Computing Results (RE: [EM] Advanced Math question)
Narins, Josh
josh.narins at lehman.com
Mon Jan 6 08:47:10 PST 2003
Firstly, thanks for the tip on "Merrill"
Unfortunately, I don't know who Merrill is. Are they on the list?
Secondly, you write "perl or other interpreted languages"
I would request you kindly cease & desist bringing your 2nd millenium
thinking into this :)
Wll written perl is, in almost all cases, just as fast as C.
Pus, i work with computers that turn into large, distributed computing
environments (beowulf clusters) by just snapping my fingers (generally).
So, there is no practical limit to the efficiency of the code, or the amount
of computers that can be used to solve the problem.
So, the brute force solution is possible.
-----Original Message-----
From: Bart Ingles [mailto:bartman at netgate.net]
Sent: Thursday, January 02, 2003 11:26 PM
To: election-methods-list at eskimo.com
Subject: Re: [EM] Advanced Math question
Merrill uses a lot of software-modeled comparisons of different systems,
some of which are presented as graphs. To generate useful models, I think
you mainly need knowledge of Statistics (other than basic algebra). For
example, some of Merrill's simulations used normal distributions of both
voters and candidates in one or more "issue space" dimensions. These were
more interesting than simulations using mere random ballots.
Examining all possible cases would not be useful in determining "typical" or
"expected" outcomes, but might be useful if you are interested, say, in
examining worst-case results for different methods.
This brute-force approach would rapidly bog down with large numbers of
candidates (especially in Perl or other interpreted languages), but you
might see some interesting trends even with a limit of 5 or 6 candidates.
There's my two cents.
Bart
"Narins, Josh" wrote:
>
> More importantly, the linear algebra "stuff" goes way past what I
> _remember_ from my only semester of linear algebra (taken at the
> tender age of 17)
>
> My background, now, is perl computer programming.
>
> With perl, easy things are trivial, and everything reasonable is
> possible.
>
> Larry, the primary author of Perl, a linguist, takes a lot of guidance
> from mathematicians, hence the built in support for arbitrary
> precision operations.
>
> Large math suites, (LAPACK, I think it is called) are freely available
> for use in perl.
>
> Now, if that's what I can do, what do I want?
>
> I want to see the different outcomes for all possible cases.
>
> When America moved to the Method of Equal Proportions for the
> Congressional District apportionment process, Congress was presented
> with five options, two tended to favor small states, two the larger,
> and MEP was in the middle (1).
>
> I want to see something similar with election methods. Some favor
> plurality candidates, some favor centrist candidates, and perhaps
> there are other mutations.
>
> I believe we won't get anywhere without being able to relay the
> tendencies of each method via VISUAL (color bar charts?) aids,
> preferably one aid per method.
>
> -josh
>
> -----Original Message-----
> From: Forest Simmons [mailto:fsimmons at pcc.edu]
> Sent: Tuesday, December 31, 2002 8:47 PM
> To: 'election-methods-list at eskimo.com'
> Subject: Re: [EM] Advanced Math question
>
> Linear algebra, graph theory, probability, statistics, measure theory,
> metric spaces, combinatorics, piecewise linear topology, linear
> programming, multivariate calculus, mathematical logic and set theory,
> theory of algorithms, etc. are all good for the tool box.
>
> Most of the minimization can be done without multivariate calculus,
> but that's where most folks get a good feel for minimization with
> constraints and for geometry with more than two dimensions.
>
> The linear algebra "stuff" may go past what you learned in the first
> semester of linear algebra.
>
> For example, moving a candidate from "Candidate Space" into "Voter
> Space" is most naturally done with the help of the "Singular Value
> Decomposition" of the matrix whose rows represent the voters and whose
> columns represent the candidates.
>
> Was the SVD part of your linear algebra course?
>
> [By the way, the lambdas that represent the eigenvalues in the SVD are
> the same lambdas that represent Lagrange multipliers in multivariate
> maximization with constraints; the similarity of notation is no
> accident.]
>
> I like this election methods field because it seems to be at the cross
> roads of all the fields of mathematics that I enjoy. Even intuitions
> from differential equations and digital filtering have helped me from
> time to time.
>
> We need people with all different kinds of backgrounds to help us find
> new ways of looking at these election methods.
>
> Forest
>
> On Mon, 30 Dec 2002, Narins, Josh wrote:
>
> >
> > What branches of mathematics are generally used when approaching
> > this topic?
> >
> > It sure seems more like Algebra than Geometry, so that's easy.
> >
> > Is it all stuff I learned in Linear Algebra, or does it go farther
> > than that?
> >
> > Here's a tidbit, in Finance, the stochastic/filtration people rely
> > somewhat on "Sigma Algebra."
> >
> > So, say, for Condorcet, is there a particular branch of Matrix or
> > Linear Algebra that anyone who hopes to speak authoritatively on
> > this subject must master?
> >
> > Pardon that English.
> >
> > What kind of math must you be great at to totally _see_ the issues
> > involved in condorcet matrix counting?
> >
> > -josh
> >
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