Strategy-resistant monotone methods
(I've tried to write this post many times, but each time, I've bogged
myself down in details. So let's try again...)
Lately (the last few months) I've been running a lot of simulations on a
class of methods to try to find a method that is Condorcet, monotone,
and around the level of resistance to strategy as Smith//IRV that James
Green-Armytage found. (The tests I've used check susceptibility to
strategic voting when the initial ballots are produced according to
impartial culture or a 4D Gaussian model.)
I don't like nonomonotonicity :-)
I have found two three-candidate methods, one simple (linear) and one
not so simple (nonlinear), that seem to meet the criteria. The linear
method is monotone, passes mono-add-top, but is not reversal symmetric.
The nonlinear method is monotone, but not reversal symmetric either. I
don't know if it passes mono-add-top, but it might.
(Furthermore, I've found out that if you want both monotonicity and
reversal symmetry, it seems you have to pay for it by the method
becoming a lot more vulnerable to strategy.
The best nonlinear monotone and reversal symmetric method I found had
about 65% susceptibility and a lot of ties on IC; now compare that to
the stats below.
I don't know why monotonicity and reversal symmetry gives such
susceptibility to strategy, however. There's of course also the
possibility that there's a magic monotone reversal-symmetric method out
there but that my tests haven't found it.)
The general class of three-candidate methods go like this.
Suppose there are three candidates and you want one winner. Then there
are three possibilities:
- There's a Condorcet winner, in which case he wins.
- There's a tie, either two-way with a Condorcet loser, or three-way;
in which case the tie is preserved (the tying candidates come first). - There's a cycle.
If there's a cycle, then for each candidate, we can consider without
loss of generality that candidate to be A, the one he beats pairwise to
be B, and the one who beats him pairwise to be C. Then we can let that
candidate's score be f(ABC, ACB, BAC, BCA, CAB, CBA) for some function
f, where the variables give the number of voters who voted that way when
the candidates have been thus relabeled. The candidate with the greatest
score then wins.
For a linear method, I let f = x_1 * ABC + x_2 * ACB + x_3 * BAC + x_4 *
BCA + x_5 * CAB + x_6 * CBA, where the x values are chosen from {-2, -1,
0, 1, 2}. That gives 5^6=15625 different methods in all; I used best-arm
multiarmed bandit search to find the top methods.
For a nonlinear method, f is defined by a mapping from integers to a
stack machine program with several primitive functions (like max, min,
addition, division, etc., as well as the six variables above). There are
as many of those as can fit in computer memory; my program found about a
million unique ones and winnowed them down to 20-30k monotone methods.
The best simple linear method I could find was this:
f = fpA - fpC
i.e. a candidate's score is the number of first preferences he has,
minus the number of first preferences for whoever is beating him pairwise.
Suppose the cycle is A>B>C>A, then
A's score = fpA - fpC
B's score = fpB - fpA
C's score = fpC - fpB
If you raise A to the first place, then A's score will increase and
whoever was in first will have his score decrease; so raising A can't
hurt A. Raising A from last to second-to-last won't do much in term of
the score, since that doesn't alter anybody's first preferences;
however, it can break a cycle, so again, that doesn't hurt A and may
help A. E.g. if C>A is very weak, a few voters raising BCA to BAC may
make A the CW.
The monotonicity check for the linear methods is somewhat strict and
might exclude methods that are monotone, so this might not be the very
best. The program also found one that was very slightly better (about
0.01% less susceptible), but it was a lot more complex, so I haven't
given that method here.
The method's susceptibility to strategy for 37-voter elections is:
IC: Susceptible 17045 times out of 100000 (17.0%); 0 ties
Gaussian: Susceptible 4142 times out of 100000 (4.1%); 0 ties
In comparison, the figures for Condorcet,IRV are:
IC: Susceptible 16385 times out of 100000 (16.4%); 0 ties
Gaussian: Susceptible 3980 times out of 100000 (4.0%); 0 ties
and for Schulze:
IC: Susceptible 75230 times out of 100000 (75.2%); 4873 ties
Gaussian: Susceptible 20497 times out of 100000 (20.5%): 348 ties
The linear method is not quite at Condorcet,IRV's level, but the
nonlinear methods can do much better. The best one I found there was:
f = A>B * min(C>A, A>B)/fpC
so in an ABCA cycle:
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
Break infinities by adding an epsilon to the denominators and let
epsilon approach zero. In practice, there will only be one infinity
anyway because otherwise two candidates would have no first preferences,
in which case the third has unanimous support and hence there'd be no cycle.
My program has only checked the monotonicity property of this method
statistically, so I can't say for sure it is monotone; however, it does
seem to be so. (If you can see any counterexample, please show it, as
that would either mean there's a bug or the tests weren't thorough enough.)
It's somewhat hard to check mathematically because the first preference
terms pull down while the minimum term pulls up unevenly.
The stats for this one are:
IC: Susceptible 12244 times out of 100000 (12.2%); 381 ties
Gaussian: Susceptible 3379 times out of 100000 (3.4%); 11 ties
which is better than Condorcet,IRV.
In theory, it would be possible to do the same kind of brute force
search for four candidates, but it gets pretty unwieldy pretty quick. A
linear four-candidate method would go
Either:
1. there's a CW (Smith set of 1)
2. there's a tie
3. there's a maximal three-cycle (Smith set of 3)
4. there's a maximal four-cycle (Smith set of 4)
Then, in case 3, without loss of generality let the candidate be A and
the cycle ABCA; then use f, as above, but with D (the loser) not
participating;
and in case 4, let the candidate be A and the cycle ABCDA; then use
g(ABCD, ACBD, ..., DCBA).
A linear four-candidate method would have 30 constants (4! + 3! = 24 +
6). 5^30 ~= 1e21 which is completely impractical, even for a bandit
search. Granted, only a few of those will be monotone, and only a few of
those will be cloneproof, but still...
(I suspect that the fpA-fpC method would generalize to something using
solid coalitions rather than something using plurality scores. But I
have nothing solid to base that suspicion on.)
It might be better to try to expand the three-candidate methods in a way
analogous to Ranked Pairs. E.g. A's score relative to B is a function of
all the 3-candidate elections where A win compared to all the
3-candidate elections where B win, both of these being ones where both A
and B are involved.
