# [EM] Poking about Heaviside methods, again

Kristofer Munsterhjelm km_elmet at t-online.de
Thu Jan 5 09:23:59 PST 2023

Happy new year!

Now that it's a new year, I've done a bit more exploring around the
low-manipulability generalizations of the contingent vote. My summary
would be this:

- I found one bug, but not all, that caused the Python vote simulator to
give different results to quadelect. As I haven't found all, the stuff
below still hasn't been fully verified.

- There's a strange pattern where every method (if not augmented by
Condorcet) seems to have almost all its strategy vulnerability as
compromising. More later.

- An interesting pattern seems to be: nonmonotone CV generalizations
work by "If A and B are at the top by first preferences, then credit A
with A>B and B with B>A". Monotone ones are about absolute comparisons
instead: "if A and B combined are above some quota, then...".

- For monotone "CW if there is one, otherwise method X", the best
impartial culture generalization for four candidates goes like this:
Let A and B be two candidates. Then if for every other candidate C, fpA
+ fpB - 2 fpC > 0, let A's score in the (A,B) comparison be A>B and B's
score be B>A. If they're all < 0, it's zero for both.
Let every candidate A's score be the maximum of any comparison involving A.
Highest score wins unless there's a CW. If there is, the CW wins.

- According to quadelect, this achieves roughly 66% strategy
vulnerability on four candidates and 97 voters. The variant with sum

- The best nonmonotone method of this form is Condorcet//contingent
vote. Its susceptibility is 60% with max and 57% with sum.

- Including terms involving first preferences after elimination can
improve the results quite a bit, but the resulting methods are
completely incomprehensible. Here's an example for four candidates, that
*should* be monotone although I haven't checked:
- Let A and B be two candidates. Then if, for every pair {C, D} of
other candidates, if fpA\C + fpB\C - fpD\C -fpD > 0, A's score is (A>B)
and B's is (B>A). Otherwise if all are <0, it's zero for both. fpX\Y
here is my terminology for "X's first preference count once Y has been
eliminated".
- The rest is as above (including the use of the max operator).

- That method has 53% strategic susceptibility under the given conditions.

----

So lots of strange things. I'm kinda understanding the limitations of
not having a theory; the simulator can come up with some generalization
that seems to work, but I have no idea why it works. Leaving elections
to a black box isn't really a good idea, either :-) What's necessary
here, I think, is some kind of explanation for just why these are
resistant to strategy and others are not, and I don't know what that
would look like.

The pattern seems to be that if A's score is a transformation of some
linear combination of first preferences, and the method is monotone,
then clearly there won't be any burial incentive because lowering A's
rank can't harm A as nobody who prefers to B as A has A as their first
preference anyway.

And obviously sums of these and maxima of these retain the property.

The surprising part is that sometimes, you can include pairwise counts
and it's still all compromise all the time. E.g. the contingent vote:

f(A) = sum over {A, B}:
H(fpA - fpB) *
product over C not in {A, B}:
H(fpB - fpC)
* A>B.

where f(A) is A's score and H is the step function.

Even the weird method
f(A) = max over B: A>B

seems to have a very low (though nonzero) rate of burial. Why? You'd
imagine that if A is tying C, and A>B is A's max pairwise victory over
someone else, then the buriers can benefit by burying A under B; and
that that would happen pretty frequently.

On the other hand,

f(A) = sum over B: H(A>B - B>A)

is just Copeland, which has *plenty* of burial incentive. Yet it looks
superficially like the other methods: monotone transformations of linear
combinations of pairwise victories. How about if we only allow a single
victory inside the monotone transformation, something like

f(A) = sum over B: (A>B)^x

with x>1? Is that also only compromise?

-km