The special cases of 3, 7, 15, 31, or 2^k-1 categories can be handled mucch more succintly: WHILE more than one category remains ... [i.e. while 3, 7, ... or 2^k-1 categories remain]... IF more than half of the judgments are better than the middle remaining category or more than half are worse than the middle remaining category.. THEN ... ... move the remaining middle category judgments and those from the minority side of the remaining middle category to the majority side category that they are nearest to.... AND apply this method recursively to the remaining categoties. ELSE ...move all judgments to the middle category among those remaining. EndWhile After execution of the above procedure the one remaining category is the MJ category. El sáb., 23 de oct. de 2021 6:35 p. m., Forest Simmons < forest.simmons21@gmail.com> escribió: > It is easier to bring MJ into compliance with Reverse Symmetry if there > are an odd number of categories 3, 5, 7, ... > > Let's call the middle category "neutral" or Greek "nu" for short because > it is the only category without either a positive or negative connotation. > > A "majority judgment" for X will be one of the judgment categories (of > which there are an odd number) and which also make up the possible voter > judgments of the candidates. > > To find the voters' Majority Judgment for candidate X, list the induvidual > voter judgments of X while respecting the natural judgment order from left > to right, say. [no slight intended, lefties!] > > If there is a median member of the list (because of an odd number of > judgments of X), then that median judgment is X's Majority Judgment (MJ). > > Or (in the even case) if the two middle judgments are the same, then that > common judgment is the MJ. > > Or if the two middle judgments are in categories situated symmetrically > about neutral, then neutral is the majority judgment. > > Otherwise, the outside one (the one furthest away from neutral in the > category order) is the MJ. > > It seems to me that these conventions respect Reverse Summetry: if all > judgments are reversed (i.e. replaced with their symmetrical opposites), > then the MJ will also move to its symmetrical opposite. > > Comments? > > FWS > > > > >

One other noteworthy case is for 3, 9, 27, etc ... 3^k judgment categories... WHILE 3,9,27, or 81 categories remain ... IF more than half of the remaining ballot judgments are in the upper third or in the lower third of the categories ... THEN ... move the judgments from the minority and middle third categories to the majority category they are closest to ... AND apply this method recursively to the remaining third of categories (from the majority side) ELSE move the outer two thirds of categories to their nearest neighbor category in the remaining middle third of caategories. END END [IF and While] Tbe one category remaining (after this procedure) holds the Majority Judgment. Remark: This version of MJ goes well with Ranked Rankings when there are lots of candidates ... typical ballot: A>B>C>>D>E>F>>G>H>I>>> ...>>>, etc. ... a version of Ranked Rankings built up recursively on the popular 3-slot approval motif that we used to play around with ... but I don't think we ever actually came up with an MJ version of 3-slot in all of our brain storms ... did we Kevin? We probably shied away from it instinctively because of the high probability of ties ... not realizing how handily the gradual removal of (well defiined) mj's can resolve those ties. Three slot approval seemed so much easier to vote sincerely than ordinary two slot Approval ... that middle category softens the high stakes Approval decisions ... just like the possibility of giving half credit makes it easier to grade math tests. Three-slot MJ fits into both the 3^k and 2^j-1 patterns. FWS El sáb., 23 de oct. de 2021 8:28 p. m., Forest Simmons < forest.simmons21@gmail.com> escribió: > The special cases of 3, 7, 15, 31, or 2^k-1 categories can be handled much > more succintly: > > WHILE more than one category remains ... [i.e. while 3, 7, ... or 2^k-1 > categories remain]... > IF more than half of the judgments are better than the middle remaining > category or more than half are worse than the middle remaining category.. > THEN ... > ... move the remaining middle category judgments and those from the > minority side of the remaining middle category to the majority side > category that they are nearest to.... > AND apply this method recursively to the remaining categoties. > ELSE ...move all judgments to the middle category among those remaining. > EndWhile > > After execution of the above procedure the one remaining category is the > MJ category. > > > > > El sáb., 23 de oct. de 2021 6:35 p. m., Forest Simmons < > forest.simmons21@gmail.com> escribió: > >> It is easier to bring MJ into compliance with Reverse Symmetry if there >> are an odd number of categories 3, 5, 7, ... >> >> Let's call the middle category "neutral" or Greek "nu" for short because >> it is the only category without either a positive or negative connotation. >> >> A "majority judgment" for X will be one of the judgment categories (of >> which there are an odd number) and which also make up the possible voter >> judgments of the candidates. >> >> To find the voters' Majority Judgment for candidate X, list the >> induvidual voter judgments of X while respecting the natural judgment >> order from left to right, say. [no slight intended, lefties!] >> >> If there is a median member of the list (because of an odd number of >> judgments of X), then that median judgment is X's Majority Judgment (MJ). >> >> Or (in the even case) if the two middle judgments are the same, then that >> common judgment is the MJ. >> >> Or if the two middle judgments are in categories situated symmetrically >> about neutral, then neutral is the majority judgment. >> >> Otherwise, the outside one (the one furthest away from neutral in the >> category order) is the MJ. >> >> It seems to me that these conventions respect Reverse Summetry: if all >> judgments are reversed (i.e. replaced with their symmetrical opposites), >> then the MJ will also move to its symmetrical opposite. >> >> Comments? >> >> FWS >> >> >> >> >>

