Method space of 3-bloc, strictly-ranked scenarios
I'll define a simple "ballot space," meaning a range of scenarios that can occur given voter, candidate, and ballot limitations. Then I'll talk about the "method space" for that ballot space, meaning the range of methods that are possible to define.
Let there be three factions, each preferring a different one of candidates labeled A, B, and C. The "A bloc" is the largest, with B being second-largest, and C being smallest. No bloc is a majority, but any two blocs would be a majority. Every voter within a bloc votes exactly the same (so that we see, total, only three unique ways of filling out the ballot). Every ballot is required to be a full, strict ranking. Additionally, each bloc will always rank their preferred candidate at the top.
Since each bloc can only fill out the ballot in two ways (e.g. A>B>C or A>C>B for the A bloc), this ballot space has only 2^3 = 8 scenarios:1. A>B B>A C>A ballots. A+B mutual majority, A is CW.2. A>B B>A C>B. A+B mutual majority, B is CW.3. A>B B>C C>A. A>B>C>A preference cycle.4. A>B B>C C>B. B+C mutual majority, B is CW.5. A>C B>A C>A. A+C mutual majority, A is CW.6. A>C B>A C>B. A>C>B>A preference cycle.7. A>C B>C C>A. A+C mutual majority, C is CW.8. A>C B>C C>B. B+C mutual majority, C is CW.
If we put a sequence of eight As, Bs, or Cs in a row to show who wins in each scenario, that sequence can define a method for this ballot space. I call this a DNA code.
I have 343-digit codes, covering a larger ballot space, for a few dozen election methods. But in this eight-scenario space these reduce to a quite small number of codes. For methods that use an approval concept, I use the assumption that only the bottom candidate is unapproved. Here then is a full list (so far) of the 8-digit codes resulting from methods that are at least mostly serious:
AAAAAAAA: FPP has its own code. Notice how it just elects A no matter what any lower preferences are.
ABABAACC: The MinMax group. This includes most MinMax methods (including also Schulze, MAM, etc.), DMC, Approval-Elimination Runoff, MAMPO, some scenarios of Kristofer's "linear" Condorcet method, a couple of obscure Woodall methods (MinGS and 3MaxGS), Condorcet//FPP, and probably a lot of other methods.
ABBBAACC: The Bucklin group. This includes Chris' IBIFA, Condorcet//Approval and ICA, MDDA, CdlA, MinMax(AWP), and probably many others. Note that, within this ballot space, these are a Condorcet method!
AAABAAAC: The "descending coalitions" group. It includes DAC, DSC, and a couple of my methods (SPST, and both pairwise and Bucklinesque versions of "Venzke Bucklin Variant" aka VBV).
ABABABAB: IRV group, including my "King of the Hill" (KH) method and IRV-without-elimination.
ABABABCC: Condorcet//IRV group, including Raynaud, Condorcet//KH, and most scenarios of Kristofer's "linear" method. Off the top of my head it should also include the "Woodall" and "Benham" methods.
ABABACAC: my QR, aka Chain Runoff.
ABABACCC: Condorcet//QR.
ABBBACCC: Forest's TACC method.
ABCBABCC: BPW (Eivind Stensholt's "Beats Plurality Winner" Condorcet method).
AAABAAAB: VDP (Venzke Disqualified Plurality, aka VFA).
That's really it... Other known codes are limited to experiments like "IRV where you eliminate the wrong candidate" or "just elect the winner of the strongest pairwise contest." (Let me know if you think there's an unusual method I've missed, and that can operate on these ballots.)
Here is an attempt at plotting the Hamming distances of these codes, so that more similar methods are plotted closer together:
. . . . . . . TACC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BPW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bucklin . . . . . . . . . . . . C//QR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ------- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ./. . . . . . . . . . . . . . . . . . . . . . . . . . . . . C//IRV. . . . . . --------- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ./. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .------ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . / . . . . . . . . . . . . . . . . . . . . . MinMax. . . . . . . . . . . . . . . . . ./. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . --Condorcet-Boundary--- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ./. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ------. . . . . . QR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ./. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .---------------------. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IRV .----. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DAC/DSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
That is 11 methods in a method space containing 3^8 = 6561 methods. Can we find any more methods?
In all of those 6561, only nine of them are Condorcet methods. I can describe all of them:
ABABAACC: MinMaxABABABCC: C//IRVABABACCC: C//QRABBBAACC: BucklinABBBABCC: (???) always elect B in a cycleABBBACCC: TACCABCBAACC: Condorcet//"IRV where you eliminate second-lowest tally"ABCBABCC: BPWABCBACCC: (???) always elect C in a cycle
Only two scenarios lack a CW, leaving not a lot of room to experiment.