It's also possible, I think, to get LIIA out of the three-candidate
methods above, but I haven't experimented with that. If there's an ABCA
cycle and A gets the highest score, then to pass LIIA, let the method
follow the cycle, i.e. rank B second and C last. Then, if C is
eliminated, the two-candidate election is A>B and A still wins.
Conversely, if A is eliminated, B>C stays and C is still last.
Another nice thing to do would be to try to construct a multiobjective
top-k bandit search and let the other axis be Bayesian Regret to find
the Pareto front. With even higher dimensionality one could make the
monotone and nonmonotone methods automatically sort themselves into
different categories; just let one axis be 0 if the method is monotone
and 1 if not; or let it be the proportion of ballot sets that exhibit
monotonicity failure. But that would probably take a lot of time and
work, and I have no idea how I'd go about to make a multiobjective pure
exploration bandit algorithm.
Hi Kristofer,
Interesting project. A few thoughts/questions on details:
- I assume your ballots are cast with full, strict rankings.
- Does the definition of susceptibility include changes caused by compromise strategy?
- Did you try any issue space models simpler than 4D?
More below...
De : Kristofer Munsterhjelm km_elmet@t-online.de
À : EM election-methods@lists.electorama.com
Envoyé le : Dimanche 7 février 2016 8h03
Objet : [EM] Strategy-resistant monotone methods
(Furthermore, I've found out that if you want both monotonicity and
reversal symmetry, it seems you have to pay for it by the method
becoming a lot more vulnerable to strategy.
The best nonlinear monotone and reversal symmetric method I found had
about 65% susceptibility and a lot of ties on IC; now compare that to
the stats below.
I don't know why monotonicity and reversal symmetry gives such
susceptibility to strategy, however. There's of course also the
possibility that there's a magic monotone reversal-symmetric method out
there but that my tests haven't found it.)
If there's a cycle, then for each candidate, we can consider without
loss of generality that candidate to be A, the one he beats pairwise to
be B, and the one who beats him pairwise to be C. Then we can let that
candidate's score be f(ABC, ACB, BAC, BCA, CAB, CBA) for some function
f, where the variables give the number of voters who voted that way when
the candidates have been thus relabeled. The candidate with the greatest
score then wins.
The best simple linear method I could find was this:
f = fpA - fpC
i.e. a candidate's score is the number of first preferences he has,
minus the number of first preferences for whoever is beating him pairwise.
I think it's clear why this works: the candidate C who beats A doesn't get
"credit" for all C>A votes but only those dedicated to C as first preference.
So, the effect of strategic B>C>A votes (where sincere is B>A>C) is limited
to causing a cycle.
You could say the opposite approach to effecting this is in Condorcet//Approval,
where a strategic lower preference for C gets evaluated (when breaking the
cycle) with equal weight to the first preference for B.
I half think the answer to your reversal symmetry question is in there:
Perhaps reversal symmetry requires too much consideration of the middle rank.
The linear method is not quite at Condorcet,IRV's level, but the
nonlinear methods can do much better. The best one I found there was:
f = A>B * min(C>A, A>B)/fpC
so in an ABCA cycle:
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
So you get the strength of your win, times the strength of the weakest contest
you were part of, divided by the FPs of the candidate who beat you. That seems
to work in a similar way, emphasizing first preferences.
From a mono-raise standpoint it does seem a little weird that if C>A drops,
it appears that that could hurt A. It's true that fpC would very likely drop
at the same rate, but I think this should still produce a slightly decreasing
ratio (i.e. the value of C>A / fpC), which A does not want.
Kevin
On 02/07/2016 09:02 PM, Kevin Venzke wrote:
Hi Kristofer,
Interesting project. A few thoughts/questions on details:
- I assume your ballots are cast with full, strict rankings.
Yes. Otherwise, I would have more variables, which would make it harder
to cover the whole space. For three candidates and truncation, you'd get
A, B, C, AB, AC, BA, BC, CA, CB + the 6 full rank ones = 21
and equal ranks add more still.
- Does the definition of susceptibility include changes caused by compromise strategy?
It includes any kind of strategy. Basically,
if the winner is W,
and there's some other candidate X different from W,
and there's some ballot that, if a subset of the voters who prefer X to
W all switch to, makes X win,
then the method is susceptible to strategy for that ballot set.
In practice, the check works by repeatedly finding an X, some voters who
prefer X to W, and trying lots of different ballots for them. If any of
those ballots make X win, then success - the method is susceptible.
Otherwise, try another X and repeat. After a while (512 tries, I think),
it'll give up on its own and mark the ballot set as not susceptible.
So it measures susceptibility to constructive strategy ("make Gore win")
rather than destructive strategy ("anybody but Bush"). It's probably
harder to resist destructive than constructive strategy since the former
can let more voters in on the strategy attempt. But within the domain of
constructive strategy, it measures every kind; it doesn't distinguish
between burial, compromising, pushover or any other form.
I have been thinking about making strategy modules that only do burial
(or compromising or pushover) so that I could get better stats on what
kind of strategy the methods are vulnerable to. James did something
similar in his paper. I haven't had the time, though.
- Did you try any issue space models simpler than 4D?
No, but it should be relatively easy to run it on simpler Gaussian
models. I don't have uniform issue space models coded yet. I also tried
writing a ballot generator for what Warren calls the Dirichlet model,
but I got very similar results to IC, so I assumed it was buggy and thus
didn't include the results for it.
If I had infinite time, I would also create a ballot generator that
takes random users and movie subsets from the Netflix data set and ranks
in that order. That would give a somewhat realistic model, although
ratings on a movie site and honest political votes might be somewhat
different anyway.
More below...
De : Kristofer Munsterhjelm km_elmet@t-online.de
À : EM election-methods@lists.electorama.com
Envoyé le : Dimanche 7 février 2016 8h03
Objet : [EM] Strategy-resistant monotone methods
The best simple linear method I could find was this:
f = fpA - fpC
i.e. a candidate's score is the number of first preferences he has,
minus the number of first preferences for whoever is beating him pairwise.
I think it's clear why this works: the candidate C who beats A doesn't get
"credit" for all C>A votes but only those dedicated to C as first preference.
So, the effect of strategic B>C>A votes (where sincere is B>A>C) is limited
to causing a cycle.
Right, it feels a bit like a restricted tactical position in chess,
where you know what you need to accomplish, but you can't get your
pieces around in time. In other words, there's not enough freedom for
the strategic voters to do everything they want to do at once.
