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

[Kristofer Munsterhjelm]

On 10/6/23 06:45, Rob Lanphier wrote:
> Hi folks,
> 
> I've made a change to electowiki's "Summability criterion" article:
> https://electowiki.org/wiki/Summability_criterion 
> <https://electowiki.org/wiki/Summability_criterion>
> 
> Here's the chunk that I added:
> 
>     For batch summability to be true, the following must be true:
> 
>       * Say that candidates "A", "B", and "C" run against each other in
>         an arbitrary election using single-winner electoral system "S"
>       * Say that candidate "A" wins in batch "X" (or precinct "X") when
>         ballots are tabulated using single-winner electoral system "S"
>       * Say that candidate "A" wins in batch "Y" (or precinct "Y") when
>         ballots are tabulated using single-winner electoral system "S"
>       * Therefore, candidate "A" must win when batches "X" and "Y" of
>         ballots are tabulated using single-winner electoral system "S"
>         for "S" to be "batch summable" (and thus, pass the "summability
>         criterion")
> 
> (end of definition)
> 
> Am I correct?  I'm trying to come up with a definition that is easy 
> enough for a layperson to understand, but is also accurate.  I realize 
> that this definition only captures a subset of elections that 
> demonstrate summability problems, but this seems (to me) like the core 
> of the problem with summability.

That sounds more like consistency, so I don't think that's it. 
Summability says nothing about what the results should be in precincts - 
what it does say is more like this:

Let a "summarizing" be a processing step that can be done to an election 
(a set of ballots).

Then for every possible election, the election method should produce the 
same winner when given a summary of that election, as when given that 
election directly;

the amount of data in the summary should not grow too quickly as the 
number of candidates increases;

and there exists a combination algorithm so that if you combine the 
summaries for two precincts, you get the same summary as if you gathered 
both precincts' ballots directly and then made a summary.

(Just what's meant by "should not grow too quickly" requires mathematics 
to explain in more detail.)

Here's an example where Minmax fails Consistency but that shows how 
summability works:

Precinct A:

1: A > B > C > D
6: A > D > B > C
5: B > C > D > A
6: C > D > B > A

with Condorcet matrix summary:

--  7  7  7
11 -- 12  6
11  6 -- 12
11 12  6 --

A wins.

Precinct B:

8: A > B > D > C
2: A > D > C > B
9: C > B > D > A
6: D > C > B > A

with Condorcet matrix:

-- 10 10 10
15 --  8 17
15 17 --  9
15  8 16 --

A wins.

If we tally both precincts as one, we get

1: A > B > C > D
6: A > D > B > C
5: B > C > D > A
6: C > D > B > A
8: A > B > D > C
2: A > D > C > B
9: C > B > D > A
6: D > C > B > A

Condorcet matrix:

-- 17 17 17
26 -- 20 23
26 23 -- 21
26 20 22 --

and C wins. So minmax fails consistency.

But note that each cell in the combined election's Condorcet matrix is 
the sum of that cell in the precincts' Condorcet matrices; and that you 
can determine the minmax social ordering by just using the summary. 
That's what summability means.

(Well, we'd also have to prove that the Condorcet matrix grows slowly 
enough. But since it's got n^2 cells and the combination operation is 
just summing each cell, that follows.)

-km