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

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

Rob Lanphier
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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Re: [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
> ----
> Election-Methods mailing list - see https://electorama.com/em for list info
----
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Re: [EM] What is the most useful definition of "monotonicity"?

Rob Lanphier
In reply to this post by Rob Lanphier
Hi Kevin,

Thanks for the prompt and helpful reply!   I think I know how I want
to go about editing the "Monotonicity" article on electowiki, but that
article needs a lot more work than just this addition.

More inline:

On Sun, Nov 15, 2020 at 6:59 PM Kevin Venzke <[hidden email]> wrote:
> For what people usually mean when discussing monotonicity (such as in
> connection with IRV) this corresponds to Woodall's Mono-raise.

Given that "Monotonicity" is more-or-less equivalent to "mono-raise"
(both by your reckoning, and by my previous understanding), I'd like
to use this as a tool to simplify the "Monotonicity" article on
electowiki, moving the more technical mathematical definitions out of
the summary of the article

> 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.

I've seen other folks use at least some of the nine criteria that
Woodall defines, so I'd like to define them somewhere on electowiki.
I'll give it some thought.

One relatively simple restatement comes from the Wikipedia article
about Arrow's theorem:
<https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem>

Here's the way that the Wikipedia article restates monotonicity:
"If any individual modifies his or her preference order by promoting a
certain option, then the societal preference order should respond only
by promoting that same option or not changing, never by placing it
lower than before. An individual should not be able to hurt an option
by ranking it higher."

More ways of thinking about it....

> 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.

Thanks for your advice! I'll give all of this more thought before I
make big changes to that electowiki article.

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

Steve Eppley
In reply to this post by Rob Lanphier
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
> ----
> Election-Methods mailing list - see https://electorama.com/em for list info

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

Rob Lanphier
In reply to this post by Rob Lanphier
Thanks Toby!  Given the private and public responses I've gotten to my
question, and the deeper understanding of "the Monotonicity criterion"
that I'm developing, I think that it makes sense to figure out how
"the Woodall nine" (as I'll start calling them) map to "monotonicity"
when talking to people who understand matrix addition.

More below...

On Mon, Nov 16, 2020 at 2:17 AM Toby Pereira <[hidden email]> wrote:
> 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.

Let's use the Woodall Nine to talk about this.  What would you say the
mapping is?  I started coming up with my own version of the list
(which was serious), but this became a self-parody.  The format of
each line is

* woodall-name - robla-named-criterion ("Proposed Wikipedia/electowiki Article")

Here's the mappings as I saw them on my first runthough:

* mono-raise - dont-muck-with-ballots-criterion ("Monotonicity (Classic)")
* mono-raise-delete - participation-criterion ("Monotonicity (Nuke ballots)")
* mono-raise-random - adding-random-ballots-causes-mayhem-criterion
("Monotonicity (Add random)")
* mono-append - fill-in-ballot-bubbles-criterion ("Monotonicity (Add
preference)")
* mono-sub-plump - replace-ballots-criterion ("Monotonicity (Erase preference)")
* mono-sub-top - replace-with-new-winner-criterion ("Monotonicity ()")
* mono-add-plump -
add-incompletely-ranked-ballots-to-cause-loss-criterion ("Monotonicity
(Add incomplete ballots)")
* mono-add-top - add-top-scored-but-mostly-random-criterion
("Monotonicity (Add mostly random)")
* mono-remove-bottom -
remove-bottom-ranked-Condorcet-loser-ballots-criterion ("Monotonicity
(Remove Condorcet loser haters)")

I think this thing needs to exist in the world, so I'm actually going
to keep working on it here:

<https://electowiki.org/wiki/User:RobLa/Woodall-to-RobLa>

Toby, I would love for you and others on this mailing list to come up
with more serious versions.  I spent about an hour trying to be
serious when coming up with my mappings, and I think it was a valuable
exercise.  My fear: there are many electoral reform advocates who will
say  "I understand the monotonicity criterion".  However, most of
those people wouldn't be able to answer this question: "what do YOU
mean when you say 'monotonicity criterion'?"  My guess: they would
deflect, and say "I know someone who knows what it means, and I've
never really understood it, but I THINK it means this: xyz", and then
their "xyz" would be an oversimplified example involving Woodall's
"mono-raise" or perhaps "Participation" or something vaguely related
to Pareto efficiency.  They probably don't understand very well, and
my hunch is that most people on this mailing list are overconfident
with THEIR understanding of "monotonicity".

