[EM] A possible low-strategy PR concept when picking n-1 winners from n candidates

[Gustav Thorzen]

On Wed, 6 May 2026 19:59:01 +0200
Kristofer Munsterhjelm <km-elmet at 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 at 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