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

[Kevin Venzke]

 Hi Rob,
For what people usually mean when discussing monotonicity (such as in connection with IRV) this corresponds to Woodall's Mono-raise.
I'd say the criterion discussed second most often is Mono-add-top, a weaker form of Participation.
I don't see how Woodall can be wrong, it's his choice to define these criteria and characterize them as dealing with monotonicity. But to me, if you say a method "fails monotonicity," or "is monotone," you're either talking about Mono-raise or else you're being unclear. Inserting these into an article called "Monotonicity" could be unhelpful unless it's made clear that these are other criteria connected to the idea, and are used (as far as I know) primarily by Woodall.
Note that (at least in other articles) Woodall is/was largely concerned with which sets of properties are compatible with each other, and he didn't usually make statements judging the relative value of criteria. This leaves the job to other people to explain why we should care about a given property.
Kevin

    Le dimanche 15 novembre 2020 à 20:32:50 UTC−6, Rob Lanphier <robla at robla.net> a écrit :  
 
 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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