F.Simmons: 4 towns A,B,C,D with respective town populations in proportion 2:1:1:1. Preference profiles
40% A>B>C>D 20% B>C>A>D 20% C>B>A>D 20% D>C>A>B Which makes D a Condorcet loser and creates the cycle A beats B beats C beats A. Warren D. Smith: Each of the following lines gives a set of coordinates for the 4 towns in the plane, such that the above preference profile happens, using EITHER Euclidean(L2), Taxicab(L1) or Linfinity (or any Lp) distance metric (all work). A=(4,0) B=(0,2) C=(2,5) D=(8,7) A=(4,0) B=(8,2) C=(6,5) D=(0,7) A=(4,7) B=(0,5) C=(2,2) D=(8,0) A=(4,7) B=(8,5) C=(6,2) D=(0,0) All 4 point-sets are really the same point set (up to coordinate-reflections). This point set is unique if x and y coords are to be integers in [0,8] and [0,7] respectively (I did an exhaustive search). Unless I screwed up. Caveat emptor. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step) and math.temple.edu/~wds/homepage/works.html ---- Election-Methods mailing list - see http://electorama.com/em for list info
