<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>Marcus,</DIV>
<DIV> </DIV>
<DIV>You wrote (8 Jan 2009):<BR></DIV>
<DIV>"Statement #1: Criterion X does not imply criterion Y.<BR>Statement #2: Criterion X and criterion Y are incompatible.<BR><BR>Statement #1 does not imply statement #2. But in your<BR>29 Dec 2008 mail, you mistakenly assume that statement #1<BR>implies statement #2."</DIV>
<DIV> </DIV>
<DIV>No I didn't. That is just your mistaken impression.<BR><BR>"You proved only that beatpath GMC does not imply mono-add-plump;<BR>but then you mistakenly concluded that this means that beatpath<BR>GMC and mono-add-plump were incompatible ("spectacularly<BR>vulnerable to mono-add-plump", "spectacular failure of<BR>mono-add-plump")."</DIV>
<DIV> </DIV>
<DIV>No, I only wrote that the beatpath GMC *concept* is "vulnerable to </DIV>
<DIV>Mono-add-Plump." </DIV>
<DIV> </DIV>
<DIV>"However, the fact, that Schulze(winning votes) satisfies beatpath GMC </DIV>
<DIV>and mono-add-plump, demonstrates that these two criteria are not incompatible."</DIV>
<DIV> </DIV>
<DIV>Yes, that is obvious. I explicitly acknowledged this in my last post.</DIV>
<DIV> </DIV>
<DIV>I think that all methods that fail Independence from Irrelevant Ballots are silly and<BR>that methods should meet the Majority criterion. The Majority *concept* is <BR>"vulnerable to Irrelevant Ballots" because candidate A can be the only candidate</DIV>
<DIV>allowed to win by the Majority criterion and then we add a handful of ballots that<BR>all plump for nobody and candidate A no longer has a majority.</DIV>
<DIV> </DIV>
<DIV>But of course I don't suggest that those two criteria are incompatible.<BR></DIV>
<DIV>The point of my Dec.29 demonstration was to refute any notion or assumption</DIV>
<DIV>that all candidates in the CDTT (i.e. those not excluded by Beatpath GMC) must<BR>be "stronger" (i.e. more representative of the voters and so more deserving of victory)</DIV>
<DIV>than any of the candidates outside the CDTT. </DIV>
<DIV> </DIV>
<DIV>This was only the first part of my argument that Beatpath GMC [compliance] is a</DIV>
<DIV>"mistaken standard". What other criterion/standard says that the winner must come</DIV>
<DIV>from set S, with S being a set that a candidate X can be kicked out of by an influx</DIV>
<DIV>of new ballots that all plump (bullet-vote) for X?</DIV>
<DIV> </DIV>
<DIV>I put it to you that the answer is "none", and that that makes Beatpath GMC uniquely<BR>weird and suspect. By itself that isn't conclusively damning because it doesn't prove<BR>that Beatpath GMC can exclude the strongest candidate. </DIV>
<DIV><BR>25: A>B<BR>26: B>C<BR>23: C>A<BR>26: C<BR></DIV>
<DIV>But I contend that here in my "situation 2" election Beatpath GMC does exclude</DIV>
<DIV>the clearly strongest candidate C. You ignored the last few paragraphs of my last post:</DIV>
<DIV> </DIV>
<DIV>.. I don't accept your suggestion that compliance with beatpath GMC is acceptably cheap </DIV>
<DIV>(let alone free), because it isn't compatible with my recently suggested "Smith- Comprehensive <BR>3-slot Ratings Winner" criterion, which I value much more.<BR></DIV>
<DIV><A href="http://lists.electorama.com/pipermail/election-methods-electorama.com/2008-December/023595.html">http://lists.electorama.com/pipermail/election-methods-electorama.com/2008-December/023595.html</A><BR><BR>In other words the CDTT set can fail to include the candidate that on overwhelming </DIV>
