A possible low-strategy PR concept when picking n-1 winners from n candidates
When picking a winner from two candidates, majority rule is
strategyproof. I was playing with ways to generalize this to multiwinner
to find at least some domain where a method can be both Droop
proportional and strategy-proof, and found this for elections with n
candidates and (n-1) seats:
Consider a set of n-1 winners {A_1, ..., A_(n-1)}, and call the loser B.
If there exists strictly more than a Droop quota of voters who prefers
A_k to B for any k, (but not necessarily the same voters for each A_k)
then say that this set is a stable set. (That is, some subset of the
voters prefers A_1 to B, another subset of the voters prefers A_2 to B
and so on, and all of these subsets contain more than a Droop quota's
woth of voters.)
If there is only a single stable set, and the method elects that set,
then it passes the DPC for that election and it's also strategyproof.
Suppose that some group of voters wants to kick out A_k and replace him
with B. These voters must necessarily (honestly) rank B over A_k. But
we've established that there exists some Droop quota of voters who rank
A_k over B. The B>A_k voters can do nothing to stop the existence of
such a contingent of voters, because they are already maximally
contributing to B over A_k, so they can't break the stable set's
condition of a Droop quota or more ranking A_k over B, and thus they
can't undo this stable set.
So the only thing they can try to do is to establish another stable set
and hope that the method switches over. Their objective is to kick out
A_k, so this other candidate set must be {A_1, ..., A_(k-1), A_(k+1),
..., A_(n-1), B}. But for such a set to exist, there must exist at least
a Droop quota's worth of voters who rank B over A_k. Since the B>A_k
voters are already ranking B over A_k, they can't contribute to the
creation of such a subset either. Hence if they create another stable
set by altering their ballots, it is not the one they want.
Since this argument holds for all A_k, there should be nothing anybody
who has an incentive to get B elected can do to get B elected.
(I'd imagine there's also a Condorcet extension for this concept for
electing k out of n candidates, but I can't quite see it yet. From a DSV
perspective, such a generalization would be "automatic vote management"
if single-winner Condorcet is "automatic compromising".
The argument above also resembles the reasoning I used for the resistant
set, so maybe there's a generalization that connects it with that.)
The problem is that there may be multiple stable sets. Impartial culture
elections are almost certain to have more than one. Here's an example:
50: A>B>C
306: A>C>B
65: B>A>C
426: B>C>A
260: C>A>B
93: C>B>A
1200 voters in total, and the quota for two seats is 400.
{A,B} is a stable set because we can assign
306 A>C>B voters,
50 A>B>C voters, and
65 B>A>C voters
to the group that prefers A to C. This is more than a Droop quota in
total. Similarly, we can assign 426 B>C>A voters to the group that
prefers B to C.
But {B, C} is also a stable set:
65 B>A>C
378 B>C>A
= 443 voters preferring B to A, and
260 C>A>B
93 C>B>A
48 B>C>A
= 401 voters preferring C to A.
So what we might wonder is: does there exist a way for a voting method
to pick a particular stable set when more than one exists, and to stick
to that stable set no matter what the voters who prefer the other stable
set may do?
Unless my simulations are wrong, Schulze STV is almost always
strategyproof for impartial culture and (n-1) out of n, so that suggests
that it's possible at least some of the time. But Schulze STV has a
nonzero coalitional manipulability rate for (n-1) out of n with other
models, say a spatial model. Is that because the method it uses to pick
a particular stable set is strategyproof almost always for IC but not in
a spatial model? What's special about it in that case? Is it something
like "for three candidates and two to be elected, it's strategyproof
whenever there are only two stable sets" and then spatial model, but not
IC, sometimes has three?
For three candidates, at least, there should always be at least one
stable set, unless there's an exact tie. I don't know if that holds for
every (n-1) of n election.
In any case, it might be something to investigate further at some point.
