On 6 Jan 2005 at 01:32 PST, Jobst Heitzig wrote: > Dear Forest! > > Your sprucing up technique is a very nice idea since it can simplify the > tallying of those methods which fulfil beat-clone-proofness and > uncoveredness. However, some of which you wrote has confused me > completely: Did I understand you right in that you claim that the > technique should reduce everything to the 3-candidate case? I wonder how > that could possibly be when, in general, there is neither a proper beat > clone set nor a covered element! Second, I don't understand how the > method I posted yesterday under the name "Condorcet Lottery" could be > the same as anything based on Random Ballot??? > > In order to study these questions I provide the following example which > is also interesting with respect to immunity: > > > The "twisted prism" example: > ---------------------------- > Six candidates A1,A2,A3,B1,B2,B3, 15 voters. Sincere preferences: > 4 A1>B2>A2>B1>B3>A3 > 4 A2>B3>A3>B2>B1>A1 > 4 A3>B1>A1>B3>B2>A2 > 1 B1>B3>B2>A1>A2>A3 > 1 B2>B1>B3>A2>A3>A1 > 1 B3>B2>B1>A3>A1>A2 > Defeats: > A1>A2>A3>A1, strength 10 (the "upper clockwise" 3-cycle) > B1<B2<B3<B1, strength 10 (the "lower counter-clockwise" 3-cycle) > Bi>Ai, strength 9 (the 3 "straight upward" beats) > Ai>Bj (i!=j), strength 8 (the 6 "diagonal downward" beats)
Jobst, you made a small mistake. I find the pairwise matrix to be the following: A1 A2 A3 B1 B2 B3 A1 - 10 5 4 8 8 A2 5 - 10 8 4 8 A3 10 5 - 8 8 4 B1 11 7 7 - 5 10 B2 7 11 7 10 - 5 B3 7 7 11 5 10 - So your defeat rankings should actually be Bi>Ai , strength 11 (the 3 "straight upward" beats) A1>A2>A3>A1 , strength 10 (the "upper clockwise" 3-cycle) B1<B2<B3<B1 , strength 10 (the "lower counter-clockwise" 3-cycle) Ai>Bj (i!=j), strength 8 (the 6 "diagonal downward" beats) I think other aspects of this example still hold, though. Is there an automated way to find the Dutta set? I've found this reference: Dutta's Minimal Covering Set and Shapley's Saddles John Duggan and Michel Le Breton Journal of Economic Theory, 1996, vol. 70, issue 1, pages 257-265 but I don't have access to the article to see if it contains an algorithm. Ted -- Send real replies to ted stern at u dot washington dot edu Frango ut patefaciam -- I break that I may reveal ---- Election-methods mailing list - see http://electorama.com/em for list info