# [EM] What is the most useful definition of "monotonicity"?

Kevin Venzke stepjak at yahoo.fr
Tue Nov 17 20:38:26 PST 2020

``` Hi Rob,
I'm not quite seeing the connection between the goal of renaming monotonicity, and the introduction (or popularization) of another eight criteria. Woodall still referred to it all as monotonicity, so how does his work help?
A couple of the criteria are so hard to satisfy that for most people it's probably not worth learning about them. Namely, Mono-raise-random and Mono-sub-top are apparently only satisfied by FPP and certain top-heavy Borda-like rules that Woodall didn't even see fit to name. His motivation for having these criteria was clearly not advocacy...
If you categorize the criteria, doing it by the operation performed gives the expected results I'd say:
modifying existing ballots: Mono-raise, Mono-raise-delete, Mono-raise-random, Mono-append. All four of these involve raising a candidate, except for the odd edge case that happens to exist within Woodall's framework, that "appending" the least liked candidate of the ballot could actually fail to raise him over anyone. If your framework doesn't support an "append" operation with that effect, then Mono-raise implies Mono-append, and Mono-raise-random implies all of the other three.
substituting ballots: Mono-sub-top and Mono-sub-plump can be categorized the same as Mono-raise because they can be conceived of as special cases (weaker forms) of Mono-raise-random and Mono-raise-delete respectively. (They aren't so much weaker though... Woodall doesn't offer any methods that satisfy the easier criterion of the pair and not the harder one.)
adding/dropping ballots: Mono-add-top, Mono-remove-bottom, Mono-add-plump. These are all implied by Participation (though Participation is much stronger than any of these). Note that you can rephrase Mono-remove-bottom to be about adding ballots.
Kevin

Le mardi 17 novembre 2020 à 02:58:35 UTC−6, Rob Lanphier <robla at robla.net> a écrit :

Thanks Toby!  Given the private and public responses I've gotten to my
question, and the deeper understanding of "the Monotonicity criterion"
that I'm developing, I think that it makes sense to figure out how
"the Woodall nine" (as I'll start calling them) map to "monotonicity"
when talking to people who understand matrix addition.

More below...

On Mon, Nov 16, 2020 at 2:17 AM Toby Pereira <tdp201b at yahoo.co.uk> wrote:
> I think the scenarios where new ballots are added have more in common with
> participation than monotonicity, in terms of how it is normally understood in
> voting theory at least.

Let's use the Woodall Nine to talk about this.  What would you say the
mapping is?  I started coming up with my own version of the list
(which was serious), but this became a self-parody.  The format of
each line is

* woodall-name - robla-named-criterion ("Proposed Wikipedia/electowiki Article")

Here's the mappings as I saw them on my first runthough:

* mono-raise - dont-muck-with-ballots-criterion ("Monotonicity (Classic)")
* mono-raise-delete - participation-criterion ("Monotonicity (Nuke ballots)")
* mono-raise-random - adding-random-ballots-causes-mayhem-criterion
* mono-append - fill-in-ballot-bubbles-criterion ("Monotonicity (Add
preference)")
* mono-sub-plump - replace-ballots-criterion ("Monotonicity (Erase preference)")
* mono-sub-top - replace-with-new-winner-criterion ("Monotonicity ()")
("Monotonicity (Add mostly random)")
* mono-remove-bottom -
remove-bottom-ranked-Condorcet-loser-ballots-criterion ("Monotonicity
(Remove Condorcet loser haters)")

I think this thing needs to exist in the world, so I'm actually going
to keep working on it here:

<https://electowiki.org/wiki/User:RobLa/Woodall-to-RobLa>

Toby, I would love for you and others on this mailing list to come up
with more serious versions.  I spent about an hour trying to be
serious when coming up with my mappings, and I think it was a valuable
exercise.  My fear: there are many electoral reform advocates who will
say  "I understand the monotonicity criterion".  However, most of
those people wouldn't be able to answer this question: "what do YOU
mean when you say 'monotonicity criterion'?"  My guess: they would
deflect, and say "I know someone who knows what it means, and I've
never really understood it, but I THINK it means this: xyz", and then
their "xyz" would be an oversimplified example involving Woodall's
"mono-raise" or perhaps "Participation" or something vaguely related
to Pareto efficiency.  They probably don't understand very well, and
my hunch is that most people on this mailing list are overconfident
with THEIR understanding of "monotonicity".

When someone asks "What is the 'monotonicity criterion'?", it seems
the correct answer is to throw the question back at them: "before I
explain: do you know what 'monotonicity' is?".  If it seems they
aren't very well-versed in advanced mathematics, and they aren't
familiar with what a monotonically-increasing sequence is, then assume
that it's time to back off the math jargon.  I think "monotonicity
criterion" seems to be a terrible name for talking about electoral
reform.

That's the reason why I want to talk about the Woodall Nine.  I'd like
to come up with layperson names for each of them, and possibly map
them to various criteria that have Wikipedia articles about them.  If
there's not yet a Wikipedia article, maybe an electowiki article.  If
there's not yet an electowiki article, maybe a section of an
electowiki article.

Responding to one last bit of your email:

> Mono-add-top is just a specific case of participation.

Are there any other mono-woodall methods that are subsets of the
Participation criterion?
<https://electowiki.org/wiki/Participation_criterion>

Rob
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