Another valuable special case is five categories because of the popular A, B, C, D, F scale. IF either the A or the F category (by itself) holds a majority of the voter judgments for candidate X, then we're done ... ELSE fold the A and F categories into the B and D categories, respectively... AND ... use the three slot version to find the MJ. This symmetrical 5-slot version of MJ is way simpler than 6-slot MJ. Why not stick with 3, 5, or 7 slots (categories) for most single winner elections ? Ordinary 6 slot MJ lacks the simplicity and symmetry of these nearby odd slot versions. FWS El sáb., 23 de oct. de 2021 9:34 p. m., Forest Simmons < forest.simmons21@gmail.com> escribió: > One other noteworthy case is for 3, 9, 27, etc ... 3^k judgment > categories... > > WHILE 3,9,27, or 81 categories remain ... > IF more than half of the remaining ballot judgments are in the upper third > or in the lower third of the categories ... > THEN ... > move the judgments from the minority and middle third categories to the > majority category they are closest to ... > AND apply this method recursively to the remaining third of categories > (from the majority side) > ELSE move the outer two thirds of categories to their nearest neighbor > category in the remaining middle third of caategories. > END END [IF and While] > > Tbe one category remaining (after this procedure) holds the Majority > Judgment. > > Remark: This version of MJ goes well with Ranked Rankings when there are > lots of candidates ... typical ballot: > > A>B>C>>D>E>F>>G>H>I>>> ...>>>, etc. > > ... a version of Ranked Rankings built up recursively on the popular > 3-slot approval motif that we used to play around with ... but I don't > think we ever actually came up with an MJ version of 3-slot in all of our > brain storms ... did we Kevin? > > We probably shied away from it instinctively because of the high > probability of ties ... not realizing how handily the gradual removal of > (well defiined) mj's can resolve those ties. > > Three slot approval seemed so much easier to vote sincerely than ordinary > two slot Approval ... that middle category softens the high stakes Approval > decisions ... just like the possibility of giving half credit makes it > easier to grade math tests. > > Three-slot MJ fits into both the 3^k and 2^j-1 patterns. > > FWS > > El sáb., 23 de oct. de 2021 8:28 p. m., Forest Simmons < > forest.simmons21@gmail.com> escribió: > >> The special cases of 3, 7, 15, 31, or 2^k-1 categories can be handled >> much more succintly: >> >> WHILE more than one category remains ... [i.e. while 3, 7, ... or 2^k-1 >> categories remain]... >> IF more than half of the judgments are better than the middle remaining >> category or more than half are worse than the middle remaining category.. >> THEN ... >> ... move the remaining middle category judgments and those from the >> minority side of the remaining middle category to the majority side >> category that they are nearest to.... >> AND apply this method recursively to the remaining categoties. >> ELSE ...move all judgments to the middle category among those remaining. >> EndWhile >> >> After execution of the above procedure the one remaining category is the >> MJ category. >> >> >> >> >> El sáb., 23 de oct. de 2021 6:35 p. m., Forest Simmons < >> forest.simmons21@gmail.com> escribió: >> >>> It is easier to bring MJ into compliance with Reverse Symmetry if there >>> are an odd number of categories 3, 5, 7, ... >>> >>> Let's call the middle category "neutral" or Greek "nu" for short because >>> it is the only category without either a positive or negative connotation. >>> >>> A "majority judgment" for X will be one of the judgment categories (of >>> which there are an odd number) and which also make up the possible voter >>> judgments of the candidates. >>> >>> To find the voters' Majority Judgment for candidate X, list the >>> induvidual voter judgments of X while respecting the natural judgment >>> order from left to right, say. [no slight intended, lefties!] >>> >>> If there is a median member of the list (because of an odd number of >>> judgments of X), then that median judgment is X's Majority Judgment (MJ). >>> >>> Or (in the even case) if the two middle judgments are the same, then >>> that common judgment is the MJ. >>> >>> Or if the two middle judgments are in categories situated symmetrically >>> about neutral, then neutral is the majority judgment. >>> >>> Otherwise, the outside one (the one furthest away from neutral in the >>> category order) is the MJ. >>> >>> It seems to me that these conventions respect Reverse Summetry: if all >>> judgments are reversed (i.e. replaced with their symmetrical opposites), >>> then the MJ will also move to its symmetrical opposite. >>> >>> Comments? >>> >>> FWS >>> >>> >>> >>> >>>