And here I have a simpler fpA-fpC generalization (haven't tested it yet): When making an X vs Y comparison with Z as a witness, X's strength against Y equals: - X's fpA-fpC score if there's a {X,Y,Z} cycle - the number of X>Y votes if there's no cycle Final strength of X against Y equals the minimum possible strength, i.e. the strength with such a witness that the strength value is minimized. Create a comparison matrix with final strength values. Proceed with a defeat-dropping Condorcet method. - Filip Ejlak

On 5/21/23 10:42, Filip Ejlak wrote: > And here I have a simpler fpA-fpC generalization (haven't tested it yet): > > When making an X vs Y comparison with Z as a witness, X's strength > against Y equals: > - X's fpA-fpC score if there's a {X,Y,Z} cycle > - the number of X>Y votes if there's no cycle > > Final strength of X against Y equals the minimum possible strength, i.e. > the strength with such a witness that the strength value is minimized. > > Create a comparison matrix with final strength values. Proceed with a > defeat-dropping Condorcet method. I've been thinking about methods that try to generalize fpA-fpC by using max, but I think similar problems may exist with methods that use min. Suppose we clone a candidate (call him A) into a three-cycle, A1>A2>A3>A1. Then if we just say, we're going to make a method that's score(A) = max over B, C: fpA-fpC(restricted to A, B, C) then it's possible to arrange the clones' first preferences so that the clones grant A a very high maximum score. (The same trick works for the method where A's score is just the margin of the greatest landslide A is involved in.) So suppose that we define a matrix D so that (X>Y)_D = min over Z: fpA-fpC restricted to X,Y,Z. Then there needs to be some additional structure so that cloning X doesn't lead fpA-fpC restricted to X1, X2, X3 to be very unfavorable to X, so that X loses. (I think an analogous observation is why Minmax fails clone dependence.) Perhaps this additional structure would come from an observation of the type: suppose X1, X2, and X3 are clones; then it may be impossible for (X1>X2)_D, (X2>X3)_D, and (X3>X1)_D to *all* be small. Thus if X used to win, at least one of the clones will succeed through Ranked Pairs (or Schulze, etc) applied to D, and thus X will keep winning. Clearly it's impossible for all the clones' inter-clone victories to be small if they have the same witness. But would it be possible to carefully craft a ballot where say (X1>X2)_D has witness A, with low score for X1 (X2>X3)_D has witness B, with low score for X2 (X3>X1)_D has witness C, with low score for X3? Perhaps at least Ranked Pairs deals with this by disregarding such low scores until the higher scores have all been locked in... -km

Well, Ranked Pairs-like completion has this important quality that X1/X2/X3 can win even if (X1>X2)-score, (X2>X3)-score, and (X3>X1)-score are all very low - it doesn't matter as long as the clones' defeats against the other candidates are strong enough. And I think that cloning X can't actually decrease X1/X2/X3's strength against any candidate Y, because if X1 and X2 are in the same pairwise relation with Y (as they should be because of how we define clones), then they cannot form a cycle, so there is no new fpA-fpC score to potentially replace the current strength value. niedz., 21 maj 2023 o 18:34 Kristofer Munsterhjelm <km_elmet@t-online.de> napisał(a): > On 5/21/23 10:42, Filip Ejlak wrote: > > And here I have a simpler fpA-fpC generalization (haven't tested it yet): > > > > When making an X vs Y comparison with Z as a witness, X's strength > > against Y equals: > > - X's fpA-fpC score if there's a {X,Y,Z} cycle > > - the number of X>Y votes if there's no cycle > > > > Final strength of X against Y equals the minimum possible strength, i.e. > > the strength with such a witness that the strength value is minimized. > > > > Create a comparison matrix with final strength values. Proceed with a > > defeat-dropping Condorcet method. > > I've been thinking about methods that try to generalize fpA-fpC by using > max, but I think similar problems may exist with methods that use min. > > Suppose we clone a candidate (call him A) into a three-cycle, > A1>A2>A3>A1. Then if we just say, we're going to make a method that's > > score(A) = max over B, C: fpA-fpC(restricted to A, B, C) > > then it's possible to arrange the clones' first preferences so that the > clones grant A a very high maximum score. (The same trick works for the > method where A's score is just the margin of the greatest landslide A is > involved in.) > > So suppose that we define a matrix D so that (X>Y)_D = min over Z: > fpA-fpC restricted to X,Y,Z. > > Then there needs to be some additional structure so that cloning X > doesn't lead fpA-fpC restricted to X1, X2, X3 to be very unfavorable to > X, so that X loses. (I think an analogous observation is why Minmax > fails clone dependence.) > > Perhaps this additional structure would come from an observation of the > type: suppose X1, X2, and X3 are clones; then it may be impossible for > (X1>X2)_D, (X2>X3)_D, and (X3>X1)_D to *all* be small. Thus if X used to > win, at least one of the clones will succeed through Ranked Pairs (or > Schulze, etc) applied to D, and thus X will keep winning. > > Clearly it's impossible for all the clones' inter-clone victories to be > small if they have the same witness. But would it be possible to > carefully craft a ballot where say > (X1>X2)_D has witness A, with low score for X1 > (X2>X3)_D has witness B, with low score for X2 > (X3>X1)_D has witness C, with low score for X3? > > Perhaps at least Ranked Pairs deals with this by disregarding such low > scores until the higher scores have all been locked in... > > -km >