Forest: How do you decompose my example (from my last email, #10873), and what do you get?:
3:A>B>C 5:A>C>B 0:C>A>B 5:C>B>A 0:B>C>A 5:B>A>C SB >From: Forest Simmons <[EMAIL PROTECTED]> >To: EM-list <[EMAIL PROTECTED]> >Subject: RE: [EM] Saari's Basic Argument > >On Tue, 18 Feb 2003, Steve Barney wrote: > >> Here is a simpler example to illustrate the difference that the order in which >> cyclic and reversal terms are canceled does not matter when using the strictly >> correct method - as opposed to the method used by Forest Simmons and Alex >> Small, and in some of Saari's popular expositions where he is merely trying to >> illustrate basic concepts to a more general audience. > >It's true that in my earlier examples I didn't take the decomposition all >of the way to its logical conclusion. But if you look at my last few >examples, you'll see that I finally hit on a simple method of adding and >subtracting the two kinds of symmetries that reduce each ballot set to a >canonical ballot set consisting of either one faction or two adjacent >factions with positive multiplicity, and all other factions zero. > >[Two factions are adjacent if one of them follows the other in the cyclic >order ABC->ACB->CAB->CBA->BCA->BAC->ABC.] > >It is easy to prove that this decomposition is unique, since it preserves >the Borda count of each candidate, and the Borda counts determine the >number in each faction when there are two or fewer adjacent factions. So >the order of application of the symmetries doesn't matter. > >As my examples show, this can be done very simply without matrices. > >Presumably Saari uses matrices because he wants to develop tools that will >generalize to more than three candidates. > >But worrying about the details of symmetry cancellations is to bark up the >wrong tree. > >The fact is that in the ballot set 65*ABC+35*BCA the candidates A, B, and >C have the same respective average ranks as they do in the simpler ballot >set 30*ABC+35*BAC , so according to Borda they are equivalent ballot sets, >and B should be the winner of the first election as decisively as in the >second (according to Borda and Saari). > >This result may make sense in the context of dispassionate decision making >such as in robotics when a robot is trying to decide what movement to make >or whether a visual image represents the letter U or V. > >But in the context of public elections, this supposed equivalence is >almost ludicrous. > >So the question is not, "Why is Borda such a great method for public >elections?" > >The question is, "Why does the symmetry argument lead us down the wrong >path?" > >At least that is the question I was trying to answer (and did answer to >my own satisfaction). > >In a nutshell the answer to this question is that the symmetries in the >distribution of ballots are at odds (more often than not) with Saari's >symmetries. > >In other words, Saari's transformations do not preserve the natural axes >of symmetry that may (and do) exist in the ballot distributions. > >In the above example, the first ballot set 65*ABC+35*BCA consists of two >factions that determine an axis of symmetry. The second ballot set >35*BAC+30*ABC also consists of two factions along an axis of symmetry, but >this axis is rotated thirty degrees relative to the original axis. > >So there is an essential change in the symmetry of the distribution that >the Borda count doesn't detect, and Saari's symmetry transformations >cannot preserve. > >[The center of gravity of the distribution is preserved, but the principal >axes of rotation and the radii of gyration are changed.] > >For me this insight is sufficient to explain the fallacy of the symmetry >arguments. > >Most non-mathematicians don't care one whit about what went wrong with the >symmetry arguments; rather than watch the gory details of an autopsy, they >prefer to move onward and upward. > >Forest Steve Barney Richard M. Hare, 1919 - 2002, In Memoriam: <http://www.petersingerlinks.com/Hare/>. Did you know there is a web site where, if you click on a button, the advertisers there will donate 2 1/2 cups of food to feed hungry people in places where there is a lot of starvation? See: <http://www.thehungersite.com>. ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em