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Graduated majority judgment (GMJ), sometimes called theusual judgment^[1] orcontinuousBucklin voting,^[2] is a single-winnerrated voting rule that selects the candidate with thehighest median score.^[1] It was first suggested as an improvement onmajority judgment by Andrew Jennings in 2010.^[3]

GMJ begins by counting all ballots for their first choice. If no candidate has a majority then later (second, third, etc.) preferences are gradually added in, continuing until one candidate reaches 50% approval. The first candidate to reach a majority of the vote is the winner.

Votes should be cast using acardinal (rated) ballot, which ask voters to give each candidate a separate grade, such as :

When counting the votes, we calculate the share of each grade for each of the votes cast. This is the candidate's "merit profile":

For each candidate, we determine themedian ormajority grade as the grade where a majority of voters would oppose giving the candidate a higher grade, but a majority would also oppose giving a lower grade.^[4] This rule means that the absolute majority of the electors judge that a candidates meritsat least its median grade, while half the electors judge that he deservesat most its median grade.^[5]

If only one candidate has the highest median grade, they are elected (as in allhighest median voting rules). Otherwise, the election uses a tie-breaking procedure.

Graduated majority judgment uses a simpleline-drawing method to break ties.^[1] This rule is easier to explain than others such asmajority judgment, and also guaranteescontinuity.

Graphically, we can represent this by drawing a plot showingthe share of voters who assign an approval less than the given score, then draw lines connecting the points on this graph. The place where this plot intersects 50% is each candidate's final score.

Consider the same election as before, but relabeling the verbal grades as numbers on a scale from 0 to 6:

Candidates A and B both cross the 50% threshold between 2 or 3, so we must invoke the tiebreaking procedure. When we do, we find that the median grades for candidates A, B, and C are 3.4, 3.1, and 2.0 respectively. Thus, Candidate A is declared the winner.

The tiebreaking rule can be explained using an analogy where every candidate is in a race. Each candidate takes 1 minute to run from one grade to the next, and they run at a constant speed when moving from one grade to the next. The winner is the first candidate to cross the finish line at 50% of the vote.

Say the median grade of a candidate${\displaystyle c}$ is${\displaystyle {\alpha }_c}$ (when there is a tie, we define the median as halfway between the neighboring grades). Let${\displaystyle p_c}$ (the share ofproponents) refer to the share of electors giving${\displaystyle c}$ a score strictly better than the median grade. The share ofopponents of${\displaystyle c}$, written${\displaystyle q_c}$, is the share of grades falling below the median. Then the complete score for GMJ is given by the following formula:^[1]$${\displaystyle n_c={\alpha }_c+\frac{1}{2}\frac{p_c-q_c}{1-p_c-q_c}}$$

In the unusual case of a tie where the formula above does not determine a single winner (if several candidates have exactly the same score), ties can be broken by binning together the 3 grades closest to the median, then repeating the tie-breaking procedure.^[1] In the example above, we would combine all "Good," "Fair," and "Passable" grades into a new "Passable to Good" grade, then apply the same tie-breaking formula as before. This process can be repeated multiple times (binning more and more grades) until a winner is found.

As an electoral system, the graduated majority judgment shares most of its advantages with otherhighest-median voting rules such asmajority judgment, including its resistance to tactical voting. It also shares most of its disadvantages (for example, it fails theparticipation criterion, and can fail themajority criterion arbitrarily badly).

The tie-breaking formula of the graduated majority judgment presents specific advantages over the otherhighest-median voting rules.

The function defined by the graduated majority judgment tie-breaking formula is acontinuous function (as well as being almost-everywheredifferentiable), whereas the functions of majority judgment and typical judgment arediscontinuous.^[1] In other words, a small change in the number of votes for each candidate is unlikely to change the winner of the election, because small changes in vote shares result in only small changes in the overall rating.

This property makes the graduated majority judgment a more robust voting method in the face of accusations of fraud or demands of a recount of all votes. As small differences of votes are less likely to change the outcome of the election, candidates are less likely to contest results.^[1]

• Bucklin voting, a similar voting rule
• Majority judgment
• Highest median voting rules
• Range voting

• Single-winner electoral systems

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