<div dir="ltr">Summability also carries with it the implication that the amount of information a given precinct would need to report is small enough that all of them can reasonably be expected to report the needed data directly to the media and public (without relying on an internet connection) so that anyone anywhere can take that data and come up with the same result as anyone else (which also implies determinism). For example, I like the idea of (Optimal) Proportional Approval Voting with 3 winners and a limit of 10 candidates (being the 10 who gather the most signatures to get on the ballot). This requires at most 120 points of data to be reported from each precinct or county, which is just barely within the realm of what can be reasonably expected without centralization. The goal of this is to maintain the robustness that comes with decentralized systems; it prevents mistakes and attacks from scaling.<div><br></div><div>Warren Smith came up with a more precise term for this concept: efficient parallelizability.</div><div><a href="https://www.rangevoting.org/cgi-bin/DoPassword.cgi">https://www.rangevoting.org/cgi-bin/DoPassword.cgi</a><br></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Fri, Oct 6, 2023 at 1:17 PM <<a href="mailto:election-methods-request@lists.electorama.com">election-methods-request@lists.electorama.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">Send Election-Methods mailing list submissions to<br>
<a href="mailto:election-methods@lists.electorama.com" target="_blank">election-methods@lists.electorama.com</a><br>
<br>
To subscribe or unsubscribe via the World Wide Web, visit<br>
<a href="http://lists.electorama.com/listinfo.cgi/election-methods-electorama.com" rel="noreferrer" target="_blank">http://lists.electorama.com/listinfo.cgi/election-methods-electorama.com</a><br>
<br>
or, via email, send a message with subject or body 'help' to<br>
<a href="mailto:election-methods-request@lists.electorama.com" target="_blank">election-methods-request@lists.electorama.com</a><br>
<br>
You can reach the person managing the list at<br>
<a href="mailto:election-methods-owner@lists.electorama.com" target="_blank">election-methods-owner@lists.electorama.com</a><br>
<br>
When replying, please edit your Subject line so it is more specific<br>
than "Re: Contents of Election-Methods digest..."<br>
<br>
<br>
Today's Topics:<br>
<br>
1. Summability criterion: do I have this right? (Rob Lanphier)<br>
2. Re: Summability criterion: do I have this right?<br>
(Michael Ossipoff)<br>
3. Re: Summability criterion: do I have this right?<br>
(Kristofer Munsterhjelm)<br>
<br>
<br>
----------------------------------------------------------------------<br>
<br>
Message: 1<br>
Date: Thu, 5 Oct 2023 21:45:15 -0700<br>
From: Rob Lanphier <<a href="mailto:roblan@gmail.com" target="_blank">roblan@gmail.com</a>><br>
To: <a href="mailto:election-methods@lists.electorama.com" target="_blank">election-methods@lists.electorama.com</a><br>
Subject: [EM] Summability criterion: do I have this right?<br>
Message-ID:<br>
<CAK9hOYnLGv0EJSg6btH2Ptfg=<a href="mailto:RLK3tV-pvRYo1SeybKp%2BwfZvA@mail.gmail.com" target="_blank">RLK3tV-pvRYo1SeybKp+wfZvA@mail.gmail.com</a>><br>
Content-Type: text/plain; charset="utf-8"<br>
<br>
Hi folks,<br>
<br>
I've made a change to electowiki's "Summability criterion" article:<br>
<a href="https://electowiki.org/wiki/Summability_criterion" rel="noreferrer" target="_blank">https://electowiki.org/wiki/Summability_criterion</a><br>
<br>
Here's the chunk that I added:<br>
<br>
> For batch summability to be true, the following must be true:<br>
><br>
> - Say that candidates "A", "B", and "C" run against each other in an<br>
> arbitrary election using single-winner electoral system "S"<br>
> - Say that candidate "A" wins in batch "X" (or precinct "X") when<br>
> ballots are tabulated using single-winner electoral system "S"<br>
> - Say that candidate "A" wins in batch "Y" (or precinct "Y") when<br>
> ballots are tabulated using single-winner electoral system "S"<br>
> - Therefore, candidate "A" must win when batches "X" and "Y" of<br>
> ballots are tabulated using single-winner electoral system "S" for "S" to<br>
> be "batch summable" (and thus, pass the "summability criterion")<br>
><br>
> (end of definition)<br>
<br>
Am I correct? I'm trying to come up with a definition that is easy enough<br>
for a layperson to understand, but is also accurate. I realize that this<br>
