Majority Judgment
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Does MJ meet Frohnmayer? My guess is "no."
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I think it is too nuanced to really answer. On one hand yes because if somebody votes on the other side of the median for all the candidates than you they effectively cancel your vote. However, it is not really clear how that is done.
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It's easy to show that MJ fails the opposite cancellation criterion.
2: A/Excellent, B/Very Good, C/Reject
1: A/Good, B/Very Good, C/RejectMedians: A/Excellent, B/Very Good, C/Reject
A is electedWe can add a pair of opposite ballots to change this result.
2: A/Excellent, B/Very Good, C/Reject
1: A/Good, B/Very Good, C/Reject
1: A/Good, B/Excellent, C/Poor
1: A/Acceptable, B/Reject, C/Very GoodMedians: A/Good, B/Very Good, C/Reject
B is electedDemonstrating that MJ fails the cancellation criterion is a bit more difficult since we must consider all possible cancelling ballots for A/Good, B/Excellent, C/Poor.
2: A/Excellent, B/Very Good, C/Reject
1: A/Good, B/Very Good, C/Reject
1: A/Good, B/Excellent, C/Poor
1: A/?, B/?, C/?Medians: A/?, B/Very Good, C/Reject
Luckily it's possible to construct an election where B and C's medians are independent of the final ballot (as I did here), so we only need to consider the possible ratings for A. An Excellent rating will lead to A keeping their median rating of Excellent and winning, but no other rating will. A Very Good rating creates a tie between A and B which is broken in B's favor, and anything lower leads to B winning as well.
Now all that remains is to find another election in which A/Good, B/Excellent, C/Poor cannot possibly be cancelled out by a ballot with an Excellent rating for A.
2: A/Poor, B/Acceptable, C/Reject
1: A/Good, B/Acceptable, C/RejectMedians: A/Poor, B/Acceptable, C/Reject
B is elected2: A/Poor, B/Acceptable, C/Reject
1: A/Good, B/Acceptable, C/Reject
1: A/Good, B/Excellent, C/Poor
1: A/Excellent, B/?, C/?Medians: A/Good, B/Acceptable, C/Reject
A is electedSo MJ fails both of these formalizations of Frohnmayer balance.