-km
On Fri, 1 May 2026 23:38:43 +0200
Kristofer Munsterhjelm via Election-Methods election-methods@lists.electorama.com wrote:
When picking a winner from two candidates, majority rule is
strategyproof. I was playing with ways to generalize this to multiwinner
to find at least some domain where a method can be both Droop
proportional and strategy-proof, and found this for elections with n
candidates and (n-1) seats:
Unless I misunderstood what is ment here,
it seems like you claim to have found a counterexample
to Duggan-Schwartz impossibility theorem (reducing to Gibbard-Satterthwaite when single winner)
https://en.wikipedia.org/wiki/Duggan-Schwartz_theorem (wikipedia summary only)
and there is also a corresponding generalization to Gibbards theorem without
the rankorder ballots requirement (though I don't know any common name for that one).
Consider a set of n-1 winners {A_1, ..., A_(n-1)}, and call the loser B.
If there exists strictly more than a Droop quota of voters who prefers
A_k to B for any k, (but not necessarily the same voters for each A_k)
then say that this set is a stable set. (That is, some subset of the
voters prefers A_1 to B, another subset of the voters prefers A_2 to B
and so on, and all of these subsets contain more than a Droop quota's
woth of voters.)
If there is only a single stable set, and the method elects that set,
then it passes the DPC for that election and it's also strategyproof.
Suppose that some group of voters wants to kick out A_k and replace him
with B. These voters must necessarily (honestly) rank B over A_k. But
we've established that there exists some Droop quota of voters who rank
A_k over B. The B>A_k voters can do nothing to stop the existence of
such a contingent of voters, because they are already maximally
contributing to B over A_k, so they can't break the stable set's
condition of a Droop quota or more ranking A_k over B, and thus they
can't undo this stable set.
So the only thing they can try to do is to establish another stable set
and hope that the method switches over. Their objective is to kick out
A_k, so this other candidate set must be {A_1, ..., A_(k-1), A_(k+1),
..., A_(n-1), B}. But for such a set to exist, there must exist at least
a Droop quota's worth of voters who rank B over A_k. Since the B>A_k
voters are already ranking B over A_k, they can't contribute to the
creation of such a subset either. Hence if they create another stable
set by altering their ballots, it is not the one they want.
Since this argument holds for all A_k, there should be nothing anybody
who has an incentive to get B elected can do to get B elected.
(I'd imagine there's also a Condorcet extension for this concept for
electing k out of n candidates, but I can't quite see it yet. From a DSV
perspective, such a generalization would be "automatic vote management"
if single-winner Condorcet is "automatic compromising".
The argument above also resembles the reasoning I used for the resistant
set, so maybe there's a generalization that connects it with that.)
The problem is that there may be multiple stable sets. Impartial culture
elections are almost certain to have more than one. Here's an example:
50: A>B>C
306: A>C>B
65: B>A>C
426: B>C>A
260: C>A>B
93: C>B>A
1200 voters in total, and the quota for two seats is 400.
{A,B} is a stable set because we can assign
306 A>C>B voters,
50 A>B>C voters, and
65 B>A>C voters
to the group that prefers A to C. This is more than a Droop quota in
total. Similarly, we can assign 426 B>C>A voters to the group that
prefers B to C.
But {B, C} is also a stable set:
65 B>A>C
378 B>C>A
= 443 voters preferring B to A, and
260 C>A>B
93 C>B>A
48 B>C>A
= 401 voters preferring C to A.
So what we might wonder is: does there exist a way for a voting method
to pick a particular stable set when more than one exists, and to stick
to that stable set no matter what the voters who prefer the other stable
set may do?
Unless my simulations are wrong, Schulze STV is almost always
strategyproof for impartial culture and (n-1) out of n, so that suggests
that it's possible at least some of the time. But Schulze STV has a
nonzero coalitional manipulability rate for (n-1) out of n with other
models, say a spatial model. Is that because the method it uses to pick
a particular stable set is strategyproof almost always for IC but not in
a spatial model? What's special about it in that case? Is it something
like "for three candidates and two to be elected, it's strategyproof
whenever there are only two stable sets" and then spatial model, but not
IC, sometimes has three?
For three candidates, at least, there should always be at least one
stable set, unless there's an exact tie. I don't know if that holds for
every (n-1) of n election.
In any case, it might be something to investigate further at some point.
