[EM] Method X, bummer

[Forest Simmons]

See if any of your worries apply to the main application I am proposing for
use of Strong Approval and Strong Disapproval:

Ballots are ranked preference style with two virtual candidates ... the
Strong Approval Cutoff and the Strong Disapproval Cutoff.

The default positions of these respective cutoffs are immediately below the
top (or equal top) position, and immediately above the bottom position that
harbors any and all candidates that do not outrank any candidate(s).

For each candidate X, let SA(X) and SD(X) be the respective number of
ballots on which X is above (respectively below) the Strong (Approval
respectively Disapproval) cutoff.

For each ballot B, let f(B) (respectively f'(B)) be the candidate strongly
(approved. resp disapproved) on ballot B that is strongly (approved resp
disapproved) on the greatest number of ballots.

We refer to f(B) and f'(B) respectively as the favorite and anti-favorite
candidates, respectively of ballot B.

The swap cost of converting one permutation P of the candidates into
another such permutation Q is the sum of the costs of the elementary swaps
needed to effect the conversion.

The swap cost of a single swap
AB --> BA is b'a, where b'=f'(B) and a=f(A).

Example.

48 C
28 A>B
24 B

What is the swap cost of converting C>A>B to its reverse order B>A>C?

The set of elementary swaps required are CA to AC, CB to BC,  and AB to BA.

The total swap cost is given by
ca'+cb'+ab' where c=48, a=28, b'=0, and
a'=24+48=72

So the swap cost is=3456.
48×72+48×0+28×0=48×72

How about converting BAC to its reverse?

ba'+bc'+ac' = 24(72+28)+28×28=3184


We see that it is more expensive to convert  CAB into BAC, than vice-versa.

Which order is the most expensive to convert into its opposite?

It could be argued the more democratically expensive to reverse an order,
the more stable the order.

Let's try reversing ABC:
The cost is ab'+ac' +bc', or ...
0+(a+b)c'=52×28=1456

Reversing back from CBA to ABC costs ... cb'+ca'+ba'=(c+b)a'=(48+24)76=5472

Reversing BCA:

b(c'+a')+ca'=2400+48×72=5856

Reversing ACB:
ac'+(a+c)b'=28×28=784

It seems that. BCA is the hardest permutation to reverse.

Note that B is not the anti-favorite of any faction and that A is the
antifavorite of every faction except its own.

Also BCA is a beatpath with defeat strengths of B>C=52 to 48 and C>A=48 to
28.

Because the C faction did not explicitly strongly disapprove B, candidate A
ended up with most of the default strong disapproval .... and B with none
of it.


On Sat, Aug 12, 2023, 2:49 PM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> On 8/8/23 01:17, Forest Simmons wrote:
>
> >> 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.
>
> Yes, that's an intuitive idea. But after further consideration, I think
> it also depends on the effort, which brings a sort of "paradox of
> voting" logic to it.
>
> Suppose that I'm an agrarian leftist voter. Suppose that candidate X has
> a strong agrarian position while candidate Y has a strong leftist
> position. On intial examination, I find that they're about equally good,
> so I would rank X and Y equally.
>
> So far so good, "very close".
>
> But let's say I were instead part of a deliberative body (something like
> a citizens' assembly) writing a reference on the different candidates'
> policies for voters to consult. Then I might investigate the candidates'
> past records, the results of policies they supported, and so on, because
> the assembly is smaller and the effect of getting it wrong is more
> serious (assuming the guide would be used by the voters). And after
> careful investigation, I might find out that, in my opinion, X is better
> than Y.
>
> Because the stakes are higher, I would make an additional effort to
> distinguish X from Y. While complete instrumental rationality is
> completely unrealistic for elections (or nobody would vote), there's
> *some* part of it to ranking otherwise very close, or very hard to tell
> apart, candidates.
>
> These candidates may not even be an epsilon apart in the limit of time
> spent scrutinizing them going to infinity. I just can't determine what
> the actual distance is at a glance. So my equal-rank is an expression
> that I trust the rest of the electorate enough, and that it would not be
> worth it to spend excessive effort trying to determine if X is really
> better than Y.
>
> I'm kind of mixing "personal preference" (i.e. what I like the most) and
> "best for society" (what candidate would be best for society), but it's
> the best I can do at getting at what my intuition says.
>
> >> 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.
>
> I think what I was trying to say is that it seems on principle very hard
> for a method to infer anything consistent from the approval cutoffs, due
> to the voters' differences in idea about where they should lie. Like I
> said in my quick and dirty STAR post, even if we assume consistent
> utilities in a vNM sense, the voter-dependent affine scaling values
> makes it very difficult to compare my expression of a cutoff to someone
> else's.
>
> With ranking, there's no problem, because the affine transformations are
> all monotone.
>
> The relative difficulty in a gradual setting (like multiparty democracy)
> also makes sense in that context. Suppose U(v, x) is voter v's utility
> if x is elected, and suppose that we have two voters with rating functions:
>
> R(v1, x) = a_1 * U(v1, x) + b_1
> R(v2, x) = a_2 * U(v2, x) + b_2
>
> and some very large threshold values A >> B so that a voter v strongly
> approves of every candidate for which if R(v, x) >= A, and strongly
> disapproves of every candidate for which R(v, x) <= B.
>
> Then if we suppose that the values of a and b are bounded in magnitude,
> something kinda like Balinski and Laraki's "common language" idea, then
> as long as candidates are easy to tell apart, then you *can* compare
> different voters' below-B/above-A statements. In addition, the voters
> can more easily classify them, particularly if the difference between
> the sides are clear; if U(v, "my side") >> U(v, "their side"), then the
> distinction is natural.
>
> On the other hand, if U is a sliding scale, then either it's very
> difficult to say just where the cutoffs should be, or the meaning won't
> be preserved.
>
> I have a kind of vague feeling that to the degree the meaning isn't
> clear, honest voters will be incentivized to strategize because there
> are multiple honest ballots. So the harder it is to understand, the
> harder it is to "just stay honest".
>
> Or in the terms of the above: if I'm only willing to disapprove of
> totalitarian dictators, and I see that nobody on the ballot is a
> totalitarian dictator, then I may start thinking "what other use could I
> put this cutoff to?". Which seems to go against the purpose of elections
> as information gathering.
>
> I'm repeating myself, but maybe it'll give a better idea of the hard to
> express intuitive feeling I have that approval cutoffs are hard and have
> very unclear interpretations. Maybe it'll give both of us a better idea,
> even!
>
> -km
>
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