How about non-Condorcet methods? Where is their diversity located? Let's ignore FPP and remove the cycle scenarios, so that we can just see where and why methods choose not to elect a CW:
AB_BA_CC: Condorcet methodsAA_BA_AB: VDPAB_BA_AB: IRVAB_BA_AC: QRAA_BA_AC: DAC/DSC
The disagreements come in scenarios 2, 7, and 8. If you check you'll see that they can be summarized in one statement: Non-Condorcet methods may prefer to elect the candidate of the larger bloc of a mutual majority, rather than the candidate of the majority who is the CW. (Note that the CW is the second preference of the faction excluded from the majority.)
In the scenarios where the CW is also from the larger bloc of the mutual majority, all methods agree (except for FPP in scenario 4). And no method outside of FPP will elect the Condorcet Loser (in this ballot space), as it would violate mutual majority. So it seems likely there are few possibilities for reasonable methods. If you don't think it's reasonable to elect C in the cycles, it might only be 2^5 = 32 possible methods.
That's all for now. I hope in the future to discuss scenarios with more ballot types.
Kevin Venzke
On 11/26/2016 07:24 AM, Kevin Venzke wrote:
I'll define a simple "ballot space," meaning a range of scenarios that
can occur given voter, candidate, and ballot limitations. Then I'll talk
about the "method space" for that ballot space, meaning the range of
methods that are possible to define.
Let there be three factions, each preferring a different one of
candidates labeled A, B, and C. The "A bloc" is the largest, with B
being second-largest, and C being smallest. No bloc is a majority, but
any two blocs would be a majority. Every voter within a bloc votes
exactly the same (so that we see, total, only three unique ways of
filling out the ballot). Every ballot is required to be a full, strict
ranking. Additionally, each bloc will always rank their preferred
candidate at the top.
Since each bloc can only fill out the ballot in two ways (e.g. A>B>C or
A>C>B for the A bloc), this ballot space has only 2^3 = 8 scenarios:
- A>B B>A C>A ballots. A+B mutual majority, A is CW.
- A>B B>A C>B. A+B mutual majority, B is CW.
- A>B B>C C>A. A>B>C>A preference cycle.
- A>B B>C C>B. B+C mutual majority, B is CW.
- A>C B>A C>A. A+C mutual majority, A is CW.
- A>C B>A C>B. A>C>B>A preference cycle.
- A>C B>C C>A. A+C mutual majority, C is CW.
- A>C B>C C>B. B+C mutual majority, C is CW.
If we put a sequence of eight As, Bs, or Cs in a row to show who wins in
each scenario, that sequence can define a method for this ballot space.
I call this a DNA code.
I have 343-digit codes, covering a larger ballot space, for a few dozen
election methods. But in this eight-scenario space these reduce to a
quite small number of codes. For methods that use an approval concept, I
use the assumption that only the bottom candidate is unapproved. Here
then is a full list (so far) of the 8-digit codes resulting from methods
that are at least mostly serious:
AAAAAAAA: FPP has its own code. Notice how it just elects A no matter
what any lower preferences are.
ABABAACC: The MinMax group. This includes most MinMax methods (including
also Schulze, MAM, etc.), DMC, Approval-Elimination Runoff, MAMPO, some
scenarios of Kristofer's "linear" Condorcet method, a couple of obscure
Woodall methods (MinGS and 3MaxGS), Condorcet//FPP, and probably a lot
of other methods.
ABBBAACC: The Bucklin group. This includes Chris' IBIFA,
Condorcet//Approval and ICA, MDDA, CdlA, MinMax(AWP), and probably many
others. Note that, within this ballot space, these are a Condorcet method!
AAABAAAC: The "descending coalitions" group. It includes DAC, DSC, and a
couple of my methods (SPST, and both pairwise and Bucklinesque versions
of "Venzke Bucklin Variant" aka VBV).
ABABABAB: IRV group, including my "King of the Hill" (KH) method and
IRV-without-elimination.
ABABABCC: Condorcet//IRV group, including Raynaud, Condorcet//KH, and
most scenarios of Kristofer's "linear" method. Off the top of my head it
should also include the "Woodall" and "Benham" methods.
So the linear methods are somewhat more Minimax-y C//IRVs. That sort of
makes sense, in particular when even Condorcet/FPP is considered
Minmaxy. Moving closer to minmax makes the methods more monotone, but
they're still similar to C//IRV in being strategy resistant Condorcet
methods.
