[EM] Summability criterion: do I have this right?

[Rob Lanphier]

Thanks everyone for weighing in on this topic, and thanks Kristofer for
fixing up my previous inaccurate summary on electowiki.  After Kristofer
made his changes, I followed up with some shuffling of the prose in the
body of the article.  It's far from perfect, but I think the article is
better than it was last week:
https://electowiki.org/wiki/Summability_criterion

We should also consider making similar changes over on English Wikipedia,
since that article hasn't gotten a lot of love since I restored it in
January 2022.
https://en.wikipedia.org/wiki/Summability_criterion

I suspect I'm not the only one who has conflated the "Summability
criterion" and the "Consistency criterion".  Here's a link to the latter:
https://electowiki.org/wiki/Consistency_criterion
If I'm right, it may be useful to explain the relationship between
"consistency" and "summability" to electowiki readers.

Even as a longtime advocate for Condorcet compliance, I have to concede
that approval voting beats strict Condorcet winner compliance with regards
to consistency and simplicity.  It's much easier to imagine real-time
results consumable by mainstream voters from talking-head newscasters as
they report on election night than it is to imagine what this looks like
with methods that fail the consistency criterion.

Rob

On Sat, Oct 7, 2023 at 2:51 AM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> On 10/7/23 06:26, James Faran wrote:
> > I'm not certain about the second item.  My worry is the following method
> > that someone must have thought of before me and rejected.
> >
> > Given any single ballot, we can use it to create the pairwise matrix.
> > The entries will be +1, 0, or -1. From that pairwise matrix we can
> > reconstruct the preference order given on the ballot. Concatenate all
> > such pairwise matrices. This "summary", an n by n by v array, has size
> > on the order of n^2*v, where n is the number of candidates and v the
> > number of voters. The combination of these is by concatenation in the
> > "v" direction of the array.  This is quadratic in n (just as good as a
> > pairwise method) and linear in v (pairwise methods are logarithmic in
> > v). Plurality is linear in n and logarithmic in v (when v gets bigger we
> > just have to increase the number of digits used to describe the totals).
> > The point is that this method transfers complete ballot information, yet
> > clearly is not a method one would want to use.  And we certainly, in
> > avoiding discussion of polynomial growth, don't want to suddenly need to
> > explain logarithmic growth.
> >
> > As v has a tendency to get bigger than n, I think "polynomial in n" and
> > "logarithmic in v" might be a good standard.  (A rough calculation leads
> > me to believe this method beats just listing the number of ballots of
> > each of the n! types when (n-2)! > v, so is only really a help when n is
> > large.  For 10^7 voters, I think n=14 might do it.)  All in all, as it
> > stands the second point is good enough. We'd just want to avoid
> > responding to a "What does that mean?" question with "You don't want to
> > know," or "You wouldn't understand." However, for studying the question,
> > a more precise definition is needed, lest every method be "summable".
> > Should the amount of calculation time needed to analyze the "summary"
> > come into it?
>
> Yes, the current mathematical standard defined in the Electowiki article
> is "polynomial in candidates, logarithmic in voters" for just the reason
> you mentioned: the number of voters increases much more quickly than the
> number of candidates. Strictly speaking, I suppose polylogarithmic could
> also work, but nobody has ever made such a summary so I shouldn't
> complicate matters too much.
>
> Perhaps something like "should grow slowly in the number of candidates
> and even more slowly in the number of voters". But even that's adding
> more detail to a brief summary.
>
> In any case, my idea was that the summary should be brief, and then
> questions like "what does *slow* mean" can be answered by pointing at
> the mathematical definition, or by a longer elaboration that first
> explains the justification (e.g. "there will be more voters than
> candidates"), and then does a rigorous definition.
>
> -km
>
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