Poking about Heaviside methods, again
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
instead of max gets 70%. -
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).
- Let A and B be two candidates. Then if, for every pair {C, D} of
-
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
On 05.01.2023 18:23, Kristofer Munsterhjelm wrote:
f(A) = sum over {A, B}:
H(fpA - fpB) *
product over C not in {A, B}:
H(fpB - fpC)
* A>B.
My bad, that should be:
f(A) = sum over {A,B}
product over C not in {A, B}:
H(fpA - fpC) * H(fpB - fpC)
* A>B.
"A has more first preferences than C, AND
B has more first preferences than C,
for all C."
-km
Hi Kristofer,
A few thoughts. I'll assume everything is just "max" because I just don't see how using the
sum could lead to a viable method no matter how good it looks in a simulation.
Le jeudi 5 janvier 2023 à 11:24:29 UTC−6, Kristofer Munsterhjelm km_elmet@t-online.de a écrit :
The results show a big nope: every such method (with the values of x
integer between 2 and -2 inclusive) seem to be 100% manipulable in the
limit. However, it did come up with this, which seemed to converge less
quickly, and might be an interesting idea on its own:
f(a) = sum (or max) over B:
A>B * H(fpA + fpB - fpC - fpD)
This is interesting. I like how Majority Favorite is ensured. The winning gross score (the
votes-for) can come either from A:B or A:C or, depending on the sizes, A:D or B:C. There is
a bias towards having more FPs, but also a step in the approval-ish direction of using gross
score.
- 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.
Do you know how many of those methods satisfy Majority Favorite? Maybe it's not that
surprising.
If we compare to methods that people actually propose, it seems true that non-Condorcet
methods don't tend to have huge burial issues. Main exceptions would be Borda, MMPO, and
iterated DSV methods that are nearly Condorcet-efficient. Methods like DSC and Chain Runoff
do have burial incentive, but their compromise incentive is probably a much bigger concern.
- 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...".
That's interesting but I have to imagine the latter approach breaks at some point (i.e.
ceases to provide monotonicity), perhaps with more than four candidates.
- 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.
So this seems the same as the December method, except you allow each candidate C to serve
as D as well. (This should make it a bit harder for some A:B to qualify.) And also you're
doing a Condorcet check here.
- 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).
The monotonicity of this method is an interesting question.
When you eliminate a candidate before checking the FP count, you arguably are weeding out
noise and trying to get to a "true" FP count after, essentially, accounting for compromise
incentive. It seems again like a somewhat arbitrary balance between FP count and the
approval-ish gross scores.
It's a little inexplicable why to compare with D and D\C instead of double D\C, but that
may just be the balance that happens to be optimal in your setting. I'm not sure you'll
really be able to explain that.
If you had a five-candidate framework I think all of your results would end up very
different. For example an expression like H(fpA + fpB - fpC - fpD) doesn't ensure that the
FP-winner (and possible majority favorite) is even included.
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.
Well, a FP emphasis is bad for compromise but can reduce burial, whereas a gross score
emphasis might reduce both but is bad for truncation incentive. I'm not sure if you have a
truncation concept though. I guess that your vulnerability metric leads you to identify
methods with some balance.
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:
If like me you're used to trying to define methods using mainly a graph of defeats, and
caring about LNH criteria, it's apparent that changing the state of the X:Y contest must
not affect (in some way) the win status of unrelated candidate Z. That's a severe
restriction. But with contingent vote you select the decisive contest without ever looking
at the graph, and no candidate outside the contest has any win odds at all. So, the issue
isn't with pairwise comparisons inherently.
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.
This is a LNHelp method with huge truncation incentive. It must depend how you look at it.
C has incentive to decrease the A>B tally (if they had been contributing to it) but not to
increase the B>A tally.
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?
It's the same surely. When you add an A preference you're just giving some positive number
of points to A. You would never wish to give points to a candidate that you don't want to
win.
Myself, I have returned to experimenting with a 4-candidate 4-bloc DNA scheme. It has 512
digits for 512 scenarios. An interesting thing I suppose is that the rate of failure of
some criterion could be found precisely. Maybe it's possible to hunt for a method that is
"as good as possible" given some other constraints.
My explicability problem is a little different from yours as I can't even say what my
generated methods are. But I've been making some progress in trying to define criteria
related to explicability, so that a generated method must be definable when limited to some
terms. If two scenarios can't be differentiated then they have to give the same winner.
Kevin
votingmethods.net
On Thu, Jan 5, 2023, 9:24 AM Kristofer Munsterhjelm km_elmet@t-online.de
wrote:
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
instead of max gets 70%. -
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.
