On Tue, 31 Dec 2002 17:46:56 -0800 (PST) Forest Simmons wrote:
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
I see three divisions:
To analyze methods anything, such as described above, goes.
The actual method had better stay with simple operations the average voter can understand.
For near ties, such as Condorcet cycles, a bit more complexity is tolerable - but the more understandable to voters with minimum requirement for mathematical understanding the better.
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Dave Ketchum 108 Halstead Ave, Owego, NY 13827-1708 607-687-5026
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