[EM] A proxy for Monroe's "strategic election of universally loathed candidates" criterion

[Closed Limelike Curves]

So, to be clear, I don't think the DH2/DH3 analyses are useless. They're
very much useful for ruling methods out (rather than in), because this
particular pathology is bad enough to be disqualifying.

I gave these as examples of pathologies that might be hiding in methods
that perform very well on criteria. I'm very sorry if I was unclear about
that—I think most of my comments on this topic went out just after you left
the mailing list, so you might not have seen them.

My broader point is I can't really recommend a voting system to the public
until I have an actual mathematical proof that nothing like DH2 or DH3 is
hiding in it. Not having found a counterexample yet isn't sufficient for a
proof. What if it turns out DH3 isn't a problem, but DH7 is? Or what if,
under realistic models of voter behavior, resistant-set methods never ever
elect the Condorcet winner in any center squeeze? It would take an
infinitely long time to evaluate all possible scenarios, so we just can't
do that.

Instead I want to start by building a useful model of strategic actors in
elections*, *then use it to prove theorems about *all* situations, given a
decent solution concept. This is the standard approach of mechanism
design: It's not enough to find situations where your method works; you
have to show it will always work.

The approach I've found most satisfying so far has been the Myerson and
Weber one, or some of Laslier's papers. Both show that positioning yourself
at the median voter maximizes your probability of winning an election. I've
also gotten quite a bit out of Balinski and Laraki's analysis of
level/score manipulation, which assumes voters' main concern is to show
support or opposition with regard to candidates. This is a very useful
framework if you think voters use their ballots to send messages to
politicians about the popularity of their positions.

On Mon, Aug 5, 2024 at 4:28 AM Kristofer Munsterhjelm <
km-elmet at munsterhjelm.no> wrote:

> On 2024-08-04 19:26, Closed Limelike Curves wrote:
> > Thank you so much for all of your hard work+analysis Kris :)
> >
> >     Smith passes DH2, but (surprisingly) Schwartz fails!
> >
> > Could you clarify what you mean by Smith/Schwartz passing/failing?
>
> Sure :-) Suppose we have
>
> N: A>B>C
> N: B>A>C
> 1: C>A>B
> 1: C>B>A
>
> The Smith and Schwartz sets are {A, B}. Now suppose that A buries:
>
> N: A>C>B
> N: B>A>C
> 1: C>A>B
> 1: C>B>A
>
> The Smith set is still {A, B}, but the Schwartz set is {A} (unless
> there's a bug in my code - and rbvote).
>
> Hence there exist Smith methods that pass DH2.[1] But there don't exist
> Schwartz methods that do so, because unilateral burial will lead the
> buriers' candidate to win, and bilateral burial makes C the CW.
>
> As I have stated, I wouldn't put *too* much into this kind of knife edge
> election. It's not going to make me throw away RP or Schulze. But if
> you're going by the letter of the criterion, then Schwartz does seem to
> imply failure.
>
> > ————
> > That said, I'd like to offer some clarification. My concern isn't
> > actually with DH2 or DH3 scenarios.
>
> [...]
>
> > Second, I reject the idea of looking at individual scenarios like DH3 or
> > Burr, then judging if a voting system fails or succeeds in such a
> > scenario. (Or rather, using this to rule electoral systems in, not out).
> > This feels like declaring a theorem proven after trying a few numbers
> > and not finding a counterexample.
> (I'll get to the rest at a later point)
>
> I find this surprising. What initially got me to investigate this was
> your reply to my post about resistant set performance, where I showed
> that even Resistant,Borda had low manipulability. You said something
> like, and I'm paraphrasing, "but how do I know that the remaining share
> of manipulable elections isn't all turkey-raising of no-hope
> candidates?", and then pointed at Monroe's analysis.
>
> If you have no concern about DH2 or DH3, then it would seem that you
> have no concern with Monroe's NIA either, because my whole point was
> that we could reduce all this complicated thinking about M-W equilibria
> in Monroe's scenario with a much simpler mechanically approximable
> criterion.
>
> But if you weren't interested in Monroe's NIA to begin with, you should
> have made that more clear when you replied to my resistant set post.
>
> > (The Burr dilemma should be the poster boy for this kind of bad
> > strategic analysis: it focuses on a single scenario, /and/ on group
> > strategy, /and/ it completely ignores the possibility of correlated
> > equilibria and mixed strategies...)
>
> I may get into why I don't think that's an accurate depiction, either,
> later. But before I do that, I would like to know if single scenarios
> concern you when it comes to ruling methods out.
>
> Do you consider concrete criteria and/or single scenario analyses
> valuable in making sure a method doesn't blow up on its users after
> implementation (or as a proxy for the risk that such an event may occur)?
>
> And do you consider statistical analyses like manipulability and VSE
> useful to determine the viability of a method? Or are you more
> fundamentally saying that we can only rule methods out, but we can't
> rule them in by either approach?
>
> -km
>
> [1] Strictly speaking, all that this argument shows is that there may
> exist such a method. But since Smith,X preserves X's DH2 resistance, we
> can just let X=IRV and so we know that one exists.
>
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