This weird fact suggests to me the following Banks efficient method:
Start a pairwise defeat chain A>B with the greatest strength defeat pair
A>B.
Then of all the X that defeat both A and B, let X1 be the one with the
greatest defeat against A, and then augment the chain with X1 to get X1>A>B.
Then ...
While some X defeats every member of the chain, augment the chain with the
X that has the strongest defeat against the newest member of the chain.
EndWhile
Elect the head of the resulting maximal chain.
Note that if there are only three candidates and those three are in a cycle
ABCA, then this method will elect
A=argmax f(X).
-Forest
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
Election-Methods mailing list - see https://electorama.com/em for list
info
On 1/7/23 02:03, Kevin Venzke wrote:
Le jeudi 5 janvier 2023 à 11:24:29 UTC−6, Kristofer Munsterhjelm km_elmet@t-online.de a écrit :
The results show a big nope: every such method (with the values of x
integer between 2 and -2 inclusive) seem to be 100% manipulable in the
limit. However, it did come up with this, which seemed to converge less
quickly, and might be an interesting idea on its own:
f(a) = sum (or max) over B:
A>B * H(fpA + fpB - fpC - fpD)
This is interesting. I like how Majority Favorite is ensured. The winning gross score (the
votes-for) can come either from A:B or A:C or, depending on the sizes, A:D or B:C. There is
a bias towards having more FPs, but also a step in the approval-ish direction of using gross
score.
A's score can come from A>B, A>C, or A>D, depending. But if no candidate
has a majority and B has a very low first preference count, B has no chance.
- 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.
Do you know how many of those methods satisfy Majority Favorite? Maybe it's not that
surprising.
If we compare to methods that people actually propose, it seems true that non-Condorcet
methods don't tend to have huge burial issues. Main exceptions would be Borda, MMPO, and
iterated DSV methods that are nearly Condorcet-efficient. Methods like DSC and Chain Runoff
do have burial incentive, but their compromise incentive is probably a much bigger concern.
Some LNHelp methods have burial problems, though it's usually easier to
just truncate instead of bury.
My general direction of thought was: it would be useful to identify just
what contributes to low burial incentive because all the strategy
resistant methods I know of have mainly low burial incentive.
Then amending them with Condorcet, e.g. going from IRV to Benham, seems
to turn some of the compromising incentive into burial instead. But it
doesn't greatly increase the strategic susceptibility as a whole, and
that seems like a good trade for avoiding center squeeze and the likes.
So if that's true, then we should find some method that has very little
burial incentive and that is strategy resistant (e.g. not Plurality,
whose compromise incentive goes to 100% very quickly).
I doubt I have the mathematical chops to do this, though. For some
methods, simple differentiation under burial with a sort of KKT strategy
will work, e.g. minmax:
f(A) = min over X != A: A>X
and
f(B) = min over X != B: B>X
then f(B) - f(A) is increased by changing a ballot from BAC to BCA if
A's minimum is A>C.
But doing so is way too nitty-gritty for larger methods and I haven't
had much luck abstracting it. So I was hoping there was something
about combinations of H() and A>B. But I don't see it at the moment.
- 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...".
That's interesting but I have to imagine the latter approach breaks at some point (i.e.
ceases to provide monotonicity), perhaps with more than four candidates.
As I mentioned in a private mail, it should retain monotonicity because
A and B share the pool that's compared against the quota. A will get A>B
from any B where A and B's first preferences combined exceed the quota,
so raising A to top on a B ballot can't harm A.
If you had a five-candidate framework I think all of your results would end up very
different. For example an expression like H(fpA + fpB - fpC - fpD) doesn't ensure that the
FP-winner (and possible majority favorite) is even included.
Strangely, that's what I get for the optimum: it's still fpA + fpB - 2
fpC. But I did a check with IRV and it seems that this is just an
artifact of the H(...) * A>B methods hitting a wall.
With five candidates and 97 voters, impartial culture (no truncation),
the results are:
Smith,(fpA + fpB - 2 fpC) is manipulable 92% of the time.
Smith,Contingent vote is manipulable 86% of the time.
Smith,IRV is manipulable 46% of the time.
(For four candidates, same number of voters:
Condorcet//(fpA + fpB - 2 fpC) is manipulable 66% of the time
Condorcet//Contingent is 60% of the time
Condorcet//fpA\C + fpB\C - fpD\C -fpD is 56% of the time[1]
Smith,IRV is 33% of the time.)
So my intuition is that as the number of candidates increases, first
preferences become increasingly useless. IRV gets around this by
reducing down to a three-candidate election using steps that themselves
pass both LNHs.
