[EM] Truncation (was re: Defeat Strength)

Forest Simmons forest.simmons21 at gmail.com
Thu Sep 15 13:04:33 PDT 2022


Kristofer,

A sophisticated voter that preferred voting ratings but was forced to use
ranked-rankings style ballots, should first express his ratings in binary
point expansions normalized between zero and one. Then conversion to dyadic
approval ballots is straightforward.

Dyadic approval ballots are just ranked-rankings where two ranking symbols
of the same strength must have a symbol of greater strength somewhere
between them. For example ...

A>B>>C>D>>>E>F>>G>H

The respective binary point expansions corresponding to A through H would
be ...

.111, .110, .101, .100, .011, .010, .001, and .000

An application of dyadic approval that appealed to Kevin was a kind of
reminiscent of Bucklin:

Imagine collapsing the dyadic ballots in stages where stronger and stronger
rank symbols are replaced by equal signs. When on each ballot all symbols
but one have been replaced by equal signs, construct the pairwise matrix
M1for these collapsed ballots.  Then uncollapse a level and construct the
pairwise matrix M2 for this stage. Then uncollapse another level to get M3,
etc.

Now list the alternatives in the pairwise order of M1, taking advantage of
the acyclic nature of M1 being based on a single rank symbol per ballot ...
i.e. equivalent to an approval ballot. Then use M2 to do sorted margins on
that list. Then use M3 to refine that sort, etc.

The final sorted list is the finish order.

You can think of this as a coarse to fine sorting process.

The method is clone free, monotone, and Smith efficient.

It is very easy to program on the basis of binary point ratings.

-Forest

On Thu, Sep 15, 2022, 2:43 AM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> On 9/15/22 05:11, Forest Simmons wrote:
> >
> >
> > On Wed, Sep 14, 2022, 2:19 AM Kristofer Munsterhjelm
> > <km_elmet at t-online.de <mailto:km_elmet at t-online.de>> wrote:
> >
> >     On 9/14/22 09:36, Juho Laatu wrote:
> >      > In addition to that, I still have some interest in the ranked
> >      > rankings style votes (A>>B>C) where one preference step is
> considered
> >      > more important than another step (forming a tree of preferences or
> >      > something like that). I have not done my homework on this (been
> lazy
> >      > for the last decade). Do you know if that approach would likely
> >      > suffer from some (strategic voting or vote counting complexity
> >      > related) problems that would make it unusable?
> >
> >     I think there would be a problem defining just what it means in the
> >     honest case. Consider ranked ballots from a utility perspective: A>B
> >     means that my utility for A is greater than my utility for B.
> >
> > I strongly doubt that quantitative considerations of utility help the
> > average voter decide between A>B,  A= B, and B>A.
> >
> > It might be relevant in a Borda election with sophisticated voters, but
> > not in a Benham election with English "ploughboys voting" as Dodgson put
> it.
>
> I might have overcomplicated things. I was simply trying to formalize
> that whether a honest voter votes A>B, A=B, or B>A depends only on his
> preferences (which are unambiguous) and neither on the method or the
> strength of that preference, apart possibly from A=B, where concerns of
> precision come into account.
>
> My point is that I'm not sure how you could make a similar
> method-agnostic unambiguous definition of what it means for an honest
> voter to vote A>>B instead of A>B.
>
> I guess I'd very much like honest voters to just be able to vote their
> preferences without having to concern themselves with what method is
> doing the counting or which honest vote is the right one.
>
> > It looks to me like you are stuck in the Borda mode with sophisticated
> > voters.
> >
> > In an ordinary  Benham election, if the voter feels ever so slightly
> > that her A>B preference is stronger than her B>C preference, it would be
> > completely appropriate to express that as A>>B>C.  Unlike in the Borda
> > context, the double chevron does not imply that >> is approximately
> > twice as strong as a single chevron preference.
>
> That suggests to me that, in a utilitarian model, you'd vote A>>B if you
> voted C>D and (utility of A - utility of B) > (utility of C - utility of
> D). And similarly that you'd vote E>>>F if there's an A>>B so that
> (utility of E - utilify of F) > (utility of A - utility of B).
>
> Which at least gives an idea of how such notation could be independently
> defined :-)
>
> It feels to me, though, like it would be easier to just ask for ratings,
> making clear that strengths of preference aren't directly taken into
> account (because it's not really a cardinal election), and then have the
> method calculate which gaps are the largest.[1]
>
> That is, in terms of user experience, it feels like asking for something
> like (A: 22, B: 11, C: 8, D: 20, E: 31, F: 0) is much easier than asking
> for
>
> E>>>>A>D>>>>B>>C>>>F
>
> where
>
>  >>>> = rating difference 9
>  >>> = rating difference 8
>  >> = rating difference 3
>  > = rating difference 2
>
> At least it was for me :-)
>
> -km
>
> [1] This poses no problems to von Neumann-Morgenstern because rankings
> of differences are unchanged over arbitrary positive affine
> transformations. Which is another way to say, like you did, that they're
> qualitative.
>
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