If I have understood correctly so far then you have not subverted the assumptions
of Duggan-Schwartz impossibility theorems and choosen:
Voter symmetry implicitly +
Candidate symmetry implicitly +
Explicitly placed a bound on the maximum number of winners
strictly lower then the number of candidates,
and therefore must be vulnerable to some form of strategic dishonesty,
in some cases.
Hopefully of interest
Gustav
On 2026-05-02 01:31, Gustav Thorzen via Election-Methods wrote:
On Fri, 1 May 2026 23:38:43 +0200
Kristofer Munsterhjelm via Election-Methods election-methods@lists.electorama.com wrote:
When picking a winner from two candidates, majority rule is
strategyproof. I was playing with ways to generalize this to multiwinner
to find at least some domain where a method can be both Droop
proportional and strategy-proof, and found this for elections with n
candidates and (n-1) seats:
Unless I misunderstood what is ment here,
it seems like you claim to have found a counterexample
to Duggan-Schwartz impossibility theorem (reducing to Gibbard-Satterthwaite when single winner)
https://en.wikipedia.org/wiki/Duggan-Schwartz_theorem (wikipedia summary only)
and there is also a corresponding generalization to Gibbards theorem without
the rankorder ballots requirement (though I don't know any common name for that one).
I might have been a bit unclear. By "some domain", I mean a subset of
elections where a method can pick an outcome (here, a set of winners) so
that there's no incentive for the voters to lie about their preferences
if that election corresponds to their honest preferences.
Single-winner methods that pass the majority criterion are strategyproof
when a majority honestly prefers a certain candidate to everybody else.
If they vote honestly, then their preferred candidate will be the
majority winner. The majority voters have no incentive to alter their
ballots because the only thing that can accomplish is to make someone
they like less win; and the minority voters are ignored entirely, so
they have no incentive to lie either.
More broadly, methods that elect from the resistant set are
strategyproof when the honest election has a "super Condorcet winner" -
a candidate who disqualifies everybody else. This explains a significant
part of IRV's resistance to coalitional manipulation.
Gibbard shows that we can't have perfection - we can't have a method
that's always strategyproof. But some methods may still be better than
others.
So what I was asking is if there's some domain where some PR methods are
strategyproof. If we could find such a domain, and it's large enough, it
might explain, for instance, why Schulze STV has such low coalitional
manipulability rates under impartial culture, but not under spatial models.
Or it could be useful for mechanism design.
For instance, a number of single-winner voting methods are all
manipulable with probability one under the impartial culture as the
number of voters approaches infinity. The Schulze method is one of
these, but its multiwinner generalization, Schulze STV, is strangely
robust to coalitional manipulation when picking (n-1) from n winners
under impartial culture.
On the other hand, single-winner IRV is resistant to strategy (IC
manipulability does not converge to unity), but its multiwinner
generalization, STV, is considerably more vulnerable to vote management
than Schulze STV, and so fares worse.
Perhaps exploring strategyproof domains for both single- and n-1-seat
elections could give some idea of how to get the best of both worlds?
-km
On Mon, 4 May 2026 20:31:26 +0200
Kristofer Munsterhjelm km-elmet@munsterhjelm.no wrote:
On 2026-05-02 01:31, Gustav Thorzen via Election-Methods wrote:
On Fri, 1 May 2026 23:38:43 +0200
Kristofer Munsterhjelm via Election-Methods election-methods@lists.electorama.com wrote:
When picking a winner from two candidates, majority rule is
strategyproof. I was playing with ways to generalize this to multiwinner
to find at least some domain where a method can be both Droop
proportional and strategy-proof, and found this for elections with n
candidates and (n-1) seats:
Unless I misunderstood what is ment here,
it seems like you claim to have found a counterexample
to Duggan-Schwartz impossibility theorem (reducing to Gibbard-Satterthwaite when single winner)
https://en.wikipedia.org/wiki/Duggan-Schwartz_theorem (wikipedia summary only)
and there is also a corresponding generalization to Gibbards theorem without
the rankorder ballots requirement (though I don't know any common name for that one).
I might have been a bit unclear. By "some domain", I mean a subset of
elections where a method can pick an outcome (here, a set of winners) so
that there's no incentive for the voters to lie about their preferences
if that election corresponds to their honest preferences.
Well, that makes my bad.
