<html><head><style type="text/css"><!-- DIV {margin:0px;} --></style></head><body><div style="font-family:times new roman, new york, times, serif;font-size:12pt"><DIV>The "Generalised Majority Criterion" says in effect that the winner</DIV>
<DIV>must come from Woodall's CDTT set, and is defined by Marcus Schulze</DIV>
<DIV>thus (October 1997):<BR><BR>"Definition ("Generalized Majority Criterion"):<BR><BR> "X >> Y" means, that a majority of the voters prefers<BR> X to Y.<BR><BR> "There is a majority beat-path from X to Y," means,<BR> that X >> Y or there is a set of candidates<BR> C[1], ..., C[n] with X >> C[1] >> ... >> C[n] >> Y.<BR><BR> A method meets the "Generalized Majority<BR> Criterion" (GMC) if and only if:<BR> If there is a majority beat-path from A to B, but<BR> no majority beat-path from B to A, then B must not<BR> be elected."</DIV>
<DIV> </DIV>
<DIV>With full strict ranking this implies Smith, and obviously <BR>"Candidates permitted to win by GMC (i.e.CDTT), Random Candidate"<BR>is much better than plain Random Candidate. Nonetheless I think that compliance</DIV>
<DIV>with GMC is a mistaken standard in the sense that the best methods should</DIV>
<DIV>fail it.</DIV>
<DIV> </DIV>
<DIV>The GMC concept is spectacularly vulnerable to Mono-add-Plump!</DIV>
<DIV> </DIV>
<DIV>25: A>B<BR>26: B>C</DIV>
<DIV>23: C>A</DIV>
<DIV>04: C</DIV>
<DIV>78 ballots (majority threshold = 40)<BR></DIV>
<DIV>B>C 51-27, C>A 53-25, A>B 48-26.</DIV>
<DIV> </DIV>
<DIV>All three candidates have a majority beat-path to each other, so GMC says that<BR>any of them are allowed to win. </DIV>
<DIV> </DIV>
<DIV>But say we add 22 ballots that plump for C:<BR><BR>25: A>B<BR>26: B>C</DIV>
<DIV>
<DIV>23: C>A</DIV>
<DIV>26: C</DIV>
<DIV>100 ballots (majority threshold = 51)<BR></DIV>
<DIV>
<DIV>B>C 51-27, C>A 75-25, A>B 48-26.</DIV>
<DIV> </DIV>
<DIV>Now B has majority beatpaths to each of the other candidates but neither of them</DIV>
<DIV>have one back to B, so the GMC says that now the winner must be B.</DIV>
<DIV> </DIV>
<DIV>The GMC concept is also naturally vulnerable to Irrelevant Ballots. Suppose we now</DIV>
<DIV>add 3 new ballots that plump for an extra candidate X.</DIV>
<DIV><BR>25: A>B<BR>26: B>C
<DIV>
<DIV>23: C>A</DIV>
<DIV>26: C<BR>03: X</DIV>
<DIV>103 ballots (majority threshold = 52)</DIV>
<DIV> </DIV>
<DIV>Now B no longer has a majority-strength beat-path to C, so now GMC says that C<BR>(along with B) is allowed to win again.<BR><BR>(BTW this whole demonstration also applies to "Majority-Defeat Disqualification"(MDD)</DIV>
<DIV>and if we pretend that the C-plumping voters are trucating their sincere preference for B</DIV>
<DIV>over A then it also applies to Eppley's "Truncation Resistance" and Ossipoff's SFC and</DIV>
<DIV>GFSC criteria.)</DIV>
<DIV> </DIV>
<DIV>If the method uses 3-slot ratings ballots and we assume that the voted 3-slot ratings are</DIV>
<DIV>sincere, then the GMC can bar the plainly highest SU candidate from winning as evidenced<BR>by its incompatibility with my recently suggested "Smith-Comprehensive 3-slot Ratings</DIV>
<DIV>Winner" criterion:</DIV>
<DIV><BR>*If no voter expresses more than three preference-levels and the ballot <BR>rules allow the expression of 3 preference-levels when there are 3 (or <BR>more) candidates, then (interpreting candidates that are voted above one<BR>or more candidates and below none as "top-rated", those voted above<BR>one or more candidates but below all the top-rated candidates as <BR>"middle-rated" and those not voted above any other candidate and below<BR>at least one other candidate as "bottom-rated", and interpreting above-<BR>bottom rating as approval) it must not be possible for candidate X to<BR>win if there is some candidate Y which has a beat-path to X and <BR>simultaneously higher Top-Ratings and Approval scores and a lower <BR>Maximum Approval-Opposition score.*<BR><BR><A
href="http://lists.electorama.com/pipermail/election-methods-electorama.com/2008-December/023548.html">http://lists.electorama.com/pipermail/election-methods-electorama.com/2008-December/023548.html</A></DIV>
<DIV><BR>25: A>B<BR>26: B>C
<DIV>
<DIV>23: C>A</DIV>
<DIV>26: C<BR><BR>TR scores: C49, B26, A25<BR>App. scores: C75, B51, A48<BR>MAO scores: C25, B49, A52</DIV>
<DIV> </DIV>
<DIV>That criterion says that C must win here. GMC says only B can win.</DIV>
<DIV> </DIV>
<DIV>Frankly I think any method needs a much better excuse than any that Winning Votes</DIV>
<DIV>can offer for not electing C here. As I discuss in another recent post, any method</DIV>
<DIV>that doesn't elect C here must be vulnerable to Push-over. So another reason not<BR>to be in love with GMC is that it is incompatible with "Pushover Invulnerability".</DIV>
<DIV> </DIV>
<DIV><A href="http://lists.electorama.com/pipermail/election-methods-electorama.com/2008-December/023543.html">http://lists.electorama.com/pipermail/election-methods-electorama.com/2008-December/023543.html</A><BR></DIV>
<DIV>As I hope some may have guessed from the spectacular failure of Mono-add-Plump, the GMC </DIV>
<DIV>concept is grossly unfair to truncators. And Winning Votes (as a GMC complying method) is</DIV>
<DIV>unfair to truncators. </DIV>
<DIV> </DIV>
<DIV>Say the 26C "we're just here to elect C and don't care about any other candidate" voters use a </DIV>
<DIV>random-fill strategy, each tossing a fair coin to decide between voting C>B or C>A; then even if as</DIV>
<DIV>few as 4 of them vote C>A they will elect C. Their chance making C the decisive winner is 99.9956% </DIV>
<DIV>(according to an online calculator <A href="http://stattrek.com/Tables/Binomial.aspx">http://stattrek.com/Tables/Binomial.aspx</A> ).</DIV>
<DIV><BR>I have some sympathy with the idea of giving up something so as to counter order-reversing buriers,</DIV>
<DIV>but not with the idea that electing a CW is obviously so wonderful that when there is no voted CW</DIV>
<DIV>we must guess that there is a "sincere CW" and if we can infer that that can only (assuming no voters</DIV>
<DIV>are order-reversing) be X then we must elect X.<BR></DIV>
<DIV> </DIV>
<DIV>Chris Benham<BR><BR></DIV><BR></DIV></DIV>
<DIV><BR> </DIV></DIV></DIV><BR><BR></DIV></DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV><BR><BR> </DIV></div><br>
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