Chris: Your range scores are a little bit wrong, so you have to add half a B vote for the example to work (or double all factions and add one B vote if you discriminate against fractional people), but yes, this is at heart a valid example where the method fails FBC.
Note that in my tendentious terminology this is only a "defensive" failure, that is, it starts from a position of a sincere condorcet cycle, which I believe will be rare enough in real elections to be discountable. In particular, this failure does not result in a stable two-party-lesser-evil-strategy self-reinforcing equilibrium. Jameson 2011/11/24 Chris Benham <cbenha...@yahoo.com.au> > Forest, > In reference to your new Condorcet method suggestion (pasted at the > bottom), which elects an > uncovered candidate and if there is none one-at-time disqualifies the > Range loser until a remaining > candidate X covers all the other remaining candidates and then elects X, > you wrote: > "Indeed, the three slot case does appear to satisfy the FBC..". > > No. Here is my example, based on that Kevin Venke proof you didn't like. > > Say sincere is > > 3: B>A > 3: A=C > 3: B=C > 2: A>C > 2: B>A > 2: C>B > 1: C > > Range (0,1,2) scores: C19, B17, A12. > C>B 8-5, B>A 10-5, A>C 7-6. > > C wins. > > Now we focus on the 3 B>A preferrers. Suppose (believing the method meets > the FBC) > they vote B=A. > > 3: B=A (sincere is B>A) > 3: A=C > 3: B=C > 2: A>C > 2: B>A > 2: C>B > 1: C > > Range (0,1,2) scores: C19, B17, A15. > > C>B 8-5, B>A 7-5, A>C 7-6. > > C still wins. > > Now suppose they instead rate their sincere favourite Middle: > > 3: A>B (sincere is B>A) > 3: A=C > 3: B=C > 2: A>C > 2: B>A > 2: C>B > 1: C > > Range (0,1,2) scores: C19, A15, B12. > > A>B 8-7, A>C 7-6, C>B 8-5 > > Now those 3 voters get a result they prefer, the election of their > compromise > candidate A. Since it is clear they couldn't have got a result for > themselves as > good or better by voting B>A or B>C or B this is a failure of the FBC. > > > Chris Benham > > > > *From:* "fsimm...@pcc.edu" <fsimm...@pcc.edu> > *Sent:* Wednesday, 23 November 2011 9:01 AM > > *Subject:* Re: An ABE solution > > You are right that although the method is defined for any number of slots, > I suggested three slots as > most practical. > > So my example of two slots was only to disprove the statement the > assertion that the method cannot be > FBC compliant, since it is obviously compliant in that case. > > Furthermore something must be wrong with the quoted proof (of the > incompatibility of the FBC and the > CC) because the winner of the two slot case can be found entirely on the > basis of the pairwise matrix. > The other escape hatch is to say that two slots are not enough to satisfy > anything but the voted ballots > version of the Condorcet Criterion. But this applies equally well to the > three slot case. > > Either way the cited "therorem" is not good enough to rule out compliance > with the FBC by this new > method. > > Indeed, the three slot case does appear to satisfy the FBC as well. It is > an open question. I did not > assert that it does. But I did say that "IF" it is strategically > equivalent to Approval (as Range is, for > example) then for "practical purposes" it satisfies the FBC. Perhaps not > the letter of the law, but the > spirit of the law. Indeed, in a non-stratetgical environment nobody > worries about the FBC, i.e. only > strategic voters will betray their favorite. If optimal strategy is > approval strategy, and approval strategy > requires you to top rate your favorite, then why would you do otherwise? > > Forest > > ----- Original Message ----- > From: Chris Benham > > Forest, > > "When the range ballots have only two slots, the method is simply > Approval, which does satisfy the > FBC." > > When you introduced the method you suggested that 3-slot ballots be used > "for simplicity". > I thought you might be open to say 4-6 slots, but a complicated algorithm > on 2-slot ballots > that is equivalent to Approval ?? > > "Now consider the case of range ballots with three slots: and suppose > that optimal strategy requires the > voters to avoid the middle slot. Then the method reduces to Approval, > which does satisfy the FBC." > > The FBC doesn't stipulate that all the voters use "optimal strategy", so > that isn't relavent. > > http://wiki.electorama.com/wiki/FBC > > http://nodesiege.tripod.com/elections/#critfbc > > Chris Benham > Forest Simmons wrote (17 Nov 2011): > > Here’s my current favorite deterministic proposal: Ballots are Range > Style, say three slot for simplicity. > > When the ballots are collected, the pairwise win/loss/tie relations are > determined among the candidates. > > The covering relations are also determined. Candidate X covers candidate > Y if X > beats Y as well as every candidate that Y beats. In other words row X of > the > win/loss/tie matrix dominates row Y. > > Then starting with the candidates with the lowest Range scores, they are > disqualified one by one until one of the remaining candidates X covers any > other > candidates that might remain. Elect X. > >
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