<DIV>common-sense (mostly positional) grounds is the strongest candidate (e.g. C in "Situation # 2").<BR> <BR>So given a method that meets what I've been recently calling "Strong Minimal Defense" </DIV>
<DIV>(and so Minimal Defense and Plurality) and Schwartz (and so fails LNHarm and meets Majority </DIV>
<DIV>for Solid Coalitions), I consider the addition of compliance with "beatpath GMC" a negative if <BR>without it the method can meet "Smith- Comprehensive 3-slot Ratings Winner" (which<BR>should be very very easy).<BR><BR>Chris Benham<BR><BR></DIV>
<DIV style="FONT-SIZE: 12pt; FONT-FAMILY: times new roman, new york, times, serif"><BR>
<DIV style="FONT-SIZE: 13px; FONT-FAMILY: arial, helvetica, sans-serif">Dear Chris Benham,<BR><BR>you wrote (29 Dec 2008):<BR><BR>> I think that compliance with GMC is a mistaken standard<BR>> in the sense that the best methods should fail it.<BR>><BR>> The GMC concept is spectacularly vulnerable to Mono-add-Plump!<BR>><BR>> [Situation #1]<BR>><BR>> 25: A>B<BR>> 26: B>C<BR>> 23: C>A<BR>> 04: C<BR>> 78 ballots (majority threshold = 40)<BR>><BR>> B>C 51-27, C>A 53-25, A>B 48-26.<BR>><BR>> All three candidates have a majority beat-path to each other,<BR>> so GMC says that any of them are allowed to win.<BR>><BR>> [Situation #2]<BR>><BR>> But say we add 22 ballots that plump for C:<BR>><BR>> 25: A>B<BR>> 26: B>C<BR>> 23: C>A<BR>> 26: C<BR>> 100 ballots (majority threshold = 51)<BR>><BR>> B>C 51-49, C>A 75-25, A>B
48-26.<BR>><BR>> Now B has majority beatpaths to each of the other candidates<BR>> but neither of them have one back to B, so the GMC says that<BR>> now the winner must be B.<BR>><BR>> The GMC concept is also naturally vulnerable to Irrelevant<BR>> Ballots. Suppose we now add 3 new ballots that plump for an<BR>> extra candidate X.<BR>><BR>> [Situation #3]<BR>><BR>> 25: A>B<BR>> 26: B>C<BR>> 23: C>A<BR>> 26: C<BR>> 03: X<BR>> 103 ballots (majority threshold = 52)<BR>><BR>> Now B no longer has a majority-strength beat-path to C,<BR>> so now GMC says that C (along with B) is allowed to win<BR>> again.<BR>><BR>> (BTW this whole demonstration also applies to "Majority-Defeat<BR>> Disqualification"(MDD) and if we pretend that the C-plumping<BR>> voters are truncating their sincere preference for B over A<BR>> then it also applies to Eppley's "Truncation Resistance"<BR>>
and Ossipoff's SFC and GFSC criteria.)<BR><BR>I wrote (29 Dec 2008):<BR><BR>> Your argumentation is incorrect. Example:<BR>><BR>> In many scientific papers, the Smith set is criticized<BR>> because the Smith set can contain Pareto-dominated<BR>> candidates. However, to these criticisms I usually<BR>> reply that the fact, that the Smith criterion doesn't<BR>> imply the Pareto criterion, is not a problem as long<BR>> as the used tie-breaker guarantees that none of these<BR>> Pareto-dominated candidates is elected. It would be<BR>> a problem only if the Smith criterion and the Pareto<BR>> criterion were incompatible.<BR>><BR>> You made the same mistake as the authors of these papers.<BR>> You didn't demonstrate that "the GMC concept is spectacularly<BR>> vulnerable to mono-add-plump". You only
demonstrated that<BR>> beatpath GMC doesn't imply mono-add-plump.<BR>><BR>> However, the fact, that Schulze(winning votes) satisfies<BR>> mono-add-plump and always chooses from the CDTT set and<BR>> isn't vulnerable to irrelevant ballots, shows that these<BR>> properties are not incompatible.<BR>><BR>> In all three situations, Schulze(winning votes) chooses<BR>> candidate B. Therefore, you demonstrated neither a<BR>> "spectacular failure of mono-add-plump" nor a "vulnerability<BR>> to irrelevant ballots" for methods that satisfy beatpath GMC.<BR>><BR>> You wrote: "All three candidates have a majority beatpath<BR>> to each other, so GMC says that any of them are allowed to<BR>> win." No! Beatpath GMC doesn't say that "any of them are<BR>> allowed to win"; beatpath GMC only doesn't exclude any of<BR>> them from winning. Similarly, the Smith criterion doesn't<BR>> say that even Pareto-dominated