For what its worth, there is a theorem from economics about mechanism design
which I don't know a common name for
(it was refered to as the "possibility theorem" when I learned of it),
which when applied to voting systems basically goes:
If there exist a system with unique strategic equilibria under
some circumstances,
then there also exist some system with the same unique equilibria,
but it is achieved by complete (strict/total/perfect) honesty.
So for just about every rankorder system satisfying the
MB-Condorcet winner criteria,
we have some other system that have a strong nash equilibrium for
complete honesty whenever there is a MB-Condorcet winner of
the true preferences.
I think this unique strong equilibria also extends to multiwinner elections
for all pairwise matchups Minority-Beat (Droop quota based) winner candidates,
that is using a minority threshold equal to the Droop quota ratio,
rather the the Majority threshold of minimal majority.
Pretty sure you need to to assign the candidate multiple, say k,
seats/units of voting power if they pairwise Majority/Minority-Beat every other
candidate a a ratio above k Droop quotas.
Single-winner methods that pass the majority criterion are strategyproof
when a majority honestly prefers a certain candidate to everybody else.
If they vote honestly, then their preferred candidate will be the
majority winner. The majority voters have no incentive to alter their
ballots because the only thing that can accomplish is to make someone
they like less win; and the minority voters are ignored entirely, so
they have no incentive to lie either.
More broadly, methods that elect from the resistant set are
strategyproof when the honest election has a "super Condorcet winner" -
a candidate who disqualifies everybody else. This explains a significant
part of IRV's resistance to coalitional manipulation.
Gibbard shows that we can't have perfection - we can't have a method
that's always strategyproof. But some methods may still be better than
others.
So what I was asking is if there's some domain where some PR methods are
strategyproof. If we could find such a domain, and it's large enough, it
might explain, for instance, why Schulze STV has such low coalitional
manipulability rates under impartial culture, but not under spatial models.
Or it could be useful for mechanism design.
For instance, a number of single-winner voting methods are all
manipulable with probability one under the impartial culture as the
number of voters approaches infinity. The Schulze method is one of
these, but its multiwinner generalization, Schulze STV, is strangely
robust to coalitional manipulation when picking (n-1) from n winners
under impartial culture.
On the other hand, single-winner IRV is resistant to strategy (IC
manipulability does not converge to unity), but its multiwinner
generalization, STV, is considerably more vulnerable to vote management
than Schulze STV, and so fares worse.
Perhaps exploring strategyproof domains for both single- and n-1-seat
elections could give some idea of how to get the best of both worlds?
I think exploring maximum possible honest is worthwhile,
and a valuable reference.
The possibility theorem about unique equilibria more or less
provides a starting point,
see which equilibria can be extended to multiwinner system
and your all good.
Hopefully I understood things correctly this time.
Gustav
On 2026-05-04 23:01, Gustav Thorzen via Election-Methods wrote:
On Mon, 4 May 2026 20:31:26 +0200
Kristofer Munsterhjelm km-elmet@munsterhjelm.no wrote:
I might have been a bit unclear. By "some domain", I mean a subset of
elections where a method can pick an outcome (here, a set of winners) so
that there's no incentive for the voters to lie about their preferences
if that election corresponds to their honest preferences.
Well, that makes my bad.
For what its worth, there is a theorem from economics about mechanism design
which I don't know a common name for
(it was refered to as the "possibility theorem" when I learned of it),
which when applied to voting systems basically goes:
If there exist a system with unique strategic equilibria under
some circumstances,
then there also exist some system with the same unique equilibria,
but it is achieved by complete (strict/total/perfect) honesty.
So for just about every rankorder system satisfying the
MB-Condorcet winner criteria,
we have some other system that have a strong nash equilibrium for
complete honesty whenever there is a MB-Condorcet winner of
the true preferences.
That's the revelation principle, I think. However, I don't think such a
method exists, and there might not be a sufficiently general strategic
equilibrium that preserves MB-Condorcet.
Consider three-candidate impartial culture with 13 voters. The optimal
majoritarian method according to my voting method generator has a
coalitional manipulability rate of 0.13567 (or exactly
8203195/60466176), with Smith,IRV slightly behind at 0.16068
(58293235/362797056).