So intuitively I can see how it works. But I was hoping it'd be possible
to derive some kind of theory of methods resistant to strategy, and that
intuition doesn't seem to help us much; unless the best way of making a
method resist strategy is reducing the freedom in a way analogous to the
above.
Similarly, I get the impression that C,IRV works in part for the reason
above (e.g. mutual dominant third and burial resistance) and in part
because it's so chaotic. Because it's chaotic and has an exponential
amplification in the worst case, an attempt at strategy can backfire
quite easily.
You could say the opposite approach to effecting this is in Condorcet//Approval,
where a strategic lower preference for C gets evaluated (when breaking the
cycle) with equal weight to the first preference for B.
I half think the answer to your reversal symmetry question is in there:
Perhaps reversal symmetry requires too much consideration of the middle rank.
That's possible, but if I give up on monotonicity, nonlinear
reversal-symmetric methods can do relatively well (around 10% on IC if
my simulation is right) . Those resistant reversal-symmetric methods
seem to look at particular ballot types (e.g. ABC, ACB, BAC, BCA) rather
than the usual aggregates (first preferences, pairwise strengths). I'm
not sure what that means, but either reversal symmetry or monotonicity
on its own is fine enough; it's when you combine them that it becomes a
lot more manipulable.
I'll have to check this more closely to see that my program's working
correctly. Linear reversal symmetric methods don't do very well; with no
constraints but reversal symmetry, they're around 40-50% on IC and 1% on
Gaussian.
The linear method is not quite at Condorcet,IRV's level, but the
nonlinear methods can do much better. The best one I found there was:
f = A>B * min(C>A, A>B)/fpC
so in an ABCA cycle:
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
So you get the strength of your win, times the strength of the weakest contest
you were part of, divided by the FPs of the candidate who beat you. That seems
to work in a similar way, emphasizing first preferences.
From a mono-raise standpoint it does seem a little weird that if C>A drops,
it appears that that could hurt A. It's true that fpC would very likely drop
at the same rate, but I think this should still produce a slightly decreasing
ratio (i.e. the value of C>A / fpC), which A does not want.
An example of that would be BCA -> BAC. That doesn't alter fpC but it
does alter C>A. Perhaps it decreases the person who would have
benefitted from it more than it decreases A's score. It's hard to tell
with nonlinearity.
Hi Kristofer,
Thanks for the additional details.
----- Mail original -----
De : Kristofer Munsterhjelm km_elmet@t-online.de
À : Kevin Venzke stepjak@yahoo.fr; EM list election-methods@electorama.com
Envoyé le : Dimanche 7 février 2016 16h52
Objet : Re: [EM] Strategy-resistant monotone methods
The best simple linear method I could find was this:
f = fpA - fpC
i.e. a candidate's score is the number of first preferences he has,
minus the number of first preferences for whoever is beating him pairwise.
I think it's clear why this works: the candidate C who beats A doesn't get
"credit" for all C>A votes but only those dedicated to C as first preference.
So, the effect of strategic B>C>A votes (where sincere is B>A>C) is limited
to causing a cycle.
Right, it feels a bit like a restricted tactical position in chess,
where you know what you need to accomplish, but you can't get your
pieces around in time. In other words, there's not enough freedom for
the strategic voters to do everything they want to do at once.
So intuitively I can see how it works. But I was hoping it'd be possible
to derive some kind of theory of methods resistant to strategy, and that
intuition doesn't seem to help us much; unless the best way of making a
method resist strategy is reducing the freedom in a way analogous to the
above.
That's an interesting way of looking at it. I'm not sure if we could easily
generalize this reduction of freedom concept though.
I see the issue much more simply, that the effect (or perhaps "independence")
of lower preferences needs to be kept low, on the assumption that they might
be mischief votes. (The possibility that they might NOT be mischief votes
usually limits my enthusiasm for these efforts!)
In this scenario, when you have decided in advance that it's going to be a
Condorcet method, you necessitate that some mischief is going to be possible.
That's inherent to Condorcet, and the nature of the vulnerability is
basically the same no matter what you do. So all you can do is try to reduce
the damage done within the method of cycle resolution.
Similarly, I get the impression that C,IRV works in part for the reason
above (e.g. mutual dominant third and burial resistance) and in part
because it's so chaotic. Because it's chaotic and has an exponential
amplification in the worst case, an attempt at strategy can backfire
quite easily.
Well, I would have said that C,IRV works because IRV works. (I think it's in
general pretty promising to take a burial-resistant non-Condorcet method and
use it to complete Condorcet, if one wants burial resistance.) And IRV works
because you can't use lower preferences for mischief.
I guess that doesn't cover all the bases, only burial. I doubt push-over is
really a big deal; in my own simulations I don't recall any sensible methods
with lots of push-over incentive.
As far as compromise, putting Condorcet on the front end helps C,IRV, I am
sure. Though IRV itself need not be terrible; the IRV exclusion rule is
often very reasonable, particularly when the model doesn't do anything to
create a "center squeeze" likelihood. IRV can somewhat "correct the course"
through eliminations, reducing the need for favorite betrayal.
(I would be curious to see though, if the fact is that you're getting most
of your compromise-related performance simply from using Condorcet.)
I think the generalization (describing why things work, wrt burial anyway)
is LNHelp, except that in your experiment there is no truncation, so the
criterion is more like "shuffling lower prefs around doesn't help higher
ones," which is quite demanding, and will probably lead you towards FPP and
IRV. It is nice to see that your generator didn't actually resolve to simple
FPP though.
Kevin
On 02/09/2016 03:57 AM, Kevin Venzke wrote:
Hi Kristofer,
Thanks for the additional details.
----- Mail original -----
De : Kristofer Munsterhjelm km_elmet@t-online.de
À : Kevin Venzke stepjak@yahoo.fr; EM list election-methods@electorama.com
Envoyé le : Dimanche 7 février 2016 16h52
Objet : Re: [EM] Strategy-resistant monotone methods
The best simple linear method I could find was this:
f = fpA - fpC
i.e. a candidate's score is the number of first preferences he has,
minus the number of first preferences for whoever is beating him pairwise.
I think it's clear why this works: the candidate C who beats A doesn't get
"credit" for all C>A votes but only those dedicated to C as first preference.
So, the effect of strategic B>C>A votes (where sincere is B>A>C) is limited
to causing a cycle.
Right, it feels a bit like a restricted tactical position in chess,
where you know what you need to accomplish, but you can't get your
pieces around in time. In other words, there's not enough freedom for
the strategic voters to do everything they want to do at once.
