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

Toby Pereira tdp201b at yahoo.co.uk
Mon Nov 16 02:17:09 PST 2020

``` 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. Mono-add-top is just a specific case of participation.
Toby
On Monday, 16 November 2020, 02:32:50 GMT, Rob Lanphier <robla at robla.net> wrote:

Hi folks,

I don't think I've ever fully prerused the "Voting matters"[1] website
before, and realized how serious of a publication it was (or rather
"is"):
[1]: <http://www.votingmatters.org.uk/>

Maybe I did, but I forgot about it.  It looks like there are several
interesting papers to read there.

Douglas Woodall published at least a couple of papers there.  It also
seems that Woodall's definition of "monotonicity"[2] is the favorite
of many folks who like to discuss election methods.  In particular, I
want to highlight Issue 3, paper 5 of "Voting matters":
[2]: http://www.votingmatters.org.uk/ISSUE3/P5.HTM

Here's Woodall's definition of monotonicity from that paper, which
breaks up monotonicity into nine different criteria:
> Monotonicity. A candidate x should not be harmed if:
>
> * (mono-raise) x is raised on some ballots without changing the orders of the other candidates;
> * (mono-raise-delete) x is raised on some ballots and all candidates now below x on those ballots are deleted from them;
> * (mono-raise-random) x is raised on some ballots and the positions now below x on those ballots are filled (or left vacant) in any way that results in a valid ballot;
> * (mono-append) x is added at the end of some ballots that did not previously contain x;
> * (mono-sub-plump) some ballots that do not have x top are replaced by ballots that have x top with no second choice;
> * (mono-sub-top) some ballots that do not have x top are replaced by ballots that have x top (and are otherwise arbitrary);
> * (mono-add-plump) further ballots are added that have x top with no second choice;
> * (mono-add-top) further ballots are added that have x top (and are otherwise arbitrary);
> * (mono-remove-bottom) some ballots are removed, all of which have x bottom, below all other candidates.

That's nine different criteria that all could be called
"monotonicity".  This raises a few questions for me:

1. Is Woodall's definition correct?
2. Is Woodall's definition the most useful?
3. Is Woodall's definition overly-complicated, or just
appropriately-complicated?
4. Does breaking up monotonicity into nine different criteria make it
easier to understand, or harder?
5. Was Woodall just copying his definition from someone else when
publishing those nine criteria?  If so, who?

The reason why I ask: I want to improve the electowiki article about
monotonicity[3], and I'm wondering if emphasis on the nine criteria
above would help make the article clearer:
[3]: https://electowiki.org/wiki/Monotonicity

Should the electowiki community use Woodall's nine criteria as the
electowiki definition of "monotonicity"?

Rob
p.s. a YouTube video[4] posted to C4ES's Discord server is what
inspired me to compose this email:
[4]: https://youtu.be/OI232JSDwDg

p.p.s.: What I mean by "C4ES's Discord Server"[5] is the Discord
server that is operated by the Center for Election Science:
[5]: https://electionscience.org/discord
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