But if it's necessarily true that you can only have burial resistance
with elimination, then a direct consequence is that we can't have burial
resistance and summability. The best we can do is to somehow string
together k-candidate elections to get summability O(n^k), where the
k-candidate election is itself strategically resistant for k candidates.
E.g. something like
f(A) = min over other candidates B,C: Friendly(A, B, C)
where Friendly(A, B, C) is the fpA-fpC method (Friendly Voting) where
everybody but A, B, and C have been eliminated.
Then the pragmatic answer would be "we aren't going to see 10-candidate
Smith sets, so strategy resistance for k<10 candidates is enough,
summability is worth more".
However, I haven't proven any of this. The simulation really could use
the other strategy-resistant summable method I found, Plurality Benham,
to see if it does any better than Smith,Contingent. But I haven't
programmed Pb in my simulator yet.
For that matter, I haven't even proven such a summability restriction.
There might still be some magical summable method; all I've done is
given evidence that it's not of the H(...) * A>B form.
(IRV itself would expand, for four candidates, to something like...
f(A) =
sum over all B, C, D, not equal to A:
product over X != D: H(fpX - fpD) <-- D is the Plurality loser
* H(fpA - fpC\D) * H(fpB - fpC\D) <-- then C is eliminated
* A>B)
Myself, I have returned to experimenting with a 4-candidate 4-bloc DNA scheme. It has 512
digits for 512 scenarios. An interesting thing I suppose is that the rate of failure of
some criterion could be found precisely. Maybe it's possible to hunt for a method that is
"as good as possible" given some other constraints.
My explicability problem is a little different from yours as I can't even say what my
generated methods are. But I've been making some progress in trying to define criteria
related to explicability, so that a generated method must be definable when limited to some
terms. If two scenarios can't be differentiated then they have to give the same winner.
I got to thinking that perhaps the next step of method brute-forcing
would be to go beyond vector enumeration (try every x for H(x) * A>B) to
abstract syntax trees. Then in theory you could iterate through them
until you find a good fit to your method, or until you find something
with high strategy resistance.
But in practice, it's pretty hard to construct a bijection between trees
and integers (with the nodes corresponding to functions that each take a
fixed number of parameters). Even harder to prove desirable things like
monotonicity in generality. And probably it would take forever to try to
find a good fit to a given DNA.
-km
[1] Note that there may be other more resistant methods of this form; I
haven't done a search through all combinations of fpX\Y terms.
Hi Kristofer,
If we compare to methods that people actually propose, it seems true that non-Condorcet
methods don't tend to have huge burial issues. Main exceptions would be Borda, MMPO, and
iterated DSV methods that are nearly Condorcet-efficient. Methods like DSC and Chain Runoff
do have burial incentive, but their compromise incentive is probably a much bigger concern.
Some LNHelp methods have burial problems, though it's usually easier to
just truncate instead of bury.
Do you say this because you conceive of burial as simultaneously adding one candidate while
lowering another, or because you include altering relative order within the ranking and not
just at the bottom?
I can imagine saying that Bucklin for example has burial incentive, but it feels a bit
off.
My general direction of thought was: it would be useful to identify just
what contributes to low burial incentive because all the strategy
resistant methods I know of have mainly low burial incentive.
I guess from my perspective I don't have a good way to merge different forms of strategic
incentive into a single resistance metric. Even if I could do exhaustive searches it seems
like different strategies are qualitatively different, for example in how intuitive they
are for a voter to apply.
Then amending them with Condorcet, e.g. going from IRV to Benham, seems
to turn some of the compromising incentive into burial instead. But it
doesn't greatly increase the strategic susceptibility as a whole, and
that seems like a good trade for avoiding center squeeze and the likes.
So if that's true, then we should find some method that has very little
burial incentive and that is strategy resistant (e.g. not Plurality,
whose compromise incentive goes to 100% very quickly).
Sure, I understand what you want to do.
- 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...".
That's interesting but I have to imagine the latter approach breaks at some point (i.e.
ceases to provide monotonicity), perhaps with more than four candidates.
As I mentioned in a private mail, it should retain monotonicity because
A and B share the pool that's compared against the quota. A will get A>B
from any B where A and B's first preferences combined exceed the quota,
so raising A to top on a B ballot can't harm A.
After playing with the H(fpA + fpB - fpC - fpD) method (max or sum, no Condorcet check) I
find a problem:
Suppose that the first preference count order goes ABCD, and the proportions are such that
the majorities are AB, AC, and AD. Say that B wins. If B obtains new first preferences at
the expense of C, the proportions could change so that fpB + fpD is now a majority. This
could allow D to be elected on the basis of a strong D>B gross score which wasn't
admissible before.
But if it's necessarily true that you can only have burial resistance
with elimination, then a direct consequence is that we can't have burial
resistance and summability.
