<div dir="auto">Another valuable special case is five categories because of the popular A, B, C, D, F scale.<div dir="auto"><br></div><div dir="auto">IF either the A or the F category (by itself) holds a majority of the voter judgments for candidate X, then we're done ...</div><div dir="auto">ELSE fold the A and F categories into the B and D categories, respectively...</div><div dir="auto">AND ... use the three slot version to find the MJ.</div><div dir="auto"><br></div><div dir="auto">This symmetrical 5-slot version of MJ is way simpler than 6-slot MJ.</div><div dir="auto"><br></div><div dir="auto">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.</div><div dir="auto"><br></div><div dir="auto">FWS</div><div dir="auto"><br></div><div dir="auto"><br></div></div><br><div class="gmail_quote"><div dir="ltr">El sáb., 23 de oct. de 2021 9:34 p. m., Forest Simmons <<a href="mailto:forest.simmons21@gmail.com">forest.simmons21@gmail.com</a>> escribió:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="auto"><div>One other noteworthy case is for 3, 9, 27, etc ... 3^k judgment categories... </div><div dir="auto"><br></div><div dir="auto">WHILE 3,9,27, or 81 categories remain ...</div><div dir="auto">IF more than half of the remaining ballot judgments are in the upper third or in the lower third of the categories ...</div><div dir="auto">THEN ...</div><div dir="auto">move the judgments from the minority and middle third categories to the majority category they are closest to ...</div><div dir="auto">AND apply this method recursively to the remaining third of categories (from the majority side)</div><div dir="auto">ELSE move the outer two thirds of categories to their nearest neighbor category in the remaining middle third of caategories.</div><div dir="auto">END END [IF and While]</div><div dir="auto"><br></div><div dir="auto">Tbe one category remaining (after this procedure) holds the Majority Judgment. </div><div dir="auto"><br></div><div dir="auto">Remark: This version of MJ goes well with Ranked Rankings when there are lots of candidates ... typical ballot:</div><div dir="auto"><br></div><div dir="auto">A>B>C>>D>E>F>>G>H>I>>> ...>>>, etc.</div><div dir="auto"><br></div><div dir="auto">... 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? </div><div dir="auto"><br></div><div dir="auto">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.</div><div dir="auto"><br></div><div dir="auto">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.</div><div dir="auto"><br></div><div dir="auto">Three-slot MJ fits into both the 3^k and 2^j-1 patterns.</div><div dir="auto"><br></div><div dir="auto">FWS<br><br><div class="gmail_quote" dir="auto"><div dir="ltr">El sáb., 23 de oct. de 2021 8:28 p. m., Forest Simmons <<a href="mailto:forest.simmons21@gmail.com" target="_blank" rel="noreferrer">forest.simmons21@gmail.com</a>> escribió:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="auto">The special cases of 3, 7, 15, 31, or 2^k-1 categories can be handled much more succintly:<div dir="auto"><br></div><div dir="auto">WHILE more than one category remains ... [i.e. while 3, 7, ... or 2^k-1 categories remain]...<br><div dir="auto">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..<br></div><div dir="auto">THEN ...<br></div><div dir="auto"> ... 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.... </div><div dir="auto">AND apply this method recursively to the remaining categoties.<br></div><div dir="auto">ELSE ...move all judgments to the middle category among those remaining.<br></div><div dir="auto">EndWhile</div><div dir="auto"><br></div><div dir="auto">After execution of the above procedure the one remaining category is the MJ category. </div><div dir="auto"><br></div><div dir="auto"> </div><div dir="auto"><br></div></div></div><br><div class="gmail_quote"><div dir="ltr">El sáb., 23 de oct. de 2021 6:35 p. m., Forest Simmons <<a href="mailto:forest.simmons21@gmail.com" rel="noreferrer noreferrer" target="_blank">forest.simmons21@gmail.com</a>> escribió:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="auto">It is easier to bring MJ into compliance with Reverse Symmetry if there are an odd number of categories 3, 5, 7, ...<div dir="auto"><br></div><div dir="auto">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.</div><div dir="auto"><br></div><div dir="auto">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. </div><div dir="auto"><br></div><div dir="auto">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!]</div><div dir="auto"><br></div><div dir="auto">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).</div><div dir="auto"><br></div><div dir="auto">Or (in the even case) if the two middle judgments are the same, then that common judgment is the MJ.</div><div dir="auto"><br></div><div dir="auto">Or if the two middle judgments are in categories situated symmetrically about neutral, then neutral is the majority judgment.</div><div dir="auto"><br></div><div dir="auto">Otherwise, the outside one (the one furthest away from neutral in the category order) is the MJ.</div><div dir="auto"><br></div><div dir="auto">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.</div><div dir="auto"><br></div><div dir="auto">Comments?</div><div dir="auto"><br></div><div dir="auto">FWS</div><div dir="auto"><br></div><div dir="auto"><br></div><div dir="auto"><br></div><div dir="auto"><br></div></div>
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