Steve Bosworth's steadfast advocacy for MJ has made me wonder if there might be a version of MJ that satisfies the Reverse Symmetry Criterion, and perhaps Participation, as well. Here's my idea ... Each voter judges each candidate with an appropriate specialized vocabulary word by writing the word next to the candidate's name on the ballot. The words must be spelled using only letters from the set {B, W}. They can be short or long, but they have to fit (left justified) into the spaces provided next to the candidate names. Fortunately, B and W are easy for optical scanner/digitzers to distinguish. For example, if a voter writes the word BBWBWWB next to the name José Jiménez, she is describing José as being among the better possible candidates... in fact among the better of the better, yet among the worse of the better of the better, etc. Each letter refines the partial judgment represented by the preceding letters ... so it's more like Romance languages where the modifiers conform to post-fix rather than pre-fix conventions ... Casa Blanca vs White House. If you are thinking that we are going to convert each word into a binary point number ... well, no ...we're going to do something much more subtle and interesting for which the arithmetic operations on binary numbers are not well adapted (though alpha numeric string operations could work, but let's stick with literature!): For convenience list all of the ballot words judging José in alphabetical (lexicographical) order... BBBBB BBBBB BBBBW ... WWBBW WWWBB ... WWWWW Suppose that most of these judgment words start with B..Then the first letter of the majority judgment (of José) will be a B. Before continuing we have to modify the losing words ... the ones that started with W ... cross out the initial W, but replace the remaining letters (if there are any) with W's to reflect these voters' opinion that the majority judgment (B) was too nice, so on the question of remaining letters, all of these voters' words will come down on the side of W. After these modifications, determine by majority rule the next letter in Jose's judgment, and make the usual adjustments to the losing words. What if there is a tie, meaning both letters get equal support. Then put a neutral letter n for "neither better nor worse" in that position of Jose's majority judgment word, and base its next letter on whether there are more W's or B's in the next position of the remaining words in the list (of voter words judging Jose). Continue as necessary until the tie is fully resolved. So we see that the majority judgment words for the various candidates will be formed with letters from the set {B, n, W} The order of finish is the dictionary order of the respective majority judgment words ... assuming a really up to date comprehensive English dictionary . Have I overlooked anything? Is Mono-Add satisfied? Reverse Symmetry? Do the rules make sense heuristically? FWS

I have a simpler approach to MJ that makes resolution of ties easier to describe ... coming soon. Preview ... a Judgment of a candidate is an ordered set [s(1), s(2), ... s(k)] for some ordinal k, where each s(i) is in the set S = {Worse, neutral, Better}, which we use interchangeably with the set of signs, {-, 0, +}, when convenient. So our MJ method converts finite sets of Judgments of each candidate into Majority Judgments of each candidate, which can then be used to obtain a complete social order (once ties are broken appropriately) satisfying Reverse Symmetry. Importantly, the input judgments and the output majority judgments are of the same type ... members of {S^m | m is an ordinal number}, in conformity with the basic social choice amalgamation goal. To be continued . . FWS El sáb., 16 de oct. de 2021 8:11 p. m., Forest Simmons < forest.simmons21@gmail.com> escribió: > Steve Bosworth's steadfast advocacy for MJ has made me wonder if there > might be a version of MJ that satisfies the Reverse Symmetry Criterion, and > perhaps Participation, as well. > > Here's my idea ... > > Each voter judges each candidate with an appropriate specialized > vocabulary word by writing the word next to the candidate's name on the > ballot. > > The words must be spelled using only letters from the set {B, W}. > > They can be short or long, but they have to fit (left justified) into the > spaces provided next to the candidate names. Fortunately, B and W are easy > for optical scanner/digitzers to distinguish. > > For example, if a voter writes the word > > BBWBWWB > > next to the name José Jiménez, she is describing José as being among the > better possible candidates... in fact among the better of the better, yet > among the worse of the better of the better, etc. Each letter refines the > partial judgment represented by the preceding letters ... so it's more like > Romance languages where the modifiers conform to post-fix rather than > pre-fix conventions ... Casa Blanca vs White House. > > If you are thinking that we are going to convert each word into a binary > point number ... well, no ...we're going to do something much more subtle > and interesting for which the arithmetic operations on binary numbers are > not well adapted (though alpha numeric string operations could work, but > let's stick with literature!): > > For convenience list all of the ballot words judging José in alphabetical > (lexicographical) order... > > BBBBB > BBBBB > BBBBW > ... > WWBBW > WWWBB > ... > WWWWW > > Suppose that most of these judgment words start with B..Then the first > letter of the majority judgment (of José) will be a B. > > Before continuing we have to modify the losing words ... the ones that > started with W ... cross out the initial W, but replace the remaining > letters (if there are any) with W's to reflect these voters' opinion that > the majority judgment (B) was too nice, so on the question of remaining > letters, all of these voters' words will come down on the side of W. > > After these modifications, determine by majority rule the next letter in > Jose's judgment, and make the usual adjustments to the losing words. > > What if there is a tie, meaning both letters get equal support. Then put a > neutral letter n for "neither better nor worse" in that position of Jose's > majority judgment word, and base its next letter on whether there are more > W's or B's in the next position of the remaining words in the list (of > voter words judging Jose). Continue as necessary until the tie is fully > resolved. > > So we see that the majority judgment words for the various candidates will > be formed with letters from the set > > {B, n, W} > > The order of finish is the dictionary order of the respective majority > judgment words ... assuming a really up to date comprehensive English > dictionary . > > Have I overlooked anything? Is Mono-Add satisfied? Reverse Symmetry? > > Do the rules make sense heuristically? > > FWS > > >