So intuitively I can see how it works. But I was hoping it'd be possible
to derive some kind of theory of methods resistant to strategy, and that
intuition doesn't seem to help us much; unless the best way of making a
method resist strategy is reducing the freedom in a way analogous to the
above.
That's an interesting way of looking at it. I'm not sure if we could easily
generalize this reduction of freedom concept though.
I see the issue much more simply, that the effect (or perhaps "independence")
of lower preferences needs to be kept low, on the assumption that they might
be mischief votes. (The possibility that they might NOT be mischief votes
usually limits my enthusiasm for these efforts!)
That'd seem intuitive, but neither C/Plurality nor C/Antiplurality do
very well, resistance wise. The simulation provides the following results:
[Condorcet],[ER-Plurality]:
Impartial Culture: susceptible 78269/100000 = 78.3% of the time, 1547 ties
Gaussian: susceptible 21595/100000 = 21.6% of the time, 80 ties
[Condorcet],[ER-Antiplurality]:
Impartial Culture: susceptible 47579/100000 = 47.6% of the time, 1957 ties
Gaussian: susceptible 20841/100000 = 20.8% of the time, 91 ties.
Plurality passes both LNHelp and LNHarm because it doesn't care about
later preferences at all. Yet it does badly even when its vulnerability
to compromising is reduced by prefixing it by Condorcet. Its IC
susceptibility is greater than the 75% of the advanced reversal
symmetric Condorcet methods even though the latter meet neither LNHelp
nor LNHarm.
In this scenario, when you have decided in advance that it's going to be a
Condorcet method, you necessitate that some mischief is going to be possible.
That's inherent to Condorcet, and the nature of the vulnerability is
basically the same no matter what you do. So all you can do is try to reduce
the damage done within the method of cycle resolution.
I agree with this. Condorcet implies some vulnerabilities (and some
incompatibilities in general, e.g. it's impossible to get both
Participation and Condorcet). So the method can be resistant in two
ways: either resistant on its own inside the cycle regime, or resistant
in a way that meshes with the Condorcet completion (i.e. strategy X
would ordinarily be possible, but trying to execute it makes someone you
don't want into the CW).
I also think that you're right about C,IRV working because IRV works.
The main thing Condorcet gives to IRV is compromising resistance (and
some degree of clone resistance, IIRC; see JGA's paper). You lose some
burial resistance and gain some compromising resistance, and in IRV's
case, the trade seems to be worth it. But the question then becomes why
IRV works yet Plurality fails.
I guess that doesn't cover all the bases, only burial. I doubt push-over is
really a big deal; in my own simulations I don't recall any sensible methods
with lots of push-over incentive.
I see push-over more as a problem that hurts honesty than a strategy as
such. If voters find out that they could have had X if only they ranked
him lower, they're understandably going to be upset. Since IRV is
chaotic, it'd be very hard to actually use push-over to cheat or do harm.
One of the reasons that I started with the whole search was that I
suspected that nonmonotonicity was something you'd just have to endure
to get strategy resistance - kind of like Random Ballot in that if you
want resistance, you're going to have to scramble the field so it's
either hard or impossible for strategists to find a consistent strategy.
But the search proved me (somewhat) wrong, since the fpA-fpC method is
monotone yet resists quite well. If I had done more thorough research
before starting, I'd also have remembered that Carey is monotone when
limited to three candidates and that C,Carey is also quite resistant.
Hi folks
I'm going to revive a REALLY old thread, which is where Kristofer
Munsterhjelm first describes "fpA-fpC", which is described here:
https://electowiki.org/wiki/fpA-fpC
At some point, I'll set myself up for simulations, but for now, I'm
going to take the REALLY LAZY approach, which is to ask on this
mailing list without really reading all of the old EM threads and
understanding what y'all were saying.
The main question I'd like to ask: what is it that makes fpA-fpC a
better method than STAR? Certainly not the name, but Kristofer hasn't
given up on it (it would seem [1]), so SOMETHING must be better about
it. What is it? I'm most interested in how it compares to STAR
because STAR seems to have a lot more traction than any
Condorcet-winner-criterion (CWC) compliant methods that I'm aware of.
I have a soft-spot for CWC-compliant methods, but I've been convinced
that strict CWC isn't necessary as long as a system is close enough.
Rob
[1]: As evidence he hasn't given up, Kristofer's still adding to the
"fpA-fpC" page on electowiki:
https://electowiki.org/w/index.php?title=FpA-fpC&curid=2805&diff=16412&oldid=15820
On Fri, Feb 19, 2016 at 12:25 PM Kristofer Munsterhjelm
km_elmet@t-online.de wrote:
On 02/09/2016 03:57 AM, Kevin Venzke wrote:
Hi Kristofer,
Thanks for the additional details.
----- Mail original -----
De : Kristofer Munsterhjelm km_elmet@t-online.de
À : Kevin Venzke stepjak@yahoo.fr; EM list election-methods@electorama.com
Envoyé le : Dimanche 7 février 2016 16h52
Objet : Re: [EM] Strategy-resistant monotone methods
The best simple linear method I could find was this:
f = fpA - fpC
i.e. a candidate's score is the number of first preferences he has,
minus the number of first preferences for whoever is beating him pairwise.
I think it's clear why this works: the candidate C who beats A doesn't get
"credit" for all C>A votes but only those dedicated to C as first preference.
So, the effect of strategic B>C>A votes (where sincere is B>A>C) is limited
to causing a cycle.
Right, it feels a bit like a restricted tactical position in chess,
where you know what you need to accomplish, but you can't get your
pieces around in time. In other words, there's not enough freedom for
the strategic voters to do everything they want to do at once.
So intuitively I can see how it works. But I was hoping it'd be possible
to derive some kind of theory of methods resistant to strategy, and that
intuition doesn't seem to help us much; unless the best way of making a
method resist strategy is reducing the freedom in a way analogous to the
above.
That's an interesting way of looking at it. I'm not sure if we could easily
generalize this reduction of freedom concept though.
I see the issue much more simply, that the effect (or perhaps "independence")
of lower preferences needs to be kept low, on the assumption that they might
be mischief votes. (The possibility that they might NOT be mischief votes
usually limits my enthusiasm for these efforts!)