When someone asks "What is the 'monotonicity criterion'?", it seems
the correct answer is to throw the question back at them: "before I
explain: do you know what 'monotonicity' is?".  If it seems they
aren't very well-versed in advanced mathematics, and they aren't
familiar with what a monotonically-increasing sequence is, then assume
that it's time to back off the math jargon.  I think "monotonicity
criterion" seems to be a terrible name for talking about electoral
reform.

That's the reason why I want to talk about the Woodall Nine.  I'd like
to come up with layperson names for each of them, and possibly map
them to various criteria that have Wikipedia articles about them.  If
there's not yet a Wikipedia article, maybe an electowiki article.  If
there's not yet an electowiki article, maybe a section of an
electowiki article.

Responding to one last bit of your email:

> Mono-add-top is just a specific case of participation.

Are there any other mono-woodall methods that are subsets of the
Participation criterion?
<https://electowiki.org/wiki/Participation_criterion>

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

Toby Pereira
I actually meant to say that mono-add-plump, not mono-add-top, that is just a specific case of participation. The new voter is only ranking one candidate, so if that causes the candidate to lose, then it is a failure of participation. But anyway, I think this is good work that you're doing. I'll have a look through your list.

Toby

On Tuesday, 17 November 2020, 08:59:30 GMT, Rob Lanphier <[hidden email]> wrote:


Thanks Toby!  Given the private and public responses I've gotten to my
question, and the deeper understanding of "the Monotonicity criterion"
that I'm developing, I think that it makes sense to figure out how
"the Woodall nine" (as I'll start calling them) map to "monotonicity"
when talking to people who understand matrix addition.

More below...

On Mon, Nov 16, 2020 at 2:17 AM Toby Pereira <[hidden email]> wrote:
> 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.

Let's use the Woodall Nine to talk about this.  What would you say the
mapping is?  I started coming up with my own version of the list
(which was serious), but this became a self-parody.  The format of
each line is

* woodall-name - robla-named-criterion ("Proposed Wikipedia/electowiki Article")

Here's the mappings as I saw them on my first runthough:

* mono-raise - dont-muck-with-ballots-criterion ("Monotonicity (Classic)")
* mono-raise-delete - participation-criterion ("Monotonicity (Nuke ballots)")
* mono-raise-random - adding-random-ballots-causes-mayhem-criterion
("Monotonicity (Add random)")
* mono-append - fill-in-ballot-bubbles-criterion ("Monotonicity (Add
preference)")
* mono-sub-plump - replace-ballots-criterion ("Monotonicity (Erase preference)")
* mono-sub-top - replace-with-new-winner-criterion ("Monotonicity ()")
* mono-add-plump -
add-incompletely-ranked-ballots-to-cause-loss-criterion ("Monotonicity
(Add incomplete ballots)")
* mono-add-top - add-top-scored-but-mostly-random-criterion
("Monotonicity (Add mostly random)")
* mono-remove-bottom -
remove-bottom-ranked-Condorcet-loser-ballots-criterion ("Monotonicity
(Remove Condorcet loser haters)")

I think this thing needs to exist in the world, so I'm actually going
to keep working on it here:

<https://electowiki.org/wiki/User:RobLa/Woodall-to-RobLa>

Toby, I would love for you and others on this mailing list to come up
with more serious versions.  I spent about an hour trying to be
serious when coming up with my mappings, and I think it was a valuable
exercise.  My fear: there are many electoral reform advocates who will
say  "I understand the monotonicity criterion".  However, most of
those people wouldn't be able to answer this question: "what do YOU
mean when you say 'monotonicity criterion'?"  My guess: they would
deflect, and say "I know someone who knows what it means, and I've
never really understood it, but I THINK it means this: xyz", and then
their "xyz" would be an oversimplified example involving Woodall's
"mono-raise" or perhaps "Participation" or something vaguely related
to Pareto efficiency.  They probably don't understand very well, and
my hunch is that most people on this mailing list are overconfident
with THEIR understanding of "monotonicity".

When someone asks "What is the 'monotonicity criterion'?", it seems
the correct answer is to throw the question back at them: "before I
explain: do you know what 'monotonicity' is?".  If it seems they
aren't very well-versed in advanced mathematics, and they aren't
familiar with what a monotonically-increasing sequence is, then assume
that it's time to back off the math jargon.  I think "monotonicity
criterion" seems to be a terrible name for talking about electoral
reform.

That's the reason why I want to talk about the Woodall Nine.  I'd like
to come up with layperson names for each of them, and possibly map
them to various criteria that have Wikipedia articles about them.  If
there's not yet a Wikipedia article, maybe an electowiki article.  If
there's not yet an electowiki article, maybe a section of an
electowiki article.

