I don't agree that "Sincere Favorite" is practically equivalent to the FBC. The FBC is about not having to lower your one favorite candidate; it is not about not having to pick a single favorite from your favorite set. As a voter, I'd regard the former as a serious dilemma, and the latter as a trivial detail.<div>
<br></div><div>Jameson<br><br><div class="gmail_quote">2011/11/23 Chris Benham <span dir="ltr"><<a href="mailto:cbenhamau@yahoo.com.au">cbenhamau@yahoo.com.au</a>></span><br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
<div><div style="color:#000;background-color:#fff;font-family:times new roman,new york,times,serif;font-size:12pt"><div><span>Forest,<div class="im"><br><br>"Furthermore something must be wrong with the quoted proof (of the incompatibility of the FBC and the <br>
CC) because the winner of the two slot case can be found entirely on the basis of the pairwise matrix."</div></span></div>
<div><span></span> </div>
<div><span>The likely explanation for some odd remarks by you and Jameson has just occurred to me. Could it be<br>that you and Jameson have mistaken a set of ballots for a "pairwise matrix" ??</span></div>
<div><span></span> </div>
<div><span>Here is that quoted proof again, with the ballots represented in the more familiar EM notation:</span></div>
<div><span><div class="im"> <br>Hello,<br><br>This is an attempt to demonstrate that Condorcet and FBC are incompatible.<br>I modified Woodall's proof that Condorcet and LNHarm are incompatible.<br>(Douglas R. Woodall, "Monotonicity of single-seat preferential election rules",<br>
Discrete Applied Mathematics 77 (1997), pages 86 and 87.)<br><br>I've suggested before that in order to satisfy FBC, it must be the case<br>that increasing the votes for A over B in the pairwise matrix can never <br>
increase the probability that the winner comes from {a,b}; that is, it must<br>
not move the win from some other candidate C to A. This is necessary because<br>if sometimes it were possible to move the win from C to A by increasing<br>v[a,b], the voter with the preference order B>A>C would have incentive to<br>
reverse B and A in his ranking (and equal ranking would be inadequate).<br><br>I won't presently try to argue that this
requirement can't be avoided somehow.<br>I'm sure it can't be avoided when the method's result is determined solely<br>from the pairwise matrix.<br><br></div><div class="im">Suppose a method satisfies this property, and also Condorcet. Consider this <br>
scenario:<br><br></div>3: A=B<br>3: A=C<br>3: B=C</span></div>
<div><span>2: A>C</span></div>
<div><span>2: B>A<br>2: C>B</span></div><div class="im">
<div><span><br>There is an A>C>B>A cycle, and the scenario is "symmetrical," as based on<br>the submitted rankings, the candidates can't be differentiated. This means<br>that an anonymous and neutral method has to elect each candidate with 33.33%<br>
probability.<br><br>Now suppose the a=b voters change their vote to a>b (thereby increasing v[a,b]).<br>This would turn A into the Condorcet winner, who would have to win with 100% <br>probability due to Condorcet.<br>
<br>But the probability that the winner comes from {a,b} has increased from 66.67%<br>to 100%, so the first property is violated.<br><br>Thus the first property and Condorcet are incompatible, and I contend that FBC<br>requires the first property.<br>
<br>Thoughts?<br><br>Kevin Venzke</span></div>
<div><span></span> </div>
</div><div><span><a href="http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-June/016410.html" target="_blank">http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-June/016410.html</a></span></div>
<div><span></span> </div>
<div><span>It is certainly a clear proof of the incompatibilty of the Condorcet criterion and Kevin's later</span></div>
<div><span>suggested "variation" of the FBC, "Sincere Favorite":</span></div>
<div><span></span> </div><span>
<div><i>Suppose a subset of the ballots, all identical, rank every candidate in S (where S contains at least two candidates) equal to each other, and above every other candidate. Then, arbitrarily lowering some candidate X from S on these ballots must not increase the probability that the winner comes from S.</i></div>
<div>A simpler way to word this would be: <i>You should never be able to help your favorites by lowering one of them.</i></div></span>
<div><span></span> </div>
<div><span><a href="http://nodesiege.tripod.com/elections/#critfbc" target="_blank">http://nodesiege.tripod.com/elections/#critfbc</a><br></span></div>
<div><span>I can't see any real difference between this and regular FBC, which probably partly explains</span></div>
<div><span>why it didn't catch on.<var></var></span></div>
<div><span></span> </div>
<div><span>Chris Benham</span></div>
<div><span></span><span></span> </div>
<div> </div>
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</div><b><span style="FONT-WEIGHT:bold">From:</span></b> "<a href="mailto:fsimmons@pcc.edu" target="_blank">fsimmons@pcc.edu</a>" <<a href="mailto:fsimmons@pcc.edu" target="_blank">fsimmons@pcc.edu</a>><br>
<b><span style="FONT-WEIGHT:bold">Sent:</span></b> Wednesday, 23 November 2011 9:01 AM<div class="im"><br><b><span style="FONT-WEIGHT:bold">Subject:</span></b> Re: An ABE solution<br></div></font><div class="im"><br>You are right that although the method is defined for any number of slots, I suggested three slots as <br>
most practical.<br><br>So my example of two slots was only to disprove the statement the assertion that the method cannot be <br>FBC compliant, since it is obviously compliant in that case. <br><br>Furthermore something must be wrong with the quoted
proof (of the incompatibility of the FBC and the <br>CC) because the winner of the two slot case can be found entirely on the basis of the pairwise matrix. <br>The other escape hatch is to say that two slots are not enough to satisfy anything but the voted ballots <br>
version of the Condorcet Criterion. But this applies equally well to the three slot case.<br><br>Either way the cited "therorem" is not good enough to rule out compliance with the FBC by this new <br>method.<br>
<br>Indeed, the three slot case does appear to satisfy the FBC as well. It is an open question. I did not <br>assert that it does. But I did say that "IF" it is strategically equivalent to Approval (as Range is, for <br>
example) then for "practical purposes" it satisfies the FBC. Perhaps not the letter of the law, but the <br>spirit of the law. Indeed, in a non-stratetgical environment nobody worries about the FBC, i.e. only <br>
strategic
voters will betray their favorite. If optimal strategy is approval strategy, and approval strategy <br>requires you to top rate your favorite, then why would you do otherwise?<br><br>Forest<br><br></div></div></div></div>
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