Re: A possible low-strategy PR concept when picking n-1 winners from n candidates

On 2026-05-04 23:01, Gustav Thorzen via Election-Methods wrote:
> On Mon, 4 May 2026 20:31:26 +0200
> Kristofer Munsterhjelm <[email protected]> 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
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