Responding to one last bit of your email:

> Mono-add-top is just a specific case of participation.

Are there any other mono-woodall methods that are subsets of the
Participation criterion?
<https://electowiki.org/wiki/Participation_criterion>


Rob

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

Toby Pereira
Just to add to this - monotonicity looks at the fate of the candidate, whereas participation looks at the fate of the voter. But where only one candidate is listed on the ballot, their fates are intertwined.

On Tuesday, 17 November 2020, 09:19:22 GMT, Toby Pereira <[hidden email]> wrote:


I actually meant to say that mono-add-plump, not mono-add-top, that is just a specific case of participation. The new voter is only ranking one candidate, so if that causes the candidate to lose, then it is a failure of participation. But anyway, I think this is good work that you're doing. I'll have a look through your list.

Toby

On Tuesday, 17 November 2020, 08:59:30 GMT, Rob Lanphier <[hidden email]> wrote:


Thanks Toby!  Given the private and public responses I've gotten to my
question, and the deeper understanding of "the Monotonicity criterion"
that I'm developing, I think that it makes sense to figure out how
"the Woodall nine" (as I'll start calling them) map to "monotonicity"
when talking to people who understand matrix addition.

More below...

On Mon, Nov 16, 2020 at 2:17 AM Toby Pereira <[hidden email]> wrote:
> 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.

Let's use the Woodall Nine to talk about this.  What would you say the
mapping is?  I started coming up with my own version of the list
(which was serious), but this became a self-parody.  The format of
each line is

* woodall-name - robla-named-criterion ("Proposed Wikipedia/electowiki Article")

Here's the mappings as I saw them on my first runthough:

* mono-raise - dont-muck-with-ballots-criterion ("Monotonicity (Classic)")
* mono-raise-delete - participation-criterion ("Monotonicity (Nuke ballots)")
* mono-raise-random - adding-random-ballots-causes-mayhem-criterion
("Monotonicity (Add random)")
* mono-append - fill-in-ballot-bubbles-criterion ("Monotonicity (Add
preference)")
* mono-sub-plump - replace-ballots-criterion ("Monotonicity (Erase preference)")
* mono-sub-top - replace-with-new-winner-criterion ("Monotonicity ()")
* mono-add-plump -
add-incompletely-ranked-ballots-to-cause-loss-criterion ("Monotonicity
(Add incomplete ballots)")
* mono-add-top - add-top-scored-but-mostly-random-criterion
("Monotonicity (Add mostly random)")
* mono-remove-bottom -
remove-bottom-ranked-Condorcet-loser-ballots-criterion ("Monotonicity
(Remove Condorcet loser haters)")

I think this thing needs to exist in the world, so I'm actually going
to keep working on it here:

<https://electowiki.org/wiki/User:RobLa/Woodall-to-RobLa>

Toby, I would love for you and others on this mailing list to come up
with more serious versions.  I spent about an hour trying to be
serious when coming up with my mappings, and I think it was a valuable
exercise.  My fear: there are many electoral reform advocates who will
say  "I understand the monotonicity criterion".  However, most of
those people wouldn't be able to answer this question: "what do YOU
mean when you say 'monotonicity criterion'?"  My guess: they would
deflect, and say "I know someone who knows what it means, and I've
never really understood it, but I THINK it means this: xyz", and then
their "xyz" would be an oversimplified example involving Woodall's
"mono-raise" or perhaps "Participation" or something vaguely related
to Pareto efficiency.  They probably don't understand very well, and
my hunch is that most people on this mailing list are overconfident
with THEIR understanding of "monotonicity".

When someone asks "What is the 'monotonicity criterion'?", it seems
the correct answer is to throw the question back at them: "before I
explain: do you know what 'monotonicity' is?".  If it seems they
aren't very well-versed in advanced mathematics, and they aren't
familiar with what a monotonically-increasing sequence is, then assume
that it's time to back off the math jargon.  I think "monotonicity
criterion" seems to be a terrible name for talking about electoral
reform.

That's the reason why I want to talk about the Woodall Nine.  I'd like
to come up with layperson names for each of them, and possibly map
them to various criteria that have Wikipedia articles about them.  If
there's not yet a Wikipedia article, maybe an electowiki article.  If
there's not yet an electowiki article, maybe a section of an
electowiki article.

Responding to one last bit of your email:

> Mono-add-top is just a specific case of participation.

Are there any other mono-woodall methods that are subsets of the
Participation criterion?
<https://electowiki.org/wiki/Participation_criterion>


Rob

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

Kevin Venzke
In reply to this post by Rob Lanphier
Hi Rob,

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?

