Graduated Majority Judgment

Like its predecessor Majority Judgment, Graduated Majority Judgment or GMJ is a single-winner, median-based voting system. Here's one way to explain it:

Ballot Explanation
The ballot will ask you to grade each candidate on a scale from A (excellent) to F (unacceptable). You may give two candidates the same grade if you wish. Any candidate whom you do not explicitly grade will get an F from you.

Conceptual
To find the winner, first the "A" votes for each candidate are counted. If no candidate gets over 50% of the voters, the "B" votes are added to the count, then "C" votes, etc. The first candidate to get over 50% is the winner. If two candidates would reach 50% at the same grade, each candidate's votes for that grade are added gradually, and the winner is the one who needs the smallest portion of those votes to reach 50%. This gradual process can be stated as a "graduated score" for each candidate. If a candidate reaches 50% using 8/10 of their "C" votes (along with all their A and B votes), then their graduated score would be 1.7 (a C-). Another candidate who needed only 2/10 of their "C" votes to reach 50% would have a graduated score of 2.3 (a C+), so between those two candidates the second would be the winner.

Two equivalent full procedures
It works as follows:


 * 1) Each voter grades each candidate on a grading scale such as A, B, C, D, F
 * 2) The top-grade (A) votes for each candidate are tallied.
 * 3) If a single candidate has a majority (that is, a number of votes greater than or equal to 50% of voters), they win.
 * 4) If no candidate has a majority, the next grade down (B) is added to the tally, and go back to step 3.
 * 5) If more than one candidate has a majority, the last grade tallied is removed from the tallies, and then re-added at the smallest fraction possible so that some candidate has a majority. This is as if the votes at that grade were added 1% at a time until one candidate gets a majority.

The above process is conceptually simple, but difficult in practice. The following process gives the same results, and is simpler to run in practice:


 * 1) Each voter grades each candidate on a grading scale such as A, B, C, D, F
 * 2) Each grade for each candidate is tallied.
 * 3) The tallies are used to find the median grade for each candidate.
 * 4) Tallies are added to find the V(>M), V(@M), and V(M) - V(<M)) / (2 * V(@M))
 * 6) The candidate with the highest adjustment among those with the highest median, wins.

If medians are converted to integers (such as 0-4), then the adjusted median scores can easily be reported alongside the full tallies.