Every Condorcet winner in an impartial culture election is an
MB-Condorcet winner because everybody fully ranks every candidate. The
proportion of (3 cand, 13 voter) impartial culture elections with a
Condorcet winner is p_IC(CW,3,13) = 0.91893 (55564279/60466176).
If it were possible to design a majoritarian method to be strategyproof
whenever an honest CW exists, then we would expect the best such method
to be manipulable no more often than 1-p_IC(CW,3,13) = 0.08106 of the
time. However, the minimally manipulable method does worse than this.
It's clear that if there is a proper majority cycle (A>B>C>A) then a
majoritarian method can't be strategyproof because, assume that A wins;
then C>A voters could all place C first and the majority criterion would
force C to win. So no method can do better than the CW probability: the
probability of a cycle presents a lower bound on the coalitional
manipulability of a majoritarian method. But it's not tight, at least
not for three candidates and 13 voters.
I suspect that for the impartial culture with three candidates, in the
limit of the number of voters approaching infinity, the manipulability
rate for a resistant set method is, essentially, a tight minimal bound.
But I haven't proven this - just observed that IRV meets this bound.
In the limit of number of voters v approaching infinity, p_IC(CW, 3, v)
= 3/4 + (3 arcsin(1/3))/(2 pi) = 0.91226... [1]
and the probability of there being a super CW, in which case a resistant
set method is strategyproof, is p_IC(Super CW, 3, v) = 3/4 + (3
arcsin(1/3))/(4 pi) = 0.83113... [2]
In any case, I think the flaw that limits the revelation principle here
is that the equilibrium model for "approval leads the CW to be elected"
isn't a fully general strategic equilibrium.
In February, after a cardinal zealot really pissed me off, I prepared a
message showing that if there is a cycle, Laslier's rule just keeps
following the cycle. I didn't post it because I later thought it was a
bit ranty, but maybe I should just to show the argument.
But let's say that it's true (that Laslier's winner will just go around
the cycle if there is one), and we have a three-candidate Approval
election where there's an honest CW (candidate A), but there's a faction
who could create a cycle by burying A. Then, if they pretend that their
strategic ballots (burying A) are their real preferences, and use
Laslier's rule consistent with those strategic preferences, then
Approval with LR would change from A winning with 100% probability, to
an equal 33% probability of any candidate winning: which candidate wins
depends on the initial Approval round and how many polls are done. If
the utilities are just right, the burial faction might prefer this
outcome, and so have an incentive to behave strategically even in the
presence of an honest CW.
I think this unique strong equilibria also extends to multiwinner elections
for all pairwise matchups Minority-Beat (Droop quota based) winner candidates,
that is using a minority threshold equal to the Droop quota ratio,
rather the the Majority threshold of minimal majority.
Pretty sure you need to to assign the candidate multiple, say k,
seats/units of voting power if they pairwise Majority/Minority-Beat every other
candidate a a ratio above k Droop quotas.
Going by Condorcet as automated compromising might produce something like:
The majority criterion means that a majority can force a candidate to
win by ranking him first.
An analogous Droop version might be: a group of voters of more than a
Droop quota can force their favored candidate to be in the outcome by
ranking him first.
Then MB-Condorcet says: if there exists a candidate A that, no matter
who else is elected, the majority could strategize to get A elected by
the majority criterion, then A should be elected.
The Droop analog might be something like: If there exists a candidate A
who, no matter what set of winners that doesn't include A is picked,
there exists at least a Droop quota's worth of voters who could, by
ranking A first, replace one of the candidates in the outcome with A...
then A should be elected.
Or something like that.
I think exploring maximum possible honest is worthwhile,
and a valuable reference.
The possibility theorem about unique equilibria more or less
provides a starting point,
see which equilibria can be extended to multiwinner system
and your all good.
I agree, though directly applying the revelation principle seems hard.