That'd seem intuitive, but neither C/Plurality nor C/Antiplurality do
very well, resistance wise. The simulation provides the following results:
[Condorcet],[ER-Plurality]:
Impartial Culture: susceptible 78269/100000 = 78.3% of the time, 1547 ties
Gaussian: susceptible 21595/100000 = 21.6% of the time, 80 ties
[Condorcet],[ER-Antiplurality]:
Impartial Culture: susceptible 47579/100000 = 47.6% of the time, 1957 ties
Gaussian: susceptible 20841/100000 = 20.8% of the time, 91 ties.
Plurality passes both LNHelp and LNHarm because it doesn't care about
later preferences at all. Yet it does badly even when its vulnerability
to compromising is reduced by prefixing it by Condorcet. Its IC
susceptibility is greater than the 75% of the advanced reversal
symmetric Condorcet methods even though the latter meet neither LNHelp
nor LNHarm.
In this scenario, when you have decided in advance that it's going to be a
Condorcet method, you necessitate that some mischief is going to be possible.
That's inherent to Condorcet, and the nature of the vulnerability is
basically the same no matter what you do. So all you can do is try to reduce
the damage done within the method of cycle resolution.
I agree with this. Condorcet implies some vulnerabilities (and some
incompatibilities in general, e.g. it's impossible to get both
Participation and Condorcet). So the method can be resistant in two
ways: either resistant on its own inside the cycle regime, or resistant
in a way that meshes with the Condorcet completion (i.e. strategy X
would ordinarily be possible, but trying to execute it makes someone you
don't want into the CW).
I also think that you're right about C,IRV working because IRV works.
The main thing Condorcet gives to IRV is compromising resistance (and
some degree of clone resistance, IIRC; see JGA's paper). You lose some
burial resistance and gain some compromising resistance, and in IRV's
case, the trade seems to be worth it. But the question then becomes why
IRV works yet Plurality fails.
I guess that doesn't cover all the bases, only burial. I doubt push-over is
really a big deal; in my own simulations I don't recall any sensible methods
with lots of push-over incentive.
I see push-over more as a problem that hurts honesty than a strategy as
such. If voters find out that they could have had X if only they ranked
him lower, they're understandably going to be upset. Since IRV is
chaotic, it'd be very hard to actually use push-over to cheat or do harm.
One of the reasons that I started with the whole search was that I
suspected that nonmonotonicity was something you'd just have to endure
to get strategy resistance - kind of like Random Ballot in that if you
want resistance, you're going to have to scramble the field so it's
either hard or impossible for strategists to find a consistent strategy.
But the search proved me (somewhat) wrong, since the fpA-fpC method is
monotone yet resists quite well. If I had done more thorough research
before starting, I'd also have remembered that Carey is monotone when
limited to three candidates and that C,Carey is also quite resistant.
Election-Methods mailing list - see http://electorama.com/em for list info
Suppose you had two methods that were indistinguishable in merit,
including simplicity and transparency, except that one was based on Score
ballots, and the other was based on Ranked Choice ballots. Which one (all
else being equal) would be the easier sell?
On Tue, Oct 18, 2022, 8:51 PM Rob Lanphier roblan@gmail.com wrote:
Hi folks
I'm going to revive a REALLY old thread, which is where Kristofer
Munsterhjelm first describes "fpA-fpC", which is described here:
https://electowiki.org/wiki/fpA-fpC
At some point, I'll set myself up for simulations, but for now, I'm
going to take the REALLY LAZY approach, which is to ask on this
mailing list without really reading all of the old EM threads and
understanding what y'all were saying.
The main question I'd like to ask: what is it that makes fpA-fpC a
better method than STAR? Certainly not the name, but Kristofer hasn't
given up on it (it would seem [1]), so SOMETHING must be better about
it. What is it? I'm most interested in how it compares to STAR
because STAR seems to have a lot more traction than any
Condorcet-winner-criterion (CWC) compliant methods that I'm aware of.
I have a soft-spot for CWC-compliant methods, but I've been convinced
that strict CWC isn't necessary as long as a system is close enough.
Rob
[1]: As evidence he hasn't given up, Kristofer's still adding to the
"fpA-fpC" page on electowiki:
https://electowiki.org/w/index.php?title=FpA-fpC&curid=2805&diff=16412&oldid=15820
On Fri, Feb 19, 2016 at 12:25 PM Kristofer Munsterhjelm
km_elmet@t-online.de wrote:
On 02/09/2016 03:57 AM, Kevin Venzke wrote:
Hi Kristofer,
Thanks for the additional details.
----- Mail original -----
De : Kristofer Munsterhjelm km_elmet@t-online.de
À : Kevin Venzke stepjak@yahoo.fr; EM list <
Envoyé le : Dimanche 7 février 2016 16h52
Objet : Re: [EM] Strategy-resistant monotone methods
The best simple linear method I could find was this:
f = fpA - fpC
i.e. a candidate's score is the number of first preferences he has,
minus the number of first preferences for whoever is beating him
pairwise.
I think it's clear why this works: the candidate C who beats A
doesn't get
"credit" for all C>A votes but only those dedicated to C as first
preference.
So, the effect of strategic B>C>A votes (where sincere is B>A>C) is
limited
to causing a cycle.
Right, it feels a bit like a restricted tactical position in chess,
where you know what you need to accomplish, but you can't get your
pieces around in time. In other words, there's not enough freedom for
the strategic voters to do everything they want to do at once.
So intuitively I can see how it works. But I was hoping it'd be
possible
to derive some kind of theory of methods resistant to strategy, and
that
intuition doesn't seem to help us much; unless the best way of making
a
method resist strategy is reducing the freedom in a way analogous to
the
above.
That's an interesting way of looking at it. I'm not sure if we could
easily
generalize this reduction of freedom concept though.
I see the issue much more simply, that the effect (or perhaps
"independence")
of lower preferences needs to be kept low, on the assumption that they
might
be mischief votes. (The possibility that they might NOT be mischief
votes
usually limits my enthusiasm for these efforts!)
That'd seem intuitive, but neither C/Plurality nor C/Antiplurality do
very well, resistance wise. The simulation provides the following
results:
[Condorcet],[ER-Plurality]:
Impartial Culture: susceptible 78269/100000 = 78.3% of the time, 1547
ties
Gaussian: susceptible 21595/100000 = 21.6% of the time, 80 ties
[Condorcet],[ER-Antiplurality]:
Impartial Culture: susceptible 47579/100000 = 47.6% of the time, 1957
ties
Gaussian: susceptible 20841/100000 = 20.8% of the time, 91 ties.
