[EM] Advanced Math question

[Bart Ingles]

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
> >
> > ----------------------------------------------------------------------
> > --------
> > This message is intended only for the personal and confidential use of the
> designated recipient(s) named above.  If you are not the intended recipient
> of this message you are hereby notified that any review, dissemination,
> distribution or copying of this message is strictly prohibited.  This
> communication is for information purposes only and should not be regarded as
> an offer to sell or as a solicitation of an offer to buy any financial
> product, an official confirmation of any transaction, or as an official
> statement of Lehman Brothers.  Email transmission cannot be guaranteed to be
> secure or error-free.  Therefore, we do not represent that this information
> is complete or accurate and it should not be relied upon as such.  All
> information is subject to change without notice.
> >
> >
> > ----
> > For more information about this list (subscribe, unsubscribe, FAQ,
> > etc), please see http://www.eskimo.com/~robla/em
> >
> >
> 
> ----
> For more information about this list (subscribe, unsubscribe, FAQ, etc),
> please see http://www.eskimo.com/~robla/em
> 
> ------------------------------------------------------------------------------
> This message is intended only for the personal and confidential use of the designated recipient(s) named above.  If you are not the intended recipient of this message you are hereby notified that any review, dissemination, distribution or copying of this message is strictly prohibited.  This communication is for information purposes only and should not be regarded as an offer to sell or as a solicitation of an offer to buy any financial product, an official confirmation of any transaction, or as an official statement of Lehman Brothers.  Email transmission cannot be guaranteed to be secure or error-free.  Therefore, we do not represent that this information is complete or accurate and it should not be relied upon as such.  All information is subject to change without notice.
> 
> ----
> For more information about this list (subscribe, unsubscribe, FAQ, etc),
> please see http://www.eskimo.com/~robla/em

----
For more information about this list (subscribe, unsubscribe, FAQ, etc), 
please see http://www.eskimo.com/~robla/em