-km
[1] KRISHNAMOORTHY, Mukkai S.; RAGHAVACHARI, Madabhushi. Condorcet
winner probabilities-a statistical perspective.
https://arxiv.org/abs/math/0511140
[2]
http://lists.electorama.com/pipermail/election-methods-electorama.com/2025-June/007015.html
On Wed, 6 May 2026 19:59:01 +0200
Kristofer Munsterhjelm km-elmet@munsterhjelm.no wrote:
On 2026-05-04 23:01, Gustav Thorzen via Election-Methods wrote:
On Mon, 4 May 2026 20:31:26 +0200
Kristofer Munsterhjelm km-elmet@munsterhjelm.no wrote:
I might have been a bit unclear. By "some domain", I mean a subset of
elections where a method can pick an outcome (here, a set of winners) so
that there's no incentive for the voters to lie about their preferences
if that election corresponds to their honest preferences.
Well, that makes my bad.
For what its worth, there is a theorem from economics about mechanism design
which I don't know a common name for
(it was refered to as the "possibility theorem" when I learned of it),
which when applied to voting systems basically goes:
If there exist a system with unique strategic equilibria under
some circumstances,
then there also exist some system with the same unique equilibria,
but it is achieved by complete (strict/total/perfect) honesty.
So for just about every rankorder system satisfying the
MB-Condorcet winner criteria,
we have some other system that have a strong nash equilibrium for
complete honesty whenever there is a MB-Condorcet winner of
the true preferences.
That's the revelation principle, I think. However, I don't think such a
method exists, and there might not be a sufficiently general strategic
equilibrium that preserves MB-Condorcet.
Looking at wikipedia for revelation principle and it looks to be that one, thanks.
Also I don't expect a unique strong nash equilibra to be sufficient for
practical applications without separate always applying honesty incentives,
like AFB and Monotonicity.
Consider three-candidate impartial culture with 13 voters. The optimal
majoritarian method according to my voting method generator has a
coalitional manipulability rate of 0.13567 (or exactly
8203195/60466176), with Smith,IRV slightly behind at 0.16068
(58293235/362797056).
Thank you for provding the exact numbers without rounding errors (so underrated).
More to the point,
is coalitional manipulability here calculated assuming strict rankorder ballots,
or does it include regular rankorder ballots allowing ties?
I ask becuase all benefits from MB-Condorcet vs PB-Condorcet comes only,
as far as I know, when ties on the ballots are allowed,
even when assuming strict rankorders for the true preferences.
Every Condorcet winner in an impartial culture election is an
MB-Condorcet winner because everybody fully ranks every candidate. The
proportion of (3 cand, 13 voter) impartial culture elections with a
Condorcet winner is p_IC(CW,3,13) = 0.91893 (55564279/60466176).
If it were possible to design a majoritarian method to be strategyproof
whenever an honest CW exists, then we would expect the best such method
to be manipulable no more often than 1-p_IC(CW,3,13) = 0.08106 of the
time. However, the minimally manipulable method does worse than this.
This was the reason for my question about the ballots earlier,
because if strict ballots are the case,
I would assume thats the reason behind this letdown.
Also I don't think a unique strong nash equilibria counts as strategyproof,
which I believe is reserved for unique dominant strategy equilibria,
which could also be a reason behind this.
It's clear that if there is a proper majority cycle (A>B>C>A) then a
majoritarian method can't be strategyproof because, assume that A wins;
then C>A voters could all place C first and the majority criterion would
force C to win. So no method can do better than the CW probability: the
probability of a cycle presents a lower bound on the coalitional
manipulability of a majoritarian method. But it's not tight, at least
not for three candidates and 13 voters.
This is more or less expected, since its conjectured the best a method using
exclusively rankorder ballots (ties allowed) is a strategic equilibra
on each member of the MB-Smith set,
with Approval being conjectured as a method with this property.
(I say conjectured becuase a proof was not presented in the paper,
and now I can't seem to find that paper or theorem at all.)
I suspect that for the impartial culture with three candidates, in the
limit of the number of voters approaching infinity, the manipulability
rate for a resistant set method is, essentially, a tight minimal bound.
But I haven't proven this - just observed that IRV meets this bound.
In the limit of number of voters v approaching infinity, p_IC(CW, 3, v)
= 3/4 + (3 arcsin(1/3))/(2 pi) = 0.91226... [1]
and the probability of there being a super CW, in which case a resistant
set method is strategyproof, is p_IC(Super CW, 3, v) = 3/4 + (3
arcsin(1/3))/(4 pi) = 0.83113... [2]
In any case, I think the flaw that limits the revelation principle here
is that the equilibrium model for "approval leads the CW to be elected"
isn't a fully general strategic equilibrium.
