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

[Steve Eppley]

Even the "most useful" monotonicity criterion isn't very important.  Monotonicity strikes me as something to write about when one has a "publish or perish" job in academia.

Much more important than monotonicity are the incentives that the voting method induces on candidates who want to win... in particular whether candidates will want to take positions similar to the positions that the voters would collectively choose in a direct democracy.  In other words, it should be a losing strategy for a candidate to take position X on some issue when there exists a position Y that a large majority prefer over X.  A voting method has a much worse problem than non-monotonicity if a candidate has a good chance to win by taking minority-preferred positions (typically via a majority coalition comprised of those minorities).

Regards,
Steve

On 11/15/2020 9:32 PM, Rob Lanphier 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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