ABABACAC: my QR, aka Chain Runoff.
ABABACCC: Condorcet//QR.
ABBBACCC: Forest's TACC method.
ABCBABCC: BPW (Eivind Stensholt's "Beats Plurality Winner" Condorcet
method).
AAABAAAB: VDP (Venzke Disqualified Plurality, aka VFA).
Stensholt also has another method (which he has called SV):
Let F(X) be the fraction of voters who gave X a first preference vote
(e.g. F(A) = 0.5 is half the voters put A first).
Let a = 1/2 - F(A), b = 1/2 - F(B), g = 1/2 - F(C).
If there's no cycle, the CW wins, otherwise WLOG suppose it's an ABCA
cycle. Then A's penalty is b/g, B's is g/a, and C's is a/b. Minimum
penalty wins.
I guess this would produce the same results as BPW, but you could check.
In my post about linear methods, I also mentioned this nonlinear method:
Again WLOG let the cycle be ABCA. Then the scores are
A's score: A>B * min(C>A, A>B)/fpC
B's score: B>C * min(A>B, B>C)/fpA
C's score: C>A * min(B>C, C>A)/fpB
and greatest score wins. fpX is the number of first preferences for X.
I kind of suspect the min() part is related to beatpaths. Isn't min(C>A,
A>B) the strength of the beatpath from C to B, i.e. from who beat A to
who A beats? Of course, the optimizer has no idea what concept it's finding.
You could also try BTR-IRV, which is IRV-like but sometimes not. It
consists of, in every round, beat whoever pairwise loses to the other of
the last two candidates sorted by Plurality count. It would be more like
the "always eliminate the second to last finisher" than plain IRV is.
That's really it... Other known codes are limited to experiments like
"IRV where you eliminate the wrong candidate" or "just elect the winner
of the strongest pairwise contest." (Let me know if you think there's an
unusual method I've missed, and that can operate on these ballots.)
Here is an attempt at plotting the Hamming distances of these codes, so
that more similar methods are plotted closer together:
[snip]
I was thinking about something similar the other day. You could do this:
Let each method's feature vector be something like:
--first letter--, --second--, --third-- ...
[is it A?, is B? is C? A?, B?, C? A?, B?, C? ]
1 if true, 0 if false. So, for instance, a method that starts in ABC
would have a feature vector starting in
[1, 0, 0, 0, 1, 0, 0, 0, 1, ....]
If you take the Euclidean distance between two different methods'
feature vectors, you should get twice the Hamming distance. (If method X
and Y agree on the first letter, that contributes 0 to the distance, but
if they disagree, that contributes 1 to the distance)
Now make a matrix of all the methods' feature vectors, run SVD on it,
and pick the two first singular vectors (corresponding to the greatest
singular values). Multiply them by their respective singular values, let
the resulting vectors give the x and y coordinates, and plot.
This can in theory be done on any dimension feature vector, so you could
collapse the 343-digit codes to two dimensions that way as well, or even
consider each particular election that way (e.g. if you test on 100k
elections, the vector would be 300k long). You could model methods that
fit partly in the camp as well, if you'd like, e.g.
[0.5, 0.5, 0, ...]
is a method that, in the first scenario set, elects A half the time, B
the other half, and never elects C.
Another possibility as far as plotting goes is to use a synthetic
coordinate algorithm to determine points on a plane that give distances
most like the distances you have already calculated. Such distances
could be sums of Kendall-tau distances between the different methods'
social orderings. I did that once, and you commented that the plot
didn't seem to make much sense beyond the rough scale: all Condorcet
methods gathered together in the middle, and then Plurality was off to
one side and Random Ballot off to another, I think.
If you have an n*n method-method distance matrix (where d[a][b] gives
some distance between methods, e.g. by the Kendall tau distance), you
could also use a hierarchical clustering algorithm to get something
analogous to a cladogram or phylogenetic tree - a dendrogram. I think I
did that once, too, and though I didn't post my results, it seemed to
make more sense than the synthetic coordinate diagram. You might end up
with something like...
+---Carey
+--IRV--+
| +----Plurality
|
all----+
|
| +---Borda
+--Sinkhorn--+
+---Schulze
That is 11 methods in a method space containing 3^8 = 6561 methods. Can
we find any more methods?
In all of those 6561, only nine of them are Condorcet methods. I can
describe all of them:
ABABAACC: MinMax
ABABABCC: C//IRV
ABABACCC: C//QR
ABBBAACC: Bucklin
ABBBABCC: (???) always elect B in a cycle
Wouldn't that one fail neutrality? Suppose the cycle is ABCA. You could
always relabel the candidates by following the cycle.