The Equilibrium model for approval is quite different from what
is noramally used, right?
In which case I would also agree that we should not expect much
aplicability from the revelation princible using Approval as a starting point.
In February, after a cardinal zealot really pissed me off, I prepared a
message showing that if there is a cycle, Laslier's rule just keeps
following the cycle. I didn't post it because I later thought it was a
bit ranty, but maybe I should just to show the argument.
I think cardinal itselfe is fine, especially when it allows subverint impossibility theorems.
Though it feels a bit silly how most of them have equivalent methods
using rankorder ballots (ties allowed/mandatory)
with at most k number of different ranks,
at which point I would have liked to know what the benefit
of going cardinal was.
But let's say that it's true (that Laslier's winner will just go around
the cycle if there is one), and we have a three-candidate Approval
election where there's an honest CW (candidate A), but there's a faction
who could create a cycle by burying A. Then, if they pretend that their
strategic ballots (burying A) are their real preferences, and use
Laslier's rule consistent with those strategic preferences, then
Approval with LR would change from A winning with 100% probability, to
an equal 33% probability of any candidate winning: which candidate wins
depends on the initial Approval round and how many polls are done. If
the utilities are just right, the burial faction might prefer this
outcome, and so have an incentive to behave strategically even in the
presence of an honest CW.
The moment polling outcomes enter utility calculation,
we might as well consider each poll its on round,
and thte system as a multi round system.
So I think, a priori, any system with significant dependence
on polling outcome (like FPTP and much less so but still Approval)
needs some justification before use.
I earlier tried to go MB-Smith//Approval with each voter providing
separate rankorder and approval ballots,
in order to accelerate convergence (assuming equilibria on each set member),
but I found it by itself still insufficient for real world conditions,
and the MB-Smith set filering (resulting in a nice ISDA compliance)
made a mess of all atempts to see if AFB/Mono/LN-Help was preserved
in some meningfull way (preferably to both ballots).
Maybe if elections were held each year or so,
but then you need something to stabilize transitions even more then now,
and there is not really anything applicable to single winner election.
I think this unique strong equilibria also extends to multiwinner elections
for all pairwise matchups Minority-Beat (Droop quota based) winner candidates,
that is using a minority threshold equal to the Droop quota ratio,
rather the the Majority threshold of minimal majority.
Pretty sure you need to to assign the candidate multiple, say k,
seats/units of voting power if they pairwise Majority/Minority-Beat every other
candidate a a ratio above k Droop quotas.
Going by Condorcet as automated compromising might produce something like:
The majority criterion means that a majority can force a candidate to
win by ranking him first.
An analogous Droop version might be: a group of voters of more than a
Droop quota can force their favored candidate to be in the outcome by
ranking him first.
Then MB-Condorcet says: if there exists a candidate A that, no matter
who else is elected, the majority could strategize to get A elected by
the majority criterion, then A should be elected.
The Droop analog might be something like: If there exists a candidate A
who, no matter what set of winners that doesn't include A is picked,
there exists at least a Droop quota's worth of voters who could, by
ranking A first, replace one of the candidates in the outcome with A...
then A should be elected.
Or something like that.
Yes, though it really ought to be possible for the same candidate to get
mutiple seats/units of voting power of they have support above multiple
quota ratios.
It is like the single most (pretended) reason to require party list,
yet things simplify greatly if we simply allow this instead.
I think exploring maximum possible honest is worthwhile,
and a valuable reference.
The possibility theorem about unique equilibria more or less
provides a starting point,
see which equilibria can be extended to multiwinner system
and your all good.
I agree, though directly applying the revelation principle seems hard.
Yeah, it is really hard in my experience.
I think I have managed to apply it for fully deterministic systems
(no random tie breakers) allowing a no winner outcome,
thus preserving candidate symmetry,
but I still lack a proof of the unique strong nash equilibrium
for MB-Condorcet winners of the true preferences
carrying over to such settings.
Gustav