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

Rob Lanphier robla at robla.net
Tue Nov 17 23:45:06 PST 2020

```Hi Kevin,

Thanks for being patient with me and humoring me with this exercise.
Your email reminded me to reply to Steve Eppley, which I did:
<http://lists.electorama.com/pipermail/election-methods-electorama.com/2020-November/002605.html>

In that email, I noted my desire to have a truly excellent article
about the "Monotonicity criterion" (either on Wikipedia or electowiki,
and preferably on both) which is approachable by the layperson.  I
number of hyperlinks and footnotes), but still doesn't bombard the
reader with acronyms, abbreviation and jargon.

The article on electowiki is here:
<https://electowiki.org/wiki/Monotonicity>

...and the corresponding English Wikipedia article is here:
<https://en.wikipedia.org/wiki/Monotonicity_criterion>

Both of those articles bombard the reader with mathematical terms, but
do a poor job of using them consistently.

More below:

On Tue, Nov 17, 2020 at 8:41 PM Kevin Venzke <stepjak at yahoo.fr> wrote:
> 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?

explicit.  "Whenver someone says 'monotonicity criterion', assume they
mean 'mono-raise'".

This whole exercise is to help me more deeply understand the Woodall
paper (in particular, the "Woodall Nine"), which is going to involve a
lot more study on my part.  Why do I want to understand the Woodall
Nine?  Well, I want to understand why he broke it up into nine
different criteria, giving them names (mono-raise, mono-raise-delete,
mono-raise-random, mono-append, mono-sub-plump, mono-sub-top,
why so many election-method experts seem to be conversant in the
distinction between these, but choose to call them all a single
criterion, as if there's a simple pass-fail relationship.

> A couple of the criteria are so hard to satisfy that for most people it's
> probably not worth learning about them.

As far as whether it's worth learning, I think that's a judgement call
I want to leave to the person trying to learn about monotonicity.  My
goal is to understand monotonicity much more deeply than I do today,
so that I'm a more effective teacher on the subject.   That's why I
appreciate how you spelled out the "couple of the criteria":

> 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.

Perfect; thank you!  Like I said, I'm trying to figure out which
criteria I'm going to spend a lot of time trying to teach others via
my writing on electowiki (and elsewhere).

The next portion of your email gives us a very useful taxonomy:

> If you categorize the criteria, doing it by the operation performed gives the
> expected results I'd say:
>
> [1] modifying existing ballots: Mono-raise, Mono-raise-delete,
> Mono-raise-random, Mono-append.
> [...]
> [2] substituting ballots: Mono-sub-top [,] Mono-sub-plump
> [...]

This is incredibly helpful!  I realize I cut out a lot of the
supporting verbiage as I'm "quoting" you, but the trimmed version
gives me a cheat sheet that helps me understand your analysis much
better, and hopefully gives me a framework I can keep in my head to
use when I'm explaining the monotonicity criterion to others.

Thanks for patiently answering my weird questions and accepting my
challenge to make sense of the Woodall Nine.  I'm going to be pretty
busy the rest of this week, but I may decide to start rewriting the
"Monotonicity" article on electowiki next week.  If I do, I'll need to
give you credit somehow.  Thank you!

Rob
```