# [EM] Witnessed Pairs (simple fpA-fpC)

Kristofer Munsterhjelm km_elmet at t-online.de
Sun May 21 09:33:38 PDT 2023

```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
```