Plurality passes both LNHelp and LNHarm because it doesn't care about
later preferences at all. Yet it does badly even when its vulnerability
to compromising is reduced by prefixing it by Condorcet. Its IC
susceptibility is greater than the 75% of the advanced reversal
symmetric Condorcet methods even though the latter meet neither LNHelp
nor LNHarm.
In this scenario, when you have decided in advance that it's going to
be a
Condorcet method, you necessitate that some mischief is going to be
possible.
That's inherent to Condorcet, and the nature of the vulnerability is
basically the same no matter what you do. So all you can do is try to
reduce
the damage done within the method of cycle resolution.
I agree with this. Condorcet implies some vulnerabilities (and some
incompatibilities in general, e.g. it's impossible to get both
Participation and Condorcet). So the method can be resistant in two
ways: either resistant on its own inside the cycle regime, or resistant
in a way that meshes with the Condorcet completion (i.e. strategy X
would ordinarily be possible, but trying to execute it makes someone you
don't want into the CW).
I also think that you're right about C,IRV working because IRV works.
The main thing Condorcet gives to IRV is compromising resistance (and
some degree of clone resistance, IIRC; see JGA's paper). You lose some
burial resistance and gain some compromising resistance, and in IRV's
case, the trade seems to be worth it. But the question then becomes why
IRV works yet Plurality fails.
I guess that doesn't cover all the bases, only burial. I doubt
push-over is
really a big deal; in my own simulations I don't recall any sensible
methods
with lots of push-over incentive.
I see push-over more as a problem that hurts honesty than a strategy as
such. If voters find out that they could have had X if only they ranked
him lower, they're understandably going to be upset. Since IRV is
chaotic, it'd be very hard to actually use push-over to cheat or do harm.
One of the reasons that I started with the whole search was that I
suspected that nonmonotonicity was something you'd just have to endure
to get strategy resistance - kind of like Random Ballot in that if you
want resistance, you're going to have to scramble the field so it's
either hard or impossible for strategists to find a consistent strategy.
But the search proved me (somewhat) wrong, since the fpA-fpC method is
monotone yet resists quite well. If I had done more thorough research
before starting, I'd also have remembered that Carey is monotone when
limited to three candidates and that C,Carey is also quite resistant.
Election-Methods mailing list - see http://electorama.com/em for list
info
Election-Methods mailing list - see https://electorama.com/em for list
info
What if both systems could use rated ballots and allowed for ties (e.g. one
could vote for multiple "5-star" candidates, and multiple "4-star"
candidates, but could also use ranked ballots and allow for ties? It seems
to me that conflating ballot type and tallying method is a mental hurdle
for folks, isn't it?
Rob
On Tue, Oct 18, 2022 at 10:22 PM Forest Simmons forest.simmons21@gmail.com
wrote:
Suppose you had two methods that were indistinguishable in merit,
including simplicity and transparency, except that one was based on Score
ballots, and the other was based on Ranked Choice ballots. Which one (all
else being equal) would be the easier sell?
On Tue, Oct 18, 2022, 8:51 PM Rob Lanphier roblan@gmail.com wrote:
Hi folks
I'm going to revive a REALLY old thread, which is where Kristofer
Munsterhjelm first describes "fpA-fpC", which is described here:
https://electowiki.org/wiki/fpA-fpC
At some point, I'll set myself up for simulations, but for now, I'm
going to take the REALLY LAZY approach, which is to ask on this
mailing list without really reading all of the old EM threads and
understanding what y'all were saying.
The main question I'd like to ask: what is it that makes fpA-fpC a
better method than STAR? Certainly not the name, but Kristofer hasn't
given up on it (it would seem [1]), so SOMETHING must be better about
it. What is it? I'm most interested in how it compares to STAR
because STAR seems to have a lot more traction than any
Condorcet-winner-criterion (CWC) compliant methods that I'm aware of.
I have a soft-spot for CWC-compliant methods, but I've been convinced
that strict CWC isn't necessary as long as a system is close enough.
Rob
[1]: As evidence he hasn't given up, Kristofer's still adding to the
"fpA-fpC" page on electowiki:
https://electowiki.org/w/index.php?title=FpA-fpC&curid=2805&diff=16412&oldid=15820
On Fri, Feb 19, 2016 at 12:25 PM Kristofer Munsterhjelm
km_elmet@t-online.de wrote:
On 02/09/2016 03:57 AM, Kevin Venzke wrote:
Hi Kristofer,
Thanks for the additional details.
----- Mail original -----
De : Kristofer Munsterhjelm km_elmet@t-online.de
À : Kevin Venzke stepjak@yahoo.fr; EM list <
Envoyé le : Dimanche 7 février 2016 16h52
Objet : Re: [EM] Strategy-resistant monotone methods
The best simple linear method I could find was this:
f = fpA - fpC
i.e. a candidate's score is the number of first preferences he has,
minus the number of first preferences for whoever is beating him
pairwise.
I think it's clear why this works: the candidate C who beats A
doesn't get
"credit" for all C>A votes but only those dedicated to C as first
preference.
So, the effect of strategic B>C>A votes (where sincere is B>A>C) is
limited
to causing a cycle.
Right, it feels a bit like a restricted tactical position in chess,
where you know what you need to accomplish, but you can't get your
pieces around in time. In other words, there's not enough freedom for
the strategic voters to do everything they want to do at once.
So intuitively I can see how it works. But I was hoping it'd be
possible
to derive some kind of theory of methods resistant to strategy, and
that
intuition doesn't seem to help us much; unless the best way of
making a
method resist strategy is reducing the freedom in a way analogous to
the
above.
That's an interesting way of looking at it. I'm not sure if we could
easily
generalize this reduction of freedom concept though.
I see the issue much more simply, that the effect (or perhaps
"independence")
of lower preferences needs to be kept low, on the assumption that
they might
be mischief votes. (The possibility that they might NOT be mischief
votes
usually limits my enthusiasm for these efforts!)
That'd seem intuitive, but neither C/Plurality nor C/Antiplurality do
very well, resistance wise. The simulation provides the following
results:
[Condorcet],[ER-Plurality]:
Impartial Culture: susceptible 78269/100000 = 78.3% of the time, 1547
ties
Gaussian: susceptible 21595/100000 = 21.6% of the time, 80 ties
[Condorcet],[ER-Antiplurality]:
Impartial Culture: susceptible 47579/100000 = 47.6% of the time, 1957
ties
Gaussian: susceptible 20841/100000 = 20.8% of the time, 91 ties.