ABBBACCC: TACC
ABCBAACC: Condorcet//"IRV where you eliminate second-lowest tally"
ABCBABCC: BPW
ABCBACCC: (???) always elect C in a cycle
Only two scenarios lack a CW, leaving not a lot of room to experiment.
Hi Kristofer,
De : Kristofer Munsterhjelm km_elmet@t-online.deÀ : Kevin Venzke stepjak@yahoo.fr; EM list election-methods@electorama.com Envoyé le : Samedi 26 novembre 2016 7h25Objet : Re: [EM] Method space of 3-bloc, strictly-ranked scenarios
ABABAACC: The MinMax group. This includes most MinMax methods (including>> also Schulze, MAM, etc.), DMC, Approval-Elimination Runoff, MAMPO, some>> scenarios of Kristofer's "linear" Condorcet method, a couple of obscure>> Woodall methods (MinGS and 3MaxGS), Condorcet//FPP, and probably a lot>> of other methods.>>> ABABABCC: Condorcet//IRV group, including Raynaud, Condorcet//KH, and>> most scenarios of Kristofer's "linear" method. Off the top of my head it>> should also include the "Woodall" and "Benham" methods.>>So the linear methods are somewhat more Minimax-y C//IRVs. That sort of>makes sense, in particular when even Condorcet/FPP is considered>Minmaxy. Moving closer to minmax makes the methods more monotone, but>they're still similar to C//IRV in being strategy resistant Condorcet>methods.
Sounds right although, within this framework, I can't even show that IRV is not monotone. I wonder if there could be a different cause of undesirablebehavior. Of the properties I have (and which make sense in the 8-scenario case), C//IRV lacks two properties compared to the MinMax group: it violatesa sort of mono-remove-bottom, and a property I can't quite generalize into words which says that the last choice of the largest faction shouldn't beelected.
(Incidentally the MinMax and C//IRV groups are the only Condorcet ones that satisfy a sort of participation property, in that a faction's lastchoice should not be elected if the withdrawal of that faction would electtheir compromise choice.)
ABCBABCC: BPW (Eivind Stensholt's "Beats Plurality Winner" Condorcet>> method).>>Stensholt also has another method (which he has called SV):>>Let F(X) be the fraction of voters who gave X a first preference vote>(e.g. F(A) = 0.5 is half the voters put A first).>Let a = 1/2 - F(A), b = 1/2 - F(B), g = 1/2 - F(C).>If there's no cycle, the CW wins, otherwise WLOG suppose it's an ABCA>cycle. Then A's penalty is b/g, B's is g/a, and C's is a/b. Minimum>penalty wins.>>I guess this would produce the same results as BPW, but you could check.>>In my post about linear methods, I also mentioned this nonlinear method:>>Again WLOG let the cycle be ABCA. Then the scores are>>A's score: A>B * min(C>A, A>B)/fpC>B's score: B>C * min(A>B, B>C)/fpA>C's score: C>A * min(B>C, C>A)/fpB>>and greatest score wins. fpX is the number of first preferences for X.>>I kind of suspect the min() part is related to beatpaths. Isn't min(C>A,>A>B) the strength of the beatpath from C to B, i.e. from who beat A to>who A beats? Of course, the optimizer has no idea what concept it's finding.
Ok. I didn't know about Stensholt's other method, and I should check it out. And I did bring up our old emails to review the nonlinear method. The troubleis that methods that are sensitive to the "absolute" size differences betweenthe factions, tend to produce more than one DNA code. That's what was goingon when you see that I put the linear method under two groups: It has two codes. Most methods (even with 343 digits) only have one. But e.g. MinMax(margins) has seven, and FPP(ER-Fractional) has twelve...
You could also try BTR-IRV, which is IRV-like but sometimes not. It>consists of, in every round, beat whoever pairwise loses to the other of>the last two candidates sorted by Plurality count. It would be more like>the "always eliminate the second to last finisher" than plain IRV is.
I forgot about BTR-IRV. I'll definitely add that one (it's unusual, and easyto code). At a glance I believe that one falls under the MinMax group, though.
I should say once we add in other ballot types the plot changes quite a bit.E.g. on the plot I just posted, DAC/DSC are a single point. But when we add inbullet votes they tend to be on opposite ends of one axis.