definition only captures a subset of elections that demonstrate summability<br>
problems, but this seems (to me) like the core of the problem with<br>
summability.<br>
<br>
Rob<br>
-------------- next part --------------<br>
An HTML attachment was scrubbed...<br>
URL: <<a href="http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20231005/d595cef3/attachment-0001.htm" rel="noreferrer" target="_blank">http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20231005/d595cef3/attachment-0001.htm</a>><br>
<br>
------------------------------<br>
<br>
Message: 2<br>
Date: Thu, 5 Oct 2023 22:25:20 -0700<br>
From: Michael Ossipoff <<a href="mailto:email9648742@gmail.com" target="_blank">email9648742@gmail.com</a>><br>
To: EM list <<a href="mailto:election-methods@electorama.com" target="_blank">election-methods@electorama.com</a>>, Rob Lanphier<br>
<<a href="mailto:roblan@gmail.com" target="_blank">roblan@gmail.com</a>><br>
Subject: Re: [EM] Summability criterion: do I have this right?<br>
Message-ID:<br>
<<a href="mailto:CAOKDY5CXseAR1xTuvsvTnM5SN1dO_kJyVxDOQmnMUCWmC0atZA@mail.gmail.com" target="_blank">CAOKDY5CXseAR1xTuvsvTnM5SN1dO_kJyVxDOQmnMUCWmC0atZA@mail.gmail.com</a>><br>
Content-Type: text/plain; charset="utf-8"<br>
<br>
All methods require counting at the precincts & at Central.<br>
<br>
Runoff & STAR require information be sent from Central back to the<br>
precincts where an additional count is done, whose results are again sent<br>
to Central for another count.<br>
<br>
RCV is like Runoff & STAR, but the process is repeated. More times, but not<br>
really a qualitative difference.<br>
<br>
On Thu, Oct 5, 2023 at 21:46 Rob Lanphier <<a href="mailto:roblan@gmail.com" target="_blank">roblan@gmail.com</a>> wrote:<br>
<br>
> Hi folks,<br>
><br>
> I've made a change to electowiki's "Summability criterion" article:<br>
> <a href="https://electowiki.org/wiki/Summability_criterion" rel="noreferrer" target="_blank">https://electowiki.org/wiki/Summability_criterion</a><br>
><br>
> Here's the chunk that I added:<br>
><br>
>> For batch summability to be true, the following must be true:<br>
>><br>
>> - Say that candidates "A", "B", and "C" run against each other in an<br>
>> arbitrary election using single-winner electoral system "S"<br>
>> - Say that candidate "A" wins in batch "X" (or precinct "X") when<br>
>> ballots are tabulated using single-winner electoral system "S"<br>
>> - Say that candidate "A" wins in batch "Y" (or precinct "Y") when<br>
>> ballots are tabulated using single-winner electoral system "S"<br>
>> - Therefore, candidate "A" must win when batches "X" and "Y" of<br>
>> ballots are tabulated using single-winner electoral system "S" for "S" to<br>
>> be "batch summable" (and thus, pass the "summability criterion")<br>
>><br>
>> (end of definition)<br>
><br>
> Am I correct? I'm trying to come up with a definition that is easy enough<br>
> for a layperson to understand, but is also accurate. I realize that this<br>
> definition only captures a subset of elections that demonstrate summability<br>
> problems, but this seems (to me) like the core of the problem with<br>
> summability.<br>
><br>
> Rob<br>
> ----<br>
> Election-Methods mailing list - see <a href="https://electorama.com/em" rel="noreferrer" target="_blank">https://electorama.com/em</a> for list<br>
> info<br>
><br>
-------------- next part --------------<br>
An HTML attachment was scrubbed...<br>
URL: <<a href="http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20231005/cae28196/attachment-0001.htm" rel="noreferrer" target="_blank">http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20231005/cae28196/attachment-0001.htm</a>><br>
<br>
------------------------------<br>
<br>
Message: 3<br>
Date: Fri, 6 Oct 2023 12:36:55 +0200<br>
From: Kristofer Munsterhjelm <<a href="mailto:km_elmet@t-online.de" target="_blank">km_elmet@t-online.de</a>><br>
To: Rob Lanphier <<a href="mailto:roblan@gmail.com" target="_blank">roblan@gmail.com</a>>,<br>
<a href="mailto:election-methods@lists.electorama.com" target="_blank">election-methods@lists.electorama.com</a><br>
Subject: Re: [EM] Summability criterion: do I have this right?<br>
Message-ID: <<a href="mailto:7b39c262-69f0-88a2-4a95-f4a3376ad43d@t-online.de" target="_blank">7b39c262-69f0-88a2-4a95-f4a3376ad43d@t-online.de</a>><br>
Content-Type: text/plain; charset=UTF-8; format=flowed<br>
<br>
On 10/6/23 06:45, Rob Lanphier wrote:<br>
> Hi folks,<br>
> <br>
> I've made a change to electowiki's "Summability criterion" article:<br>