candidates must be allowed<BR>> to win; that would have meant that the Smith criterion and<BR>> the Pareto criterion were incompatible; the Smith criterion<BR>> only doesn't imply the Pareto criterion.<BR><BR>You wrote (8 Jan 2009):<BR><BR>> I can't see that the distinction between "allowed to win" and <BR>> "not excluded from winning" is anything more than that between<BR>> "the glass is half full" and "the glass is half empty", so I<BR>> reject your semantic quibble. Any candidate that a criterion C<BR>> doesn't exclude from winning is (as far as C is concerned)<BR>> "allowed to win".<BR><BR>Statement #1: Criterion X does not imply criterion Y.<BR>Statement #2: Criterion X and criterion Y are incompatible.<BR><BR>Statement #1 does not imply statement #2. But in your<BR>29 Dec 2008 mail, you mistakenly assume that statement #1<BR>implies statement #2.<BR><BR>Example:<BR><BR> X = Smith criterion.<BR> Y = Pareto
criterion.<BR> Then statement #1 is true and statement #2 is false.<BR><BR>The fact, that statement #1 does not imply statement #2,<BR>is not "semantic quibble".<BR><BR>You proved only that beatpath GMC does not imply mono-add-plump;<BR>but then you mistakenly concluded that this means that beatpath<BR>GMC and mono-add-plump were incompatible ("spectacularly<BR>vulnerable to mono-add-plump", "spectacular failure of<BR>mono-add-plump"). However, the fact, that Schulze(winning votes)<BR>satisfies beatpath GMC and mono-add-plump, demonstrates that<BR>these two criteria are not incompatible.<BR><BR>Example:<BR><BR> The Smith criterion does not imply the Pareto criterion; that<BR> means that it can happen that a Pareto-dominated candidate is<BR> not excluded from winning by the Smith criterion. However,<BR> this doesn't mean that the Smith criterion implies that even<BR> Pareto-dominated candidates must be allowed to
win.<BR><BR>************<BR><BR>You wrote (8 Jan 2009):<BR><BR>> Perhaps you misunderstand my use of the word "concept".<BR>> Beatpath GMC says that the winner must come from a certain<BR>> set S, but a candidate X can fall out of S if a relatively<BR>> large number of new ballots are added, all plumping<BR>> (bullet-voting) for X. Is there any other criterion with<BR>> that absurd feature?<BR><BR>Your argumentation doesn't make any sense.<BR><BR>Example:<BR><BR> The fact, that the Borda method satisfies monotonicity<BR> and violates independence of clones, demonstrates that<BR> monotonicity doesn't imply independence of clones. You<BR> rejected beatpath GMC because it doesn't imply mono-add-plump.<BR> But with the same logic, you could reject the "concept"<BR> of monotonicity for the "absurd feature" of not implying<BR> independence of clones or for being "spectacularly<BR> vulnerable"
to clones.<BR><BR>The fact, that Schulze(winning votes) satisfies beatpath GMC<BR>and mono-add-plump, demonstrates that these two criteria are<BR>not incompatible. But you claim that already the fact, that<BR>beatpath GMC doesn't imply mono-add-plump, was an "absurd<BR>feature" of beatpath GMC.<BR><BR>Markus Schulze<BR><BR><BR></DIV></DIV></div><br>
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