electorate.

Disadvantages

Participation failure

Highest median rules violate the participation criterion; in other words, a candidate may lose because they have "too many supporters." For example, in the below election, notice how adding the two ballots labeled "+" causes A (the initial winner) to lose to B:

		+	+	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. highest mean instead of highest median) is the unique voting system satisfying the participation criterion and independence of irrelevant alternatives.

Majority and near-unanimity failure

Highest median rules violate the majority criterion (thus failing both the Condorcet winner and Condorcet loser criteria), as well as a stronger version of the unanimity criterion. While many scholars disagree that the Condorcet criterion is desirable in a voting system (arguing a minority with strong preferences should be able to override a majority with weak preferences), highest medians methods fail the majority criterion in spectacular fashion.

The following example shows it is possible for a candidate X to defeat another candidate Y in an election, even if:

1. Y is the Condorcet winner, while X is the Condorcet loser
2. An arbitrarily large share of the electorate prefers B to A (i.e. up to, but not including, 100% of voters),
3. Only a single voter prefers X to Y, and
4. This single voter is almost-indifferent between candidates A and B (assigns neighboring grades to both).

Ballots (Bolded medians)

# ballots	A's score	B's score	C's score
n	100/100	52/100	0/100
1	50/100	51/100	1/100
n	49/100	0/100	100/100

In this election, note that candidate B has the highest median score (51), despite 2n voters supporting A over B. Moreover, note that only a single voter supports B over A. Thus, highest median methods fail the slightly stronger criterion of near-unanimity (there exist situations where a single voter's weak preference overrides the strong preferences of the rest of the electorate).

The above example restricted to candidates A and B also serves as an example of highest medians failing the majority criterion.

Perceived complexity

Voters may have difficulty understanding the tiebreaking rules of highest median methods, or perceive them as too complex. A poll of French voters found a majority would be opposed to implementing majority judgment (due to its complexity), but a majority would support conducting elections by score voting.

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 degenerate case where there are only two possible ratings: approval and disapproval. In this case, all tie-breaking rules are equivalent.

See also

• Cardinal voting
• Majority judgment
• Bucklin voting
• Electoral System
• Comparison of electoral systems

Further reading

• Baujard, Antoinette; Gavrel, Frédéric; Igersheim, Herrade; Laslier, Jean-François; Lebon, Isabelle (September 2017). "How voters use grade scales in evaluative voting". European Journal of Political Economy 55: 14–28. doi:10.1016/j.ejpoleco.2017.09.006. ISSN 0176-2680. https://halshs.archives-ouvertes.fr/halshs-01618039/file/1729.pdf. 

External links

• R package implementing different highest median rules, as well as range voting: HighestMedianRules.