Plurality passes both LNHelp and LNHarm because it doesn't care about
later preferences at all. Yet it does badly even when its vulnerability
to compromising is reduced by prefixing it by Condorcet. Its IC
susceptibility is greater than the 75% of the advanced reversal
symmetric Condorcet methods even though the latter meet neither LNHelp
nor LNHarm.
In this scenario, when you have decided in advance that it's going to
be a
Condorcet method, you necessitate that some mischief is going to be
possible.
That's inherent to Condorcet, and the nature of the vulnerability is
basically the same no matter what you do. So all you can do is try to
reduce
the damage done within the method of cycle resolution.
I agree with this. Condorcet implies some vulnerabilities (and some
incompatibilities in general, e.g. it's impossible to get both
Participation and Condorcet). So the method can be resistant in two
ways: either resistant on its own inside the cycle regime, or resistant
in a way that meshes with the Condorcet completion (i.e. strategy X
would ordinarily be possible, but trying to execute it makes someone you
don't want into the CW).
I also think that you're right about C,IRV working because IRV works.
The main thing Condorcet gives to IRV is compromising resistance (and
some degree of clone resistance, IIRC; see JGA's paper). You lose some
burial resistance and gain some compromising resistance, and in IRV's
case, the trade seems to be worth it. But the question then becomes why
IRV works yet Plurality fails.
I guess that doesn't cover all the bases, only burial. I doubt
push-over is
really a big deal; in my own simulations I don't recall any sensible
methods
with lots of push-over incentive.
I see push-over more as a problem that hurts honesty than a strategy as
such. If voters find out that they could have had X if only they ranked
him lower, they're understandably going to be upset. Since IRV is
chaotic, it'd be very hard to actually use push-over to cheat or do
harm.
One of the reasons that I started with the whole search was that I
suspected that nonmonotonicity was something you'd just have to endure
to get strategy resistance - kind of like Random Ballot in that if you
want resistance, you're going to have to scramble the field so it's
either hard or impossible for strategists to find a consistent strategy.
But the search proved me (somewhat) wrong, since the fpA-fpC method is
monotone yet resists quite well. If I had done more thorough research
before starting, I'd also have remembered that Carey is monotone when
limited to three candidates and that C,Carey is also quite resistant.
Election-Methods mailing list - see http://electorama.com/em for list
info
Election-Methods mailing list - see https://electorama.com/em for list
info
On 19.10.2022 05:50, Rob Lanphier wrote:
Hi folks
I'm going to revive a REALLY old thread, which is where Kristofer
Munsterhjelm first describes "fpA-fpC", which is described here:
https://electowiki.org/wiki/fpA-fpC
At some point, I'll set myself up for simulations, but for now, I'm
going to take the REALLY LAZY approach, which is to ask on this
mailing list without really reading all of the old EM threads and
understanding what y'all were saying.
The main question I'd like to ask: what is it that makes fpA-fpC a
better method than STAR? Certainly not the name, but Kristofer hasn't
given up on it (it would seem [1]), so SOMETHING must be better about
it. What is it? I'm most interested in how it compares to STAR
because STAR seems to have a lot more traction than any
Condorcet-winner-criterion (CWC) compliant methods that I'm aware of.
I have a soft-spot for CWC-compliant methods, but I've been convinced
that strict CWC isn't necessary as long as a system is close enough.
One way of looking at it is that Range and STAR are about honest
performance, whereas fpA-fpC is about resistance to strategy.[1] So
whether one is better than the other depends on what you prioritize.
More concretely, fpA-fpC and its variants attempt to retain IRV's
strategy resistance while fixing its chaos and monotonicity problems and
making it Smith in a natural way. They resist strategy by passing
DMTCBR, which seems to grant considerable burial resistance in practice.
(As a bit of a happy coincidence, both Friendly Cover and Forest's
Friendly Voting also happen to be summable, which Smith-IRV is not. I
know that rb-j has emphasized summability in his Condorcet advocacy.)
The fpA-fpC variants are mainly research methods. But research methods
can turn out to be useful in practice: Tideman's Ranked Pairs was also
originally a research method. A suitably advanced fpA-fpC variant could
be useful where strategy resistance is very important but IRV is
recognized as just too flawed.
As examples of strategy resistance, fpA-fpC and its generalizations pass
the chicken dilemma criterion, thus avoiding the Burr dilemma of plain
Range (though one could argue that chicken dilemmas aren't as severe to
begin with for ranked methods). Friendly Cover is also independent of
twins - cloning candidates one at a time - while STAR is not.[2]
Finally, I'm limiting myself to ranked ballots partly because I think
rated ones don't have good enough a foundation. There may be better
strategy-resistant rated methods, but I haven't looked. [3]
-km
[1] James Green-Armytage shows that Range is considerably more
susceptible to strategy than Smith-IRV (Condorcet-Hare), see
http://jamesgreenarmytage.com/strategy-utility.pdf page 18.
(Unfortunately he didn't include STAR.)
[2] This probably holds for Friendly Voting too, but I haven't checked
it yet.
[3] For instance, Smith//Range or Smith,Range would be cloneproof by
rated standards; perhaps there exists a monotonicity-fixed version of
renormalized Smith//Range, the way fpA-fpC is a monotonicity-fixed
version of Smith-IRV. Or something based off SARVO-Range.
On Tue, Oct 18, 2022, 10:53 PM Rob Lanphier roblan@gmail.com wrote:
What if both systems could use rated ballots and allowed for ties (e.g.
one could vote for multiple "5-star" candidates, and multiple "4-star"
candidates, but could also use ranked ballots and allow for ties? It seems
to me that conflating ballot type and tallying method is a mental hurdle
for folks, isn't it?
Rob
That would be wonderful! ...and people that double down and insist on only
allowing ratings or only allowing rankings would be revealing their true
colors.
Judgment and Grade style ballots (as in MJ, ASM, and STAR descriptions)
should also be options.
Ratings/scores, judgments, grades, approvals, etc. can easily be converted
to rankings (by extracting the ordinal information inherent in the
ballots), as long as equal ranks are allowed. So any good method designed
with rankings in mind can also allow ratings ballot ... no excuse not to
... really a basic good faith requirement, when you think about it.
