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

[Gustav Thorzen]

On Mon, 4 May 2026 20:31:26 +0200
Kristofer Munsterhjelm <km-elmet at 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 at 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