It seems to me that there is a simpler way to compare candidates with the same median grade in majority judgement voting than the method described in the Wikipedia page for majority judgement. Why isn't this simpler way used?
Every voter grades every candidate. Elect the candidate with the highest median grade (the highest grade for which more than 50% of voters grade the candidate equal to or higher than that grade.) If there are two or more candidates with the same highest median grade, elect the candidate with the highest score of those with the highest median grade. A candidate's score is equal to the the number of voters that grade the candidate higher than the median grade plus the number of voters that grade to candidate equal to or higher than the median grade. This is equivalent to giving one point to each candidate for each voter who grades the candidate its median grade and two points for each voter who grades the candidate higher than its median grade. Motivation: voters who vote median grade instead of something lower should increase the score for the candidate by the same amount as voters who vote above the median grade instead of equal to the median grade. With this scoring, going from less than median to median increases the candidate score by one point and going from median to higher than median also increases the candidate score by one point. Example using same example from Wikipedia's majority judgement entry: 26% of voters grade Nashville as Excellent and 42% of voters grade Nashville as Good. Nashville's median grade is Good and its score is 26+26+42 = 94 15% of voters grade Chattanooga as Excellent and 43% of voters grade Chattanooga as Good. Chattanooga's median grade is Good and its score is 15+15+43 = 73. Nashville wins.
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