[EM] (no subject)

Wed Aug 25 12:58:00 PDT 2010

To everyone, I get these messages in digest form, so sometimes I miss 
things.  If you have comments on my manuscript, go to my blog:

http://votingmath.blogspot.com/2010/08/open-thread-for-comments-on-manuscript.html

Warren,

Thanks for the comments and the corrections.  I have changed the paper to spell all names properly and give credit to Kevin Venzke.  I now include a link to the Electowiki entry on MDDA as well as the longer article by you and Mike Ossipoff at RangeVoting.org.  A table of contents is also a great suggestion, and I've added it.

I don't have time to address everything now, but on the general issue of style, well, I plead guilty to many errors and thank you for the feedback.  I'm a theoretical physicist, not a mathematician, and this is the first time I've tried to write something that involves a proof rather than a derivation or calculation.  I'll work on it over time, but I can't promise immediate changes in the next draft.  Not being a mathematician, I tried to emulate a lot of Saari's style from his book "Basic Geometry of Voting" since he is a pro and I use some geometrical ideas from his book.  But clearly I have a long way to go.

The idea of the linearity condition is simply that the statements being checked are all linear inequalities.  In the very simplest case of plurality voting, candidate A would win if:
f(votes for A) - f(votes for B) > 0
AND
f(votes for A) - f(votes for C) > 0
AND
etc. for all other candidates

where f(votes for A) is the fraction of voters casting ballots for A.  If we were doing a more complicated method, we'd specify more elaborate inequalities, regarding (perhaps) the fraction of voters giving A a certain number of points, or the fraction ranking A above B, or whatever.

The key point is that those inequalities are all linear in the fraction casting each ballot type.  However, somebody could always come along and say "Hey, I want a rule where we check whether (f(votes for A))^2 - f(votes for B) > 0!"  Why would somebody write down that rule?  I dunno.  Seems illogical.  But they could.  I wanted to exclude it from analysis, since every method that I'm aware of involves conditions that are linear in ballot types.

A table summarizing methods is an excellent idea.  A table explaining Type 1, Type 1b, Type 2, and Type 3 might still not make a lot of sense to a newbie, since these definitions all depend on properties of normal vectors, i.e. it is not immediately obvious what they have to do with practical statements about election methods like "Whoever has the most points wins" or "Whoever beats all others pairwise wins" or "If there's no majority first choice then you eliminate the candidate with the fewest first-choice votes and transfer his ballots..." etc.

However, I might still put in a table to at least summarize these things for the technical reader trying to keep track, and a separate table with examples of SFBC-compliant methods in each category so that the non-technical reader gets a flavor for it.  Thanks for the suggestion.

As for "*" vs. "x" for multiplication, all I can say is that when I was writing my thesis I frequently used "x" and my advisor (a physicist) chided me gently for my "unorthodox" choice of multiplication symbol.  Remember that "x" means "cross product" to many scientists and engineers.  I have replaced * with \cdot, since it means "scalar multiplied by" to many people.  Now somebody will no doubt accuse me of mixing in a dot product, and I'll change it to \times, and then somebody will complain that I'm doing topological products :)

More later.

Alex

--- On Mon, 8/23/10, election-methods-request at lists.electorama.com <election-methods-request at lists.electorama.com> wrote:
Message: 2
Date: Mon, 23 Aug 2010 16:43:38 -0400
From: Warren Smith <warren.wds at gmail.com>
To: election-methods <election-methods at electorama.com>
Subject: [EM] (no subject)
Message-ID:
    <AANLkTim8snHT88T_Si9=n-0MTpw=fHn-8AgS1do5LN3b at mail.gmail.com>
Content-Type: text/plain; charset=ISO-8859-1

Dear Alex Small

your FBC manuscript looks interesting.  The typesetting is sometimes
annoying (use of * for multiply).

Kevin Venzke is quite right he invented MDDA not me.
Ossipoff has 2 Fs.  Warren D. Smith has a "D."

Your paper is long.  It needs to be written to be more accessible.
Think how to provide fast-access routes for the reader who wants to
know certain things (make a list of what things various typical readers
might want to know, and find a way to make them be able to find it fast).

Like put a table of contents, table of FBC-complaint methods, index,
I dunno.   It is not easy for a newbie to quickly assimilate what's
important in your paper.

