I'll take a crack at it:<div><br></div><div>In general a voting method is "monotone" if certain kinds of modifications to the ballot set do not decrease the winning probability of the winner. What these kinds of modifications have in common is that a reasonable person would not expect any of them to hurt the winner.</div><div><br></div><div>The taxonomy of different kinds of monotonicity is informed by the specific kinds of modifications contemplated.</div><div><br></div><div> Something like that ... you can do better than that Rob, but something along those lines for an introduction to the concept of monotonicity.</div><div><div><br>On Wednesday, November 18, 2020, <<a href="mailto:election-methods-request@lists.electorama.com">election-methods-request@lists.electorama.com</a>> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">Send Election-Methods mailing list submissions to<br>
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Today's Topics:<br>
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1. Re: What is the most useful definition of "monotonicity"?<br>
(Rob Lanphier)<br>
<br>
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Message: 1<br>
Date: Tue, 17 Nov 2020 23:45:06 -0800<br>
From: Rob Lanphier <<a href="mailto:robla@robla.net">robla@robla.net</a>><br>
To: Kevin Venzke <<a href="mailto:stepjak@yahoo.fr">stepjak@yahoo.fr</a>><br>
Cc: "<a href="mailto:election-methods@lists.electorama.com">election-methods@lists.<wbr>electorama.com</a>"<br>
<<a href="mailto:election-methods@lists.electorama.com">election-methods@lists.<wbr>electorama.com</a>><br>
Subject: Re: [EM] What is the most useful definition of<br>
"monotonicity"?<br>
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<<a href="mailto:CAK9hOYn_OLK7PDxdV6gCJr2aeg85mZAf3dm-M0J7JbQQFZQxJA@mail.gmail.com">CAK9hOYn_<wbr>OLK7PDxdV6gCJr2aeg85mZAf3dm-<wbr>M0J7JbQQFZQxJA@mail.gmail.com</a>><br>
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<br>
Hi Kevin,<br>
<br>
Thanks for being patient with me and humoring me with this exercise.<br>
Your email reminded me to reply to Steve Eppley, which I did:<br>
<<a href="http://lists.electorama.com/pipermail/election-methods-electorama.com/2020-November/002605.html" target="_blank">http://lists.electorama.com/<wbr>pipermail/election-methods-<wbr>electorama.com/2020-November/<wbr>002605.html</a>><br>
<br>
In that email, I noted my desire to have a truly excellent article<br>
about the "Monotonicity criterion" (either on Wikipedia or electowiki,<br>
and preferably on both) which is approachable by the layperson. I<br>
want an article that the reader to learn more (with an appropriate<br>
number of hyperlinks and footnotes), but still doesn't bombard the<br>
reader with acronyms, abbreviation and jargon.<br>
<br>
The article on electowiki is here:<br>
<<a href="https://electowiki.org/wiki/Monotonicity" target="_blank">https://electowiki.org/wiki/<wbr>Monotonicity</a>><br>
<br>
...and the corresponding English Wikipedia article is here:<br>
<<a href="https://en.wikipedia.org/wiki/Monotonicity_criterion" target="_blank">https://en.wikipedia.org/<wbr>wiki/Monotonicity_criterion</a>><br>
<br>
Both of those articles bombard the reader with mathematical terms, but<br>
do a poor job of using them consistently.<br>
<br>
More below:<br>
<br>
On Tue, Nov 17, 2020 at 8:41 PM Kevin Venzke <<a href="mailto:stepjak@yahoo.fr">stepjak@yahoo.fr</a>> wrote:<br>
> I'm not quite seeing the connection between the goal of renaming monotonicity,<br>
> and the introduction (or popularization) of another eight criteria. Woodall<br>
> still referred to it all as monotonicity, so how does his work help?<br>
<br>
Admittedly, "renaming monotonicity" isn't as helpful as being more<br>
explicit. "Whenver someone says 'monotonicity criterion', assume they<br>
mean 'mono-raise'".<br>
<br>
This whole exercise is to help me more deeply understand the Woodall<br>
paper (in particular, the "Woodall Nine"), which is going to involve a<br>
lot more study on my part. Why do I want to understand the Woodall<br>
Nine? Well, I want to understand why he broke it up into nine<br>
different criteria, giving them names (mono-raise, mono-raise-delete,<br>
mono-raise-random, mono-append, mono-sub-plump, mono-sub-top,<br>
mono-add-plump, mono-add-top, mono-remove-bottom). I'm also wondering<br>
why so many election-method experts seem to be conversant in the<br>
distinction between these, but choose to call them all a single<br>
criterion, as if there's a simple pass-fail relationship.<br>
<br>
> A couple of the criteria are so hard to satisfy that for most people it's<br>
> probably not worth learning about them.<br>
<br>
As far as whether it's worth learning, I think that's a judgement call<br>
I want to leave to the person trying to learn about monotonicity. My<br>
goal is to understand monotonicity much more deeply than I do today,<br>
so that I'm a more effective teacher on the subject. That's why I<br>
appreciate how you spelled out the "couple of the criteria":<br>
<br>
> Namely, Mono-raise-random and Mono-sub-top are apparently only satisfied by<br>
> FPP and certain top-heavy Borda-like rules that Woodall didn't even see fit<br>
> to name. His motivation for having these criteria was clearly not advocacy.<br>
<br>
Perfect; thank you! Like I said, I'm trying to figure out which<br>
criteria I'm going to spend a lot of time trying to teach others via<br>
my writing on electowiki (and elsewhere).<br>
<br>
The next portion of your email gives us a very useful taxonomy:<br>
<br>
> If you categorize the criteria, doing it by the operation performed gives the<br>
> expected results I'd say:<br>
><br>
> [1] modifying existing ballots: Mono-raise, Mono-raise-delete,<br>
> Mono-raise-random, Mono-append.<br>
> [...]<br>
> [2] substituting ballots: Mono-sub-top [,] Mono-sub-plump<br>
> [...]<br>
> [3] adding/dropping ballots: Mono-add-top, Mono-remove-bottom, Mono-add-plump.<br>
<br>
This is incredibly helpful! I realize I cut out a lot of the<br>
supporting verbiage as I'm "quoting" you, but the trimmed version<br>
gives me a cheat sheet that helps me understand your analysis much<br>
better, and hopefully gives me a framework I can keep in my head to<br>
use when I'm explaining the monotonicity criterion to others.<br>
<br>
Thanks for patiently answering my weird questions and accepting my<br>
challenge to make sense of the Woodall Nine. I'm going to be pretty<br>
busy the rest of this week, but I may decide to start rewriting the<br>
"Monotonicity" article on electowiki next week. If I do, I'll need to<br>
give you credit somehow. Thank you!<br>
<br>
Rob<br>
<br>
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End of Election-Methods Digest, Vol 196, Issue 16<br>
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