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

If you categorize the criteria, doing it by the operation performed gives the expected results I'd say:

modifying existing ballots: Mono-raise, Mono-raise-delete, Mono-raise-random, Mono-append. All four of these involve raising a candidate, except for the odd edge case that happens to exist within Woodall's framework, that "appending" the least liked candidate of the ballot could actually fail to raise him over anyone. If your framework doesn't support an "append" operation with that effect, then Mono-raise implies Mono-append, and Mono-raise-random implies all of the other three.

substituting ballots: Mono-sub-top and Mono-sub-plump can be categorized the same as Mono-raise because they can be conceived of as special cases (weaker forms) of Mono-raise-random and Mono-raise-delete respectively. (They aren't so much weaker though... Woodall doesn't offer any methods that satisfy the easier criterion of the pair and not the harder one.)

adding/dropping ballots: Mono-add-top, Mono-remove-bottom, Mono-add-plump. These are all implied by Participation (though Participation is much stronger than any of these). Note that you can rephrase Mono-remove-bottom to be about adding ballots.

Kevin



Le mardi 17 novembre 2020 à 02:58:35 UTC−6, Rob Lanphier <[hidden email]> a écrit :


Thanks Toby!  Given the private and public responses I've gotten to my
question, and the deeper understanding of "the Monotonicity criterion"
that I'm developing, I think that it makes sense to figure out how
"the Woodall nine" (as I'll start calling them) map to "monotonicity"
when talking to people who understand matrix addition.

More below...

On Mon, Nov 16, 2020 at 2:17 AM Toby Pereira <[hidden email]> wrote:
> 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.

Let's use the Woodall Nine to talk about this.  What would you say the
mapping is?  I started coming up with my own version of the list
(which was serious), but this became a self-parody.  The format of
each line is

* woodall-name - robla-named-criterion ("Proposed Wikipedia/electowiki Article")

Here's the mappings as I saw them on my first runthough:

* mono-raise - dont-muck-with-ballots-criterion ("Monotonicity (Classic)")
* mono-raise-delete - participation-criterion ("Monotonicity (Nuke ballots)")
* mono-raise-random - adding-random-ballots-causes-mayhem-criterion
("Monotonicity (Add random)")
* mono-append - fill-in-ballot-bubbles-criterion ("Monotonicity (Add
preference)")
* mono-sub-plump - replace-ballots-criterion ("Monotonicity (Erase preference)")
* mono-sub-top - replace-with-new-winner-criterion ("Monotonicity ()")
* mono-add-plump -
add-incompletely-ranked-ballots-to-cause-loss-criterion ("Monotonicity
(Add incomplete ballots)")
* mono-add-top - add-top-scored-but-mostly-random-criterion
("Monotonicity (Add mostly random)")
* mono-remove-bottom -
remove-bottom-ranked-Condorcet-loser-ballots-criterion ("Monotonicity
(Remove Condorcet loser haters)")

I think this thing needs to exist in the world, so I'm actually going
to keep working on it here:


Toby, I would love for you and others on this mailing list to come up
with more serious versions.  I spent about an hour trying to be
serious when coming up with my mappings, and I think it was a valuable
exercise.  My fear: there are many electoral reform advocates who will
say  "I understand the monotonicity criterion".  However, most of
those people wouldn't be able to answer this question: "what do YOU
mean when you say 'monotonicity criterion'?"  My guess: they would
deflect, and say "I know someone who knows what it means, and I've
never really understood it, but I THINK it means this: xyz", and then
their "xyz" would be an oversimplified example involving Woodall's
"mono-raise" or perhaps "Participation" or something vaguely related
to Pareto efficiency.  They probably don't understand very well, and
my hunch is that most people on this mailing list are overconfident
with THEIR understanding of "monotonicity".

When someone asks "What is the 'monotonicity criterion'?", it seems
the correct answer is to throw the question back at them: "before I
explain: do you know what 'monotonicity' is?".  If it seems they
aren't very well-versed in advanced mathematics, and they aren't
familiar with what a monotonically-increasing sequence is, then assume
that it's time to back off the math jargon.  I think "monotonicity
criterion" seems to be a terrible name for talking about electoral
reform.

That's the reason why I want to talk about the Woodall Nine.  I'd like
to come up with layperson names for each of them, and possibly map
them to various criteria that have Wikipedia articles about them.  If
there's not yet a Wikipedia article, maybe an electowiki article.  If
there's not yet an electowiki article, maybe a section of an
electowiki article.

Responding to one last bit of your email:

> Mono-add-top is just a specific case of participation.

Are there any other mono-woodall methods that are subsets of the
Participation criterion?