>> Here is an attempt at plotting the Hamming distances of these codes, so>> that more similar methods are plotted closer together:>>[snip]>>I was thinking about something similar the other day. You could do this:>>Let each method's feature vector be something like:>--first letter--, --second--, --third-- ...>[is it A?, is B? is C? A?, B?, C? A?, B?, C? ]>>1 if true, 0 if false. So, for instance, a method that starts in ABC>would have a feature vector starting in>>[1, 0, 0, 0, 1, 0, 0, 0, 1, ....]>>If you take the Euclidean distance between two different methods'>feature vectors, you should get twice the Hamming distance. (If method X>and Y agree on the first letter, that contributes 0 to the distance, but>if they disagree, that contributes 1 to the distance)>>Now make a matrix of all the methods' feature vectors, run SVD on it,>and pick the two first singular vectors (corresponding to the greatest>singular values). Multiply them by their respective singular values, let>the resulting vectors give the x and y coordinates, and plot.
Uh oh. I had to google SVD.
This can in theory be done on any dimension feature vector, so you could>collapse the 343-digit codes to two dimensions that way as well, or even>consider each particular election that way (e.g. if you test on 100k>elections, the vector would be 300k long). You could model methods that>fit partly in the camp as well, if you'd like, e.g.
I can say part of the motivation for the 343 is to not have to do simulations, and be able to exhaustively investigate a method. (Or atleast its representation within the framework...)
[0.5, 0.5, 0, ...]>>is a method that, in the first scenario set, elects A half the time, B>the other half, and never elects C.>>Another possibility as far as plotting goes is to use a synthetic>coordinate algorithm to determine points on a plane that give distances>most like the distances you have already calculated. Such distances>could be sums of Kendall-tau distances between the different methods'>social orderings. I did that once, and you commented that the plot>didn't seem to make much sense beyond the rough scale: all Condorcet>methods gathered together in the middle, and then Plurality was off to>one side and Random Ballot off to another, I think.
Hmm, I think I remember that.
The way the plotting works now is that it finds the Hamming distances(using however many digits of DNA) and then plots methods one-by-one in2D space, trying to reduce the sum of squares of difference between theHamming distance and the apparent Euclidean distance. The whole thing isdone a number of times and with the methods in randomized order, keepingthe best result.
If you have an n*n method-method distance matrix (where d[a][b] gives>some distance between methods, e.g. by the Kendall tau distance), you>could also use a hierarchical clustering algorithm to get something>analogous to a cladogram or phylogenetic tree - a dendrogram. I think I>did that once, too, and though I didn't post my results, it seemed to>make more sense than the synthetic coordinate diagram. You might end up>with something like...>> +---Carey> +--IRV--+> | +----Plurality> |>all----+> |> | +---Borda> +--Sinkhorn--+> +---Schulze>
That's a neat idea. I'll need to read up on how this algorithm can workthough. Sometimes a strange method gets plotted out in space and I don'tknow how arbitrary its placement is wrt the methods it appears to be closestto. A tree would probably be helpful there.
In all of those 6561, only nine of them are Condorcet methods. I can>> describe all of them:>> >> ABABAACC: MinMax>> ABABABCC: C//IRV>> ABABACCC: C//QR>> ABBBAACC: Bucklin>> ABBBABCC: (???) always elect B in a cycle>Wouldn't that one fail neutrality? Suppose the cycle is ABCA. You could>always relabel the candidates by following the cycle.
Not neutrality, because the candidates are labeled by faction size. The problem is that the A faction may wish they had received fewer votes sothat their candidate could be the one labeled "B."
I actually made an unexpected amount of progress today evaluating theproperties of all 6561 methods. I think probably only a handful more areworth mentioning...
Kevin
On 11/27/2016 12:00 AM, Kevin Venzke wrote:
I can say part of the motivation for the 343 is to not have to do
simulations, and be able to exhaustively investigate a method. (Or at
least its representation within the framework...)
You could do this in reverse, too. The problem would be something like:
given 100000 (or however many) different ballot sets and a bunch of
methods' results on these ballot sets, find eight of these so that the
methods, evaluated on these ten ballot sets, give as few duplicate
signatures as possible.
More formally, suppose the rows of a matrix M corresponds to methods and
the columns correspond to ballot sets, and M[i][j] is 0 if A won, 1 if B
won, and 2 if C won. Then pick eight (or 343 or ...) columns so that
there are as few duplicate row vectors as possible when you remove all
columns but those eight from the matrix.
This is properly speaking a set cover problem (and thus NP-hard), but it
can be approximated in many ways. The simplest greedy algorithm would
be: pick the column that breaks as many ties as possible, then pick the
next column that breaks as many ties of those that remain, etc.