I have recently found a couple of exceptions to the first part although not the second.
That's to say LNHarm+LNHelp methods that can't be explained in terms of eliminations. I'm
not sure it will be any use to you, but it's still interesting. I'm sort of checking my
work and looking for any "nearby" findings before posting.
Kevin
votingmethods.net
On 1/14/23 01:18, Kevin Venzke wrote:
Hi Kristofer,
If we compare to methods that people actually propose, it seems true that non-Condorcet
methods don't tend to have huge burial issues. Main exceptions would be Borda, MMPO, and
iterated DSV methods that are nearly Condorcet-efficient. Methods like DSC and Chain Runoff
do have burial incentive, but their compromise incentive is probably a much bigger concern.
Some LNHelp methods have burial problems, though it's usually easier to
just truncate instead of bury.
Do you say this because you conceive of burial as simultaneously adding one candidate while
lowering another, or because you include altering relative order within the ranking and not
just at the bottom?
Perhaps an example is better, for Bucklin:
Suppose that every ballot is complete (i.e. no truncation) and A wins
after first and second preferences are counted, being ahead of C by just
one point. And suppose that if it went to the third round (counting all
three ranks), then C would have won.
Then some C>A>... voters could change their vote to C>...>A and C would
win. That's just plain burial of A.
For Bucklin it's possible to do the same thing by just bullet-voting for
C. From the perspective of my simulator, that doesn't mean that this
election isn't susceptible to burial, but rather that it's also
susceptible to truncation. When it's checking for burial in isolation,
it doesn't know if there's also a possibility to truncate.
My general direction of thought was: it would be useful to identify just
what contributes to low burial incentive because all the strategy
resistant methods I know of have mainly low burial incentive.
I guess from my perspective I don't have a good way to merge different forms of strategic
incentive into a single resistance metric. Even if I could do exhaustive searches it seems
like different strategies are qualitatively different, for example in how intuitive they
are for a voter to apply.
Yes, I'm just considering all strategies to count equally much at the
moment. This is probably overly generous and makes methods look worse
than they are, but I don't know how to add different weighting in a way
that won't introduce too much subjectivity. E.g. whether a party can
coordinate all of its voters to rank its candidate top and the honest
winner bottom, whether all the voters would go along with this, and
whether the party gets the honest winner right, and so on.
As I mentioned in a private mail, it should retain monotonicity because
A and B share the pool that's compared against the quota. A will get A>B
from any B where A and B's first preferences combined exceed the quota,
so raising A to top on a B ballot can't harm A.
After playing with the H(fpA + fpB - fpC - fpD) method (max or sum, no Condorcet check) I
find a problem:
Suppose that the first preference count order goes ABCD, and the proportions are such that
the majorities are AB, AC, and AD. Say that B wins. If B obtains new first preferences at
the expense of C, the proportions could change so that fpB + fpD is now a majority. This
could allow D to be elected on the basis of a strong D>B gross score which wasn't
admissible before.
That's a good point. As a shortcut, I was looking at how f(A) grows when
A is raised. What I really should be looking at is f(A) - f(B) for any
other B. Since I didn't, I didn't see that monotonicity violation. (It
also suggests that there are monotone methods that I don't detect, too,
because while dA/d(raise) < 0, it might be that dA/d(raise) -
dB/d(raise) >= 0 for all other B -- to misuse differential notation a bit.)
But with the way things are set up, evaluating the growth of f(A) - f(B)
would be very hard. I probably have to implement a monotonicity checker
and either prescreen methods by checking them against it or combine the
checker with the bandit system so that the moment a monotonicity
violation is detected, it disqualifies the method in question.
But if it's necessarily true that you can only have burial resistance
with elimination, then a direct consequence is that we can't have burial
resistance and summability.
I have recently found a couple of exceptions to the first part although not the second.
That's to say LNHarm+LNHelp methods that can't be explained in terms of eliminations. I'm
not sure it will be any use to you, but it's still interesting. I'm sort of checking my
work and looking for any "nearby" findings before posting.
Now that I think about it... of course. I must've been not thinking far
enough ahead. Plain old Plurality qualifies: it has zero burial
resistance as it passes both LNHarm and LNHelp, and it's summable. For
that matter, the f(A) = max X: A>X method has low burial resistance and
is also summable.
The difficult part is that neither has low strategy resistance in
general. I guess there has to be some kind of "compromise resistance"
criterion that's weaker than the strong FBC, and that IRV passes but the
other methods don't.
IRV passes mutual majority, which would at first seem like a good
candidate. But Woodall showed that we can't have all of mutual majority,
LNHelp, LNHarm, and monotonicity.
-km