Forest W Simmons wrote:
>A candidate X covers a >candidate Y if and only if X (pairwise) defeats both Y and each >candidate that Y defeats. > >So if X covers Y, then in a pairwise sense X dominates Y. > >Now for UncAAO: > >1. Write abbreviations for all of the candidate names on a big sheet of >butcher paper. > >2. For each candidate X, if X is covered by some candidate, then draw >an arrow from X to the candidate (among those that cover X) >against which X has the least approval opposition. > >3. If X is not covered by any candidate, then do not draw any arrow >from X to another candidate. > >4. Once all of the arrows have been drawn, start at the candidate A >with the most approval, and follow the arrows until you reach (the name >of) an uncovered candidate. This candidate is the winner. > >When there are only three candidates, UncAAO is the same as Smith >Approval. > Forest, How is this supposed to be better than ASM and DMC? In April 2002 Adam Tarr discussed Condorcet//Approval (which he calls "ACC", Approval-Completed Condorcet). Since his examples all have three candidates, it applies equally to Smith//Approval and UncAOO. http://lists.electorama.com/pipermail/election-methods-electorama.com/2002-April/008013.html > I'm now going to compare ACC to margins and winning votes Condorcet > methods, using the example that has become my signature example on this > list. The following are the sincere preferences of my example electorate: > > 49: Bush>Gore>Nader > 12: Gore>Bush>Nader > 12: Gore>Nader>Bush > 27: Nader>Gore>Bush > > If everyone votes sincerely, then Gore is the Condorcet winner. The > problem arises when the Bush voters swap Nader and Gore on their ballots > (in margins they can achieve the same effect by truncating, but I'll > ignore > that for this analysis). So the new "preferences" are > > 49: Bush>Nader>Gore > 12: Gore>Bush>Nader > 12: Gore>Nader>Bush > 27: Nader>Gore>Bush > > In margins-based methods, the only way for Gore to still win the election > is for the Nader voters to bury Nader behind Gore. The stable > equilibrium > ballots become: > > 49: Bush>Nader>Gore > 12: Gore>Bush>Nader > 39: Gore>Nader>Bush > > And this allows Gore to still carry the election. This sort of > equilibrium > is what Mike is talking about when he says that margins methods are > "falsifying". > > In winning votes methods, the Nader camp can vote equal first-place > rankings rather than swap Gore and Nader entirely. The stable result is > therefore: > > 49: Bush>Nader>Gore > 12: Gore>Bush>Nader > 12: Gore>Nader>Bush > 27: Nader=Gore>Bush > > In ACC... we first have to define where the approval cutoffs on the > ballots > are. Since the approval tally is only used to break cyclic ties, clearly > the Bush camp has no incentive to Approve of anyone except Bush. I'm > going > to make the assumption that since Gore and Bush are the apparent front > runners in this race (the only two with a decent shot at election), every > voter will approve one and not the other. This is the logical approval > cutoff to use, based on the approval strategy threads that have been > circulating on the list of late. So the ballots could look something > like > this: (>> denotes approval cutoff) > > 49: Bush>>Nader>Gore > 12: Gore>>Bush>Nader > 6: Gore>>Nader>Bush > 6: Gore>Nader>>Bush > 27: Nader>Gore>>Bush > > In this case, Gore wins the approval runoff 51-49-33. So not only did > ACC > avoid the need for defensive order-reversal like margins methods, but it > avoided the need for defensive equal-ranking like winning votes > methods. This is a super result: totally strategy-free voting for the > majority side. > > There is a dark side to this result, though. Say that some of the > Gore>Bush>Nader voters were extremely non-strategic and decided to > approve > both Bush and Gore. So the votes now look like: > > 49: Bush>>Nader>Gore > 6: Gore>Bush>>Nader > 6: Gore>>Bush>Nader > 6: Gore>>Nader>Bush > 6: Gore>Nader>>Bush > 27: Nader>Gore>>Bush > > Now, Bush wins the approval runoff 55-51-33. This is where ACC's > favorite > betrayal scenario comes in. Since Bush wins the approval vote, the only > way the majority can guarantee a Gore win is to make Gore the initial > Condorcet winner, which requires that the Nader camp vote Gore in > first place: > > 49: Bush>>Nader>Gore > 6: Gore>Bush>>Nader > 6: Gore>>Bush>Nader > 6: Gore>>Nader>Bush > 33: Gore>Nader>>Bush > > So this is more or less the same as the margins method equilibrium. > > In summary, if the voters are fairly logical in the placement of their > approval cutoff, then ACC seems almost uniquely free of strategy > considerations. If the underlying approval votes do not back up the > sincere Condorcet winner, however, then ACC becomes just as vulnerable to > strategic manipulation as the margins methods are, if not more so. Note that in his "dark side" example, ASM and DMC have no problem. >Here's another classical example: > >49 C >24 B>A >27 A>B > >Under wv, this is not a Nash Equilibrium, because B can unilaterally >gain by truncating. > >But if the direct supporters of the CW strategically put their approval >cutoff just below A, then we end up with a Nash equilibrium, no matter >where the B faction puts its approval cutoff. > >49 C >24 B>A >27 A>>B > >As in wv, no defensive strategy is needed under zero info conditions. >But if you suspect that X is the CW, and you could live with X, then a >prudent move would be to approve X and above. > For what it's worth, this all applies at least as well to ASM and DMC. Of course some of the sincere B>A preferrers have to at least truncate for A not to be alone in the Smith set. When the ballot-style allows voters to rank among unapproved candidates ASM and DMC are my co-equal favourites, and when it doesn't I prefer ASM. http://wiki.electorama.com/wiki/Approval_Sorted_Margins Chris Benham ---- election-methods mailing list - see http://electorama.com/em for list info
