[EM] Method X, bummer

[Forest Simmons]

On Sun, Aug 6, 2023, 4:57 AM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> On 8/6/23 03:11, Forest Simmons wrote:
> > The reason I'm willing to consider Implicit Approval at all is because
> > so far it's the only simple UD method we know of for generating a
> > monotone, clone free agenda for agenda based methods.
> >
> > [The Ranked Pair finish order might work, but surely there's a simpler
> > solution than that!]
> >
> > I do not think IA has any special burial resistance ... the burial
> > resistance is mostly if not entirely from the fact that in the three
> > candidate Smith case (the most common case by far when there is no
> > ballot CW) the lowest approval Smith candidate is the one most likely to
> > have been buried.
> >
> >  From my point of view your comments about truncation are a little off
> > base because nothing would substantially change strategically if
> > truncations were not allowed at all, because IA should be defined as
> > total number of ballots minus the equal bottom count, and (in such a way
> > that) equal bottom candidates can be either ranked equal bottom or all
> > truncated together without affecting ting the IA scores.
>
> Sorry about that, I must've misunderstood. My impression


Your impression is the traditional flawed meaning of IA that Kevin, Chris,
and I have been fighting against for quite a while ... without making too
big a deal about it. DMC was the first context where we started making the
distinction, since DMC is based on eliminating candidates from the bottom
of the IA order until a pairwise unbeaten candidate emerges ... Benham
based on IA elimination.

of IA was that
> you'd basically count candidates that were explicitly ranked, so e.g. for
>
> 12: A>B>C
> 11: B>C>A
> 10: C>A>B
>
> every candidate would have an IA count of 33, but if you did
>
> 12: A
> 11: B>C>A
> 10: C>A>B
>
> then A would have an IA of 33, and the other two would have 21.
>
> > Candidate X's bottom count is the number of ballots on which X out ranks
> > no candidate, and her top count is the number of ballots on which she is
> > not outranked.
> >
> > X's implicit approval score is best defined as the total number of
> > ballots minus its bottom count plus epsilon times its top count.
> >
> > The epsilon term is the built in tie breaker that makes the method
> > highly decisive in public elections even when complete rankings are
> > required as in Australia.
> >
> > Keep in mind that the only purpose of the method, as far as we are
> > concerned is to get an agenda order that is both monotone and clone free
> > without going outside of UD.
>
> That's good: these methods should be testable without having to model
> where voters would put their approval cutoffs. That should give more
> evidence to whether methods using these orders are cloneproof, monotone,
> and burial resistant.[1]
>
> However, there's a slight complication. As I first noticed back when JGA
> was doing his simulations, impartial culture is overly nice to
> Antiplurality-type methods; IC simulations will say they're extremely
> good at resisting strategy. I never found out why - I think it's an
> artifact of the distribution, but I don't know what. But what it means
> is that I should probably create a spatial model before I start testing
> methods that use bottom counts :-)
>
> Hopefully I'll get to it, eventually, but I'm also probably going to
> think about whether there are ways to salvage method X, first.
>

I wonder if my old idea about max A>B restricted to A=Head and B=Tail would
work.  In ther words elect the head of the strongest minimal covering
chain. The strength of a chain is the strength of the Head>Tail defeat. The
covering requirement is that no candidate defeats every member of the
chain. A minimal covering is one that ceases to cover when any of its
members is removed.

That requirement takes the place of the 1/n quota requirement.

The method is NP hard in general, but in practice (public elections) no
minimal covering chain has more than two members.