It is not so obvious how to convert ordinal ballots into the other styles
in a standard way.
That's the main reason I have been working on ordinal or "Universal Domain"
methods lately.
-Forest
On Tue, Oct 18, 2022 at 10:22 PM Forest Simmons <
forest.simmons21@gmail.com> wrote:
Suppose you had two methods that were indistinguishable in merit,
including simplicity and transparency, except that one was based on Score
ballots, and the other was based on Ranked Choice ballots. Which one (all
else being equal) would be the easier sell?
On Tue, Oct 18, 2022, 8:51 PM Rob Lanphier roblan@gmail.com wrote:
Hi folks
I'm going to revive a REALLY old thread, which is where Kristofer
Munsterhjelm first describes "fpA-fpC", which is described here:
https://electowiki.org/wiki/fpA-fpC
At some point, I'll set myself up for simulations, but for now, I'm
going to take the REALLY LAZY approach, which is to ask on this
mailing list without really reading all of the old EM threads and
understanding what y'all were saying.
The main question I'd like to ask: what is it that makes fpA-fpC a
better method than STAR? Certainly not the name, but Kristofer hasn't
given up on it (it would seem [1]), so SOMETHING must be better about
it. What is it? I'm most interested in how it compares to STAR
because STAR seems to have a lot more traction than any
Condorcet-winner-criterion (CWC) compliant methods that I'm aware of.
I have a soft-spot for CWC-compliant methods, but I've been convinced
that strict CWC isn't necessary as long as a system is close enough.
Rob
[1]: As evidence he hasn't given up, Kristofer's still adding to the
"fpA-fpC" page on electowiki:
https://electowiki.org/w/index.php?title=FpA-fpC&curid=2805&diff=16412&oldid=15820
On Fri, Feb 19, 2016 at 12:25 PM Kristofer Munsterhjelm
km_elmet@t-online.de wrote:
On 02/09/2016 03:57 AM, Kevin Venzke wrote:
Hi Kristofer,
Thanks for the additional details.
----- Mail original -----
De : Kristofer Munsterhjelm km_elmet@t-online.de
À : Kevin Venzke stepjak@yahoo.fr; EM list <
Envoyé le : Dimanche 7 février 2016 16h52
Objet : Re: [EM] Strategy-resistant monotone methods
The best simple linear method I could find was this:
f = fpA - fpC
i.e. a candidate's score is the number of first preferences he
has,
minus the number of first preferences for whoever is beating him
pairwise.
I think it's clear why this works: the candidate C who beats A
doesn't get
"credit" for all C>A votes but only those dedicated to C as first
preference.
So, the effect of strategic B>C>A votes (where sincere is B>A>C)
is limited
to causing a cycle.
Right, it feels a bit like a restricted tactical position in chess,
where you know what you need to accomplish, but you can't get your
pieces around in time. In other words, there's not enough freedom
for
the strategic voters to do everything they want to do at once.
So intuitively I can see how it works. But I was hoping it'd be
possible
to derive some kind of theory of methods resistant to strategy, and
that
intuition doesn't seem to help us much; unless the best way of
making a
method resist strategy is reducing the freedom in a way analogous
to the
above.
That's an interesting way of looking at it. I'm not sure if we could
easily
generalize this reduction of freedom concept though.
I see the issue much more simply, that the effect (or perhaps
"independence")
of lower preferences needs to be kept low, on the assumption that
they might
be mischief votes. (The possibility that they might NOT be mischief
votes
usually limits my enthusiasm for these efforts!)
That'd seem intuitive, but neither C/Plurality nor C/Antiplurality do
very well, resistance wise. The simulation provides the following
results:
[Condorcet],[ER-Plurality]:
Impartial Culture: susceptible 78269/100000 = 78.3% of the time, 1547
ties
Gaussian: susceptible 21595/100000 = 21.6% of the time, 80 ties
[Condorcet],[ER-Antiplurality]:
Impartial Culture: susceptible 47579/100000 = 47.6% of the time, 1957
ties
Gaussian: susceptible 20841/100000 = 20.8% of the time, 91 ties.
Plurality passes both LNHelp and LNHarm because it doesn't care about
later preferences at all. Yet it does badly even when its vulnerability
to compromising is reduced by prefixing it by Condorcet. Its IC
susceptibility is greater than the 75% of the advanced reversal
symmetric Condorcet methods even though the latter meet neither LNHelp
nor LNHarm.
In this scenario, when you have decided in advance that it's going
to be a
Condorcet method, you necessitate that some mischief is going to be
possible.
That's inherent to Condorcet, and the nature of the vulnerability is
basically the same no matter what you do. So all you can do is try
to reduce
the damage done within the method of cycle resolution.
I agree with this. Condorcet implies some vulnerabilities (and some
incompatibilities in general, e.g. it's impossible to get both
Participation and Condorcet). So the method can be resistant in two
ways: either resistant on its own inside the cycle regime, or resistant
in a way that meshes with the Condorcet completion (i.e. strategy X
would ordinarily be possible, but trying to execute it makes someone
you
don't want into the CW).
I also think that you're right about C,IRV working because IRV works.
The main thing Condorcet gives to IRV is compromising resistance (and
some degree of clone resistance, IIRC; see JGA's paper). You lose some
burial resistance and gain some compromising resistance, and in IRV's
case, the trade seems to be worth it. But the question then becomes why
IRV works yet Plurality fails.
I guess that doesn't cover all the bases, only burial. I doubt
push-over is
really a big deal; in my own simulations I don't recall any sensible
methods
with lots of push-over incentive.
I see push-over more as a problem that hurts honesty than a strategy as
such. If voters find out that they could have had X if only they ranked
him lower, they're understandably going to be upset. Since IRV is
chaotic, it'd be very hard to actually use push-over to cheat or do
harm.
One of the reasons that I started with the whole search was that I
suspected that nonmonotonicity was something you'd just have to endure
to get strategy resistance - kind of like Random Ballot in that if you
want resistance, you're going to have to scramble the field so it's
either hard or impossible for strategists to find a consistent
strategy.
But the search proved me (somewhat) wrong, since the fpA-fpC method is
monotone yet resists quite well. If I had done more thorough research
before starting, I'd also have remembered that Carey is monotone when
limited to three candidates and that C,Carey is also quite resistant.
Election-Methods mailing list - see http://electorama.com/em for list
info
Election-Methods mailing list - see https://electorama.com/em for list
info