Too much hurry ...starting at *** below, typos abound ... x-y should be x-z, then y-z should be y-x, and z-x should be z-y. On Tuesday, March 16, 2021, Forest Simmons <fsimmons@pcc.edu> wrote: > In his book, The Geometry of Voting, Saari approaches the problem of a > resolving a Condorcet cycle in the three candidate case by first canceling > ABC ballots against CBA ballots, BCA ballots against ACB ballots, and CAB > ballots against BAC ballots. This first step cannot eliminate the Condorcet > cycle, but it reduces the number of factions to three: either ABC, BCA, & > CAB or else > CBA, BAC, & ACB. > > Without loss in generality suppose the three factions are > > x: ABC > y: BCA & > z: CAB > > > Now suppose that y = min(x,y,z). Saari now removes y ballots from each > faction, which eliminates the middle faction entirely: > > (x-y): ABC > (z-y): CAB > > With only two factions there can no longer be a Condorcet cycle. > Majoritarians would say that A or C must win depending on which of the two > remaining factions is larger. A should win if (x-y) is larger than (z-y), > which is true iff x - z is positive. > *** x-y---> x-z, etc. I'm going to make the changes... > > Saari continues his analysis to argue in general (no matter which of the > three cyclical factions is smallest) the largest difference among (x -z), > (y -x), (z-y) should determine whether A, B, or C wins, respectively. > > [Note x-z is fpA - fpC] > > Saari goes on to show that this result is equivalent to the Borda Count, > and rests his case that Borda is the best election method (to make a long > story short). > > To be continued .... > > > > > > ************************************************ >> >