Highest median voting rules
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.
[1] Then, for each candidate, we calculate what percentage of voters assigned them each grade, e.g.: This is presented graphically in the form of a cumulative histogram whose total corresponds to 100% of the votes cast: For each candidate, we then determine the majority (or median) grade (shown here in bold).
When different candidates share the same median rating, a tie-breaking rule is required, analogous to interpolation.
[4] Voters can choose between a wide variety of options for rating candidates, allowing for nuanced judgments of quality.
[4][5] Because highest median methods ask voters to evaluate candidates rather than rank them, they escape Arrow's impossibility theorem, and satisfy both unanimity and independence of irrelevant alternatives.
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).
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: 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 irrelevant alternatives, as a corollary of the VNM utility theorem.
The above example restricted to candidates Alice and Bob also serves as an example of highest median rules failing the majority criterion, although highest medians can pass the majority criterion with normalized ballots (i.e. ballots scaled to use the whole 0-100 range).