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

Rob Lanphier
In reply to this post by Rob Lanphier
Hi Steve,

Great hearing from you!  Weirdly, I didn't receive your email from the
mailing list, but I saw your note in the archive.[a]  Since you were
frequently a lonely voice of sanity in the early days of the EM list,
I was baffled as to how your email might have been spam filtered.
Anyway, I've reconstructed your message the best I could from the
archive so that I can reply to it.  More inline below....

[a]: http://lists.electorama.com/pipermail/election-methods-electorama.com/2020-November/002598.html

On 11/16/2020 11:02 PST, Steve Eppley wrote:
> 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.

Do you think it's really that unimportant?  It seems that the folks
that like to make a big deal about the alleged superiority of
"cardinal voting systems" like to talk about this one, and not talk
about any of the Condorcet criteria (Condorcet winner, Condorcet
loser, or any of the other criteria named after the 18th century dead
guy).  "Condorcet Winner" is not that hard to explain, and I think
it's (by far and away) the most important criterion that failed in
Burlington in the 2009 mayoral race[b].

[b]: https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election

To be clear, I fully agree with the remainder of your message about
minority coalitions to form a majority..  We may want to start a
separate thread about that.

But I'd like to keep discussing the monotonicity, since I keep hearing
about it in places I don't expect it. For example, it often comes up
in presentations made by folks at the Center for Election Science, and
prominently their website [c][d).

[c]: https://electionscience.org/learn/library/
[d]: https://electionscience.org/library/monotonicity/

I fancy myself an election methods expert, and I want to be able to go
toe-to-toe with any "publish or perish" tenure-track professor who
tries to gish gallop the readers of an online forum (like this one, or
like electowiki, or Wikipedia) by deliberately introducing ambiguous
criteria to introduce confusion.  There are a lot of people editing
Wikipedia, and some of them have an undisclosed conflict-of-interest
when they do it.

I'll say more in my response to Kevin Venzke (who has also been a
consistent voice of sanity).  In short, I really appreciate that other
criteria are probably more important (in addition to guarding against
other ways of gaming the system).  But I want to make sure there's a
truly excellent article about the "Monotonicity criterion" (either on
Wikipedia or electowiki, and preferably on both) which is approachable
by the layperson.  I want an article that the reader to learn more
(with an appropriate number of hyperlinks and footnotes), and that
doesn't implicitly tell the reader "you don't need to know that. Only
NERDS care about that".  As of right now, I fear that such an article
doesn't exist.

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

Rob Lanphier
In reply to this post by Kevin Venzke
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
want an article that the reader to learn more (with an appropriate
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 <[hidden email]> 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?

Admittedly, "renaming monotonicity" isn't as helpful as being more
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,
mono-add-plump, mono-add-top, mono-remove-bottom).  I'm also wondering
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
> [...]
> [3] adding/dropping ballots: Mono-add-top, Mono-remove-bottom, Mono-add-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
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Re: [EM] What is the most useful definition of "monotonicity"?

Kevin Venzke
Hi Rob,

I see the wiki pages clearly talk about Mono-raise... Maybe the discussion is confusing, but won't it be worse if you discuss another eight criteria, or favor a less common term for the concept? I could see fleshing out the Woodall section at the bottom, calling out the ones that are Mono-raise variants vs. the other three that are implied by Participation (and so not as obviously relevant to the article's main topic).

A couple more responses:

>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,
>mono-add-plump, mono-add-top, mono-remove-bottom). 

(KV) Rereading the articles I have to concede that he did, at least, think they were important... He states in particular a preference for DAC over IRV, due to these criteria, with DAC satisfying 7 of them vs. IRV's 3. But it certainly appears he devised the criteria before inventing DAC.

>I'm also wondering
>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.

(KV) I don't think that's what's happening, I think it's just code-switching based on what terminology will be familiar to the audience one is talking to at that moment. One can't possibly lump them all together as one criterion because basically FPP is the only method that satisfies them all.

Kevin


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

Kristofer Munsterhjelm-3
In reply to this post by Rob Lanphier
On 16/11/2020 03.32, Rob Lanphier wrote:

> 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?

"Monotonicity" is almost always mono-raise. So maybe the Monotonicity
article could detail mono-raise and then have a disambiguation
section or something like it for the other types of monotonicity; kind
of like how the clone independence page is mainly about Tideman's
concept, although Woodall split the clone independence criterion into
three of his own (clone-no-help, clone-no-harm, clone-in).

I'm not myself aware of any paper that uses unqualified "monotonicity"
to mean something else than mono-raise. Authors that refer to e.g.
mono-add-top tend to say mono-add-top.
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