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 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.

In addition to faithfully reflecting the voter's opinion, highest median rules satisfy the independence of irrelevant alternatives and do not fall within the scope of Arrow's impossibility theorem.

Definition and Notations

Let ${𝒞}$ be a finite set of candidates, ${𝒱}$ a finite set of voters, and ${𝒢}$ an ordered finite set of ratings. To simplify the exposition, we assume that the number of voters is odd and force each voter to rate each candidate.

For any candidate $c\in {𝒞}$, $c$'s median rating ${\alpha }_c\in {𝒢}$ is the median rating among the ratings that $c$ received from voters.

If, for any candidate $i\ne c$, ${\alpha }_i<{\alpha }_c$, then $c$ obtained a higher median rating than all other candidates, and $c$ 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.

Before giving examples of such rules, it is useful to introduce two concepts. For any candidate $c$, call (somewhat abusively) the share of proponents to $c$, noted $p_c$, the proportion of voters attributing to $c$ a rating greater than its median ${\alpha }_c$. Similarly, the share of opponents to $c$, noted $q_c$, refers to the proportion of voters who attribute to $c$ a rating lesser than its median.

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: $d=\alpha +p-q$ (the indices $c$ are omitted for simplicity).
• The usual judgment is the rule said to offer the best properties, but it orders the candidates according to a slightly more complex formula: $n=\alpha +\frac{1}{2}\frac{p-q}{1-p-q}$.
• 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: $s=\alpha +\frac{1}{2}\frac{p-q}{p+q+ϵ}$ (where $ϵ$ is an arbitrarily small number that simply allows the denominator to remain positive).
• The majority judgment considers the highest share between that of proponents and that of opponents, and amounts to ordering the candidates according to their score $mj$, defined by the following formula (the symbol $\mathbf{𝟏}$ denotes the indicator function) : $mj=\alpha +{\mathbf{𝟏}}_{p>q}p-{\mathbf{𝟏}}_{p\le q}q$.
• 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: $b=\alpha -q$. 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 corresponds to the degenerate case where there are only two possible rating: approval and disapproval. In this case, all the tie-breaking rules are equivalent.

See also

Bibliography

• Fabre, Adrien (2020). "Tie-breaking the Highest Median: Alternatives to the Majority Judgment" (PDF). Social Choice and Welfare. doi:10.1007/s00355-020-01269-9.CS1 maint: Date and year (link)
• R package implementing different highest median rules, as well as range voting: HighestMedianRules.
• 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.

Related Articles

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

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