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. Proponents of highest median rules argue that they provide the most faithful reflection of the voters' opinion. They note that highest median methods are among the few that can simultaneously satisfy a wide variety of voting criteria, including independence of irrelevant alternatives, monotonicity, and the Condorcet winner criterion, so long as elections do not result in ties. 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 . The typical judgment orders the candidates according to the largest difference between their share of proponents and opponents, i.