See also the end of http://rangevoting.org/FBCsurvey.html
where the Smith-Simmons theorem is mentioned, see
http://rangevoting.org/SimmonsSmithPf.html

somehow I feel this theorem has heavy importance and you ought to
discuss it to some degree.

Among your SFBC compliant methods, you might want to compare.  Which
should we like and why.

But I haven't really read the thing yet :)

-- 
Warren D. Smith
http://RangeVoting.org  <-- add your endorsement (by clicking
"endorse" as 1st step)
and
math.temple.edu/~wds/homepage/works.html

------------------------------

------------------------------

Message: 4
Date: Mon, 23 Aug 2010 17:14:40 -0400
From: Warren Smith <warren.wds at gmail.com>
To: election-methods <election-methods at electorama.com>
Subject: Re: [EM] Alex Small SFBC paper draft
Message-ID:
    <AANLkTinUi-5kWgQJGJxe7709DR2YVREOS2jGKUJ_+E78 at mail.gmail.com>
Content-Type: text/plain; charset=windows-1252

On 8/23/10, Warren Smith <warren.wds at gmail.com> wrote:
> Dear Alex Small
>
> your FBC manuscript looks interesting.  The typesetting is sometimes
> annoying (use of * for multiply).
>
> Kevin Venzke is quite right he invented MDDA not me.
> Ossipoff has 2 Fs.  Warren D. Smith has a "D."
>
> Your paper is long.  It needs to be written to be more accessible.
> Think how to provide fast-access routes for the reader who wants to
> know certain things (make a list of what things various typical readers
> might want to know, and find a way to make them be able to find it fast).
>
> Like put a table of contents, table of FBC-complaint methods, index,
> I dunno.   It is not easy for a newbie to quickly assimilate what's
> important in your paper.
>
> See also the end of http://rangevoting.org/FBCsurvey.html
> where the Smith-Simmons theorem is mentioned, see
> http://rangevoting.org/SimmonsSmithPf.html
>
> somehow I feel this theorem has heavy importance and you ought to
> discuss it to some degree.
>
> Among your SFBC compliant methods, you might want to compare.  Which
> should we like and why.
>
> But I haven't really read the thing yet :)
>
>
> --
> Warren D. Smith
> http://RangeVoting.org  <-- add your endorsement (by clicking
> "endorse" as 1st step)
> and
> math.temple.edu/~wds/homepage/works.html

4. Linearity: The conditions for a candidate to win can be expressed
by a series simple inequalities that are linear in the tallies of the
ballot types.

--that was bogus english grammar.   Do you mean "boolean combination
(via AND, OR, NOT) of linear inequalities"??  Note the result would be
nonlinear

anyway, since I don't know what you meant on key axiom, this paper is
currently dead right there.

And quit with the * for multiply.

5. "Generic decisiveness" might be better name.   E.g. with random
real numbers as the vote counts, prob=1 get a unique winner.  Or full
measure, if like measures not probs.

Based on the geometric properties of the boundaries, we will clas-
sify SFBC-compliant election methods into 4 distinct categories.

OK, how about right here:
I. ...
II. ...
III. ...
IV. ...

Because your bleeping discussion right here now, is super annoying.

etc.
ditions are expressed as sets of linear inequalities, we get: IF    (p,u11)>0

seems like a major typo here.

Elec- tion methods can generally be specified as procedure defined by a series
10
of if-then-else statements regarding the numbers of voters submitting
different ballot types.

--well, that was vague.  If you had a precise axiom back there (like
my boolean AND suggestion above) you could
here make a precise claim.  As it is, you are just asserting something
with no proof
which I see no reason to buy, nor is it even clear exactly what is meant.

We will refer to a set of conditions related by symmetry operations as
a ?stage? of an election method.
Definition 3 Given...

--that also was vague.  This is not  living up to the precision+clarity
it takes to write a math paper "definition."

...hum.  Well, looking further, I'm not happy about the writing as it
stands now.
I think it has a long way to go to make the writing get good, tight,
clear, precise.
Try to make your theorems & defns standalone readable, or if not, at least with
backpointers to the stuff needed to comprehend it.
As for the correctness and importance of your results, I
currently have no opinion since I still don't know what they are.

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