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

[Richard Lung]

Monotonicity.

The transfer of surpluses in STV is monotonic. Not so the redistribution 
of excluded candidates votes.

FAB STV is completely monotonic, because the votes are transferable, in 
a (symmetric) exclusion count, as well as an election count. Hence the B 
in FAB for a bi-nomial count. This does indeed make use of the binomial 
theorem, for higher order counts. Normally, in an ordinary election, a 
first order count would do. That is to say just an election count and an 
exclusion count, unqualified by any higher order binomial theorem 
refinements.

Regards,

Richard L.



On 16/11/2020 02:32, 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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