> <a href="https://electowiki.org/wiki/Summability_criterion" rel="noreferrer" target="_blank">https://electowiki.org/wiki/Summability_criterion</a> <br>
> <<a href="https://electowiki.org/wiki/Summability_criterion" rel="noreferrer" target="_blank">https://electowiki.org/wiki/Summability_criterion</a>><br>
> <br>
> Here's the chunk that I added:<br>
> <br>
> For batch summability to be true, the following must be true:<br>
> <br>
> * Say that candidates "A", "B", and "C" run against each other in<br>
> an arbitrary election using single-winner electoral system "S"<br>
> * Say that candidate "A" wins in batch "X" (or precinct "X") when<br>
> ballots are tabulated using single-winner electoral system "S"<br>
> * Say that candidate "A" wins in batch "Y" (or precinct "Y") when<br>
> ballots are tabulated using single-winner electoral system "S"<br>
> * Therefore, candidate "A" must win when batches "X" and "Y" of<br>
> ballots are tabulated using single-winner electoral system "S"<br>
> for "S" to be "batch summable" (and thus, pass the "summability<br>
> criterion")<br>
> <br>
> (end of definition)<br>
> <br>
> Am I correct?? I'm trying to come up with a definition that is easy <br>
> enough for a layperson to understand, but is also accurate.? I realize <br>
> that this definition only captures a subset of elections that <br>
> demonstrate summability problems, but this seems (to me) like the core <br>
> of the problem with summability.<br>
<br>
That sounds more like consistency, so I don't think that's it. <br>
Summability says nothing about what the results should be in precincts - <br>
what it does say is more like this:<br>
<br>
Let a "summarizing" be a processing step that can be done to an election <br>
(a set of ballots).<br>
<br>
Then for every possible election, the election method should produce the <br>
same winner when given a summary of that election, as when given that <br>
election directly;<br>
<br>
the amount of data in the summary should not grow too quickly as the <br>
number of candidates increases;<br>
<br>
and there exists a combination algorithm so that if you combine the <br>
summaries for two precincts, you get the same summary as if you gathered <br>
both precincts' ballots directly and then made a summary.<br>
<br>
(Just what's meant by "should not grow too quickly" requires mathematics <br>
to explain in more detail.)<br>
<br>
Here's an example where Minmax fails Consistency but that shows how <br>
summability works:<br>
<br>
Precinct A:<br>
<br>
1: A > B > C > D<br>
6: A > D > B > C<br>
5: B > C > D > A<br>
6: C > D > B > A<br>
<br>
with Condorcet matrix summary:<br>
<br>
-- 7 7 7<br>
11 -- 12 6<br>
11 6 -- 12<br>
11 12 6 --<br>
<br>
A wins.<br>
<br>
Precinct B:<br>
<br>
8: A > B > D > C<br>
2: A > D > C > B<br>
9: C > B > D > A<br>
6: D > C > B > A<br>
<br>
with Condorcet matrix:<br>
<br>
-- 10 10 10<br>
15 -- 8 17<br>
15 17 -- 9<br>
15 8 16 --<br>
<br>
A wins.<br>
<br>
If we tally both precincts as one, we get<br>
<br>
1: A > B > C > D<br>
6: A > D > B > C<br>
5: B > C > D > A<br>
6: C > D > B > A<br>
8: A > B > D > C<br>
2: A > D > C > B<br>
9: C > B > D > A<br>
6: D > C > B > A<br>
<br>
Condorcet matrix:<br>
<br>
-- 17 17 17<br>
26 -- 20 23<br>
26 23 -- 21<br>
26 20 22 --<br>
<br>
and C wins. So minmax fails consistency.<br>
<br>
But note that each cell in the combined election's Condorcet matrix is <br>
the sum of that cell in the precincts' Condorcet matrices; and that you <br>
can determine the minmax social ordering by just using the summary. <br>
That's what summability means.<br>
<br>
(Well, we'd also have to prove that the Condorcet matrix grows slowly <br>
enough. But since it's got n^2 cells and the combination operation is <br>
just summing each cell, that follows.)<br>
<br>
-km<br>
<br>
<br>
------------------------------<br>
<br>
Subject: Digest Footer<br>
<br>
_______________________________________________<br>
Election-Methods mailing list<br>
<a href="mailto:Election-Methods@lists.electorama.com" target="_blank">Election-Methods@lists.electorama.com</a><br>
<a href="http://lists.electorama.com/listinfo.cgi/election-methods-electorama.com" rel="noreferrer" target="_blank">http://lists.electorama.com/listinfo.cgi/election-methods-electorama.com</a><br>
<br>
<br>
------------------------------<br>
<br>
End of Election-Methods Digest, Vol 231, Issue 12<br>
*************************************************<br>
</blockquote></div>