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I'm not certain about the second item. My worry is the following method that someone must have thought of before me and rejected.</div>
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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.</div>
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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?<br>
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In item 3., perhaps "combined in any order" would work. In my mind that covers both associativity and commutativity.</div>
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Sorry if none of this makes sense. Late night thoughts.</div>
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Jim Faran<br>
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<div id="divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" style="font-size:11pt" color="#000000"><b>From:</b> Election-Methods <election-methods-bounces@lists.electorama.com> on behalf of Kristofer Munsterhjelm <km_elmet@t-online.de><br>
<b>Sent:</b> Friday, October 6, 2023 8:05 PM<br>
<b>To:</b> Rob Lanphier <roblan@gmail.com><br>
<b>Cc:</b> election-methods@lists.electorama.com <election-methods@lists.electorama.com><br>
<b>Subject:</b> Re: [EM] Summability criterion: do I have this right?</font>
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<div class="PlainText">On 10/7/23 01:44, Rob Lanphier wrote:<br>
> Hi Kristofer,<br>
> <br>
> Thanks for the thoughtful responses. It was a long time ago that I <br>
> learned (not to my surprise) that most Wikipedia users only read the <br>
> summary of an article (i.e. the portion before the table of contents), <br>
> and frequently they only skim that. So, here's the Wikipedia article <br>
> about Summability:<br>
> <br>
> ...and here's what is stated in the summary:<br>
> <br>
> /Each vote should be able to be mapped onto a summable array, such<br>
> that its size at most grows polynomially with respect to the amount<br>
> of candidates, the summation operation is associative and<br>
> commutative and the winner could be determined from the array sum<br>
> for all votes cast alone./<br>
> <br>
> /<br>
> /<br>
> The good news: that's almost certainly more accurate than what I added <br>
> to the summary on electowiki (which apparently more accurately describes <br>
> the "Consistency criterion" than it does the Summability criterion). <br>
> The bad news: most peoples' eyes glaze over when they hear math nerds <br>
> explain the difference between "grows polynomially" and "grows <br>
> exponentially". By the time we start talking about big-O notation, <br>
> they've already fled.<br>
> <br>
> Is there a TERSE way of describing the summability criterion that is <br>
> both accurate and doesn't use jargon like "summable array", "associative <br>
> and commutative", and "grows polynomially". It's OKAY if one uses <br>
> jargon if one can explain that jargon to a layperson in a single <br>
> sentence, but it's not great. Just assume you're explaining the concept <br>
> to someone that barely passed American high-school algebra.<br>
<br>
That's what I'm having trouble thinking up. I would say what needs to be <br>
conveyed is that a method is summable if:<br>
<br>
1. Any election can be boiled down to a summary that contains all the <br>
data the method requires to call the election,<br>
2. The storage space required for the summary doesn't grow too quickly <br>
as you add new candidates or voters,<br>
3. and the summaries can be combined (added up) to get the summary for <br>
the combined election.<br>
<br>
That's about as short as I could make the conditions. The "grows <br>
polynomially" part is a formalization of the second condition; the <br>
"commutative and associative" part relates to the third condition.<br>
<br>
My language is definitely far from perfect, but perhaps it'll help guide <br>
others who'd like to try to write a good summary of what summability means.<br>
<br>
-km<br>
----<br>
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