>
> > If grade ballots or other judgment ballots are preferred, that would
> > suit me fine ... but it would be exterior to UD.
> >
> > My dream would be to have RCV ballots with optional strong approval and
> > strong disapproval annotations.
> >
> > To me it is much easier to make those heart felt decisions than to put
> > in one all purpose cutoff that is supposed to separate the generally
> > approved from the unapproved.
> >
> > The history of mathematics bears out this psychological observation
> > (about cutoff decisions): what we now call "calculus" was originally
> > "The Calculus of Infinitesimals" which involved distinguishing from
> > ordinary numbers those very close to zero and those very far from zero.
> >
> > That calculus was the basis of all of the progress in mathematics from
> > the time of Newton, Leibniz,Euler; the Bernoullis, Laplace, Gauss, etc
> > ... until the time of Cauchy, Weirstrauss and eventually Cantor, when
> > the logical foundations of "infinities" of various kinds came under
> > close scrutiny ... resulting in a reformulation of analysis in terms of
> > limits and other set theoretic constructs. Infinitesimals were put on
> > hold until set theoreticians and other mathematical logicians
> > (especially Abraham Robinson in the 1960's) finally advanced enough to
> > put infinitesimal calculus on a rigorous footing ... a system as
> > consistent as modern set theory itself ... which Euler and company had
> > long ago navigated flawlessly with their unerring intuition.
> >
> > This ability to have the top approval and bottom disapproval while still
> > distinguishing the rankswould be a great improvement over current
> > implicit approval that requires collapsing to equal top or equal bottom
> > for the ability express respective approval or disapproval .... the
> > agonizing decision of whether sacrificing ordinal information for
> > approval/ disapproval information is worth it.
> >
> > It seems to me that the decision of where to put these cutoffs would be
> > no harder than the current corresponding decisions about equal rankings
> > and truncations.
> >
> > Am I the only one that feels that way?
>
> I think it depends on the person. Myself, I find ranking easier than
> rating, because I'm always trying (and failing) to find some natural
> calibrated scale when rating, but ranking is easy: just "do I prefer a
> world with X to one with Y?". And then if it's below my JND, equal-rank.
>

To me "Below my JND" is the same as "infinitely close." And "I strongly
approve X" means I consider X to be infinitely close to my favorite.  "I
strongly disapprove Z" means I consider Z to be infinitely close to my
anti-favorite.


> I seem to recall that you said you get a feeling for a natural scale
> after rating for long enough, in reference to grading papers.


Try to rank all of the student solutions to problem one, then to problem
two, then to problem three, etc and then use those rankings to get a finish
order among the students ... and you still have to decide where in the
finish order to put the cutoffs for the different grades.

My other source of intuition for "infinitely close" to ballot favorite or
ballot anti-favorite ... that is strongly approved or disapproved ... is
the Internal Set Theory formulation of "infinitely close" in any standard
topological space whether metrizeable or not:

If P is a standard point of a standard topological space S, then point X is
infinitely close to P
if and only if
X is in every standard neighborhood of P.

Perhaps
> that is true; perhaps most people find a natural rating scale and I'm
> the odd one out.
>
> Some cases are clear cut: if I were faced with an election with a bunch
> of contemporary candidates, and then Stalin and Hitler, I know where I
> would put my cutoff. But generalizing it in a more nuanced multiparty
> environment is hard. For instance, the Norwegian parties currently
> represented in Parliament are, from left to right:
>
> Red Party
> Socialist Left
> Green Party
> Labor
> Patient Focus
> Center Party
> Christian Democrats
> Liberal Party
> Conservative Party
> Progress Party
>
> These are all democratic parties in the sense that they support the
> continuation of parliamentary democracy. There are no Orban-style
> autocrats, and thus nobody to really "intensely disapprove of" as such.
> Sure, there are some whose policies I'd rather not have be enacted, but
> not on that level.
>
> Perhaps I would disapprove of the other end of the scale from where my
> preferences lie, but if you were to add a (hypothetical) Stalinist party
> and a Norwegian NSDAP (to mirror the Stalin and Hitler example above),
> then my disapproval thresholds would probably change so that I would
> disapprove of those two and approve of all the democratic parties.
>
> And what that suggests to me is that when multiparty rule happens and
> there's more of a gradual scale, then it gets harder to place dividing
> lines,



You seem to be forgetting that strong approval and strong disapproval are
optional designations. If you do not feel strongly about approving or
disapproving a candidate, then you cannot honestly use those designations.

In infinitesimal calculus, you are not required to classify every number
you use as infinitely large, infinitesimal, or neither ...  but it is nice
to have those options.

and that it's difficult to create an approval expression that
> doesn't inherently violate the spirit of IIA due to calibration issues.
>
> But it might just be me!
>
> -km
>
> [1] I'm not sure how it could be cloneproof though? Neither top nor
> bottom preferences are cloneproof.
>
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