Because an inequality can be made into an equation by adding an unspecified number, as when A>B is written as A=B+N, for some unspecified positive N, and if these N are a lot bigger than one vote, then it seems reasonable to expect that the values of the variables can be changed enough to change B or Bc by one vote, , while still satisfying the original inequalities, if we wanted to. So if we get a solution of a system of equations that gives B & Bc the same vote total, then we'd be able to make enough change in the other variables to change B or Bc by one vote, so that Bc has one more vote than B. Here are my names for the relevant vote totals: A: people at A position who rank A in 1st place. Ac: people at A position who rank Bc in 1st place B: people in B position who rank B in 1st place Bc people in B position who rank Bc in 1st place. C: people in C position In this example outline, if there were just one candidate at the B position, he'd be the middle Condorcet candidate, and the smallest of the 3. That means that everyone at the A position + everyone at the B position > everyone at the C position. And everyone at the C position + everyone at the B position > everyone at the A position. So: A+Ac+B+Bc > C C+B+Bc > A And, in this example outline, the B position is the least populated of the 3 positions: A+Ac>B+Bc C>B+Bc And, so that, when B gets eliminated and transfers to Bc, Bc will have more votes than A: B+Bc+Ac > A Lastly, B & Bc are supposed to have equal vote totals: Bc+Ac = B These inequalities can be made into equations: A+Ac = B+Bc+N1 C = B+Bc+N2 A+Ac+B+Bc = C+N3 C+B+Bc = A+Ac+N4 B+Bc+Ac = A+N5 Bc+Ac = B Those are the equations that I'd solve to find the numbers for the IRV UUCC badexample. 6 linear equations in 10 unknowns. Since there are more equations than unknowns, and the equations don't appear to contradict eachother, then doesn't that mean that there are more than one solution to the system of equations? I realize that there are systems of equations that have no solution, like: A+B = 2 A+B = 3 But these 6 don't seem to obviously contradict eachother. Because there are lots of solutions, and because the N can be much larger than 1 vote, it seems that if we had a solution where Bc+Ac=B, we could change the other variables enough to make Bc+Ac=B+1, and still satisfy the original inequalities. When I can take the time, and when I have several big pieces of paper, I'll try the solution of those 6 equations. But in the meantime, it certainly seems that they have solutions, and that there are IRV UUCC failure examples. Again, I realize that there are mathematicians on this list who would deal with this problem in a quicker way, but they don't always have the inclination to do the same subjects that I'm inclined to do. Anyway, Blake asked me for an example where IRV fails UUCC. I intend to find one, but for now, haven't I shown that it looks as if there are such examples? Mike Ossipoff There are _________________________________________________________________________ Get Your Private, Free E-mail from MSN Hotmail at http://www.hotmail.com. Share information about yourself, create your own public profile at http://profiles.msn.com.
