Pressing Toward the Prize

Posts Tagged ‘linear algebra

Spring Break has arrived, and I will be using this time to write the first draft of my Capstone paper that is due the Wednesday after we return to school. I met with my professor this past week to get some much-needed direction, and he showed me how to use concepts I learned in Linear Algebra to find and look at election results. For a 3-candidate election, a 3 x 6 matrix can be used to transform a voting profile vector from \mathbb{R}^6 into an election outcome vector in \mathbb{R}^3 (I think I got that right??). Anyway, what he said made perfect sense, even though my Linear Algebra skills are a bit rusty. So the plan for this week is to review transformations, among other things, see how to apply Linear Algebra to Mr. Saari’s concepts, and get as much as I can on paper before the week is over.

I am continuing to read Chaotic Elections, and I still find this entire subject of “elections gone wrong” quite fascinating. How can something we often take for granted wreak such havoc in our lives – without us even realizing it is happening? And then when “we” think we have found a wonderful strategy to bend an election in “our” favor, we create another whole set of problematic outcomes. Amazing!

So far, the book has been part history lesson, part civics lesson, part statistics, and a fair amount of surprises. Donald Saari does a great job of putting the voting paradoxes in terms that can be easily understood, such as using allegories to school grades and class standings for various voting methods, before he gets to the actual mathematics he used to analyze them. His mathematical analysis will be the meat of my Capstone presentation, but I also intend to use some simpler examples to aid in understanding.

This week I will be working on the outline as I continue to read Chaotic Elections, all the while referencing my linear algebra textbook, as needed.

On Wednesday I met with my faculty liaison, Dr. Edgar, about my Capstone project. He encouraged me to continue reading Chaotic Elections with an eye for linear algebra. Donald Saari uses vectors, specifically what he calls voting vectors, to analyze the various voting methods, do manipulations of outcomes, and support his argument that the Borda Count is the superior voting method whenever three or more candidates or issues are on the ballot. My goal for this week, then, is to continue reading and refining my understanding.

The time has come for me to submit the proposal for my Capstone Senior Project. I have decided that I would like to explore the complexities of voting, including the paradoxes and problems that can arise when there are three or more candidates (or issues) on the ballot. There is a rich history of voting analysis that dates back to 1770 with French mathematician JC Borda, who, concerned with the outcomes produced by plurality voting, developed a weighted voting system called the Borda Count. A decade or so later, the Marquis de Condorcet attempted to discredit the Borda Count by demonstrating flaws in the procedure, and presented his method based on pair-wise counting, which had some problems of its own.

Then in the 1950’s, Kenneth Arrow, perhaps unaware of this 18th century conflict, analyzed similar problems with voting. He began by defining basic conditions that should be met in a voting procedure, and then attempted to find a voting method that satisfied these conditions. His conclusion was that with three or more candidates, the only procedure that satisfies all the conditions is a dictatorship! So if we are left to choose “between a dictatorship or a paradox” (per Donald G. Saari), what are we to do? Saari uses mathematics to show that there is a more reasonable option, and in fact shows mathematically that the Borda Count is the most reasonable option. I would like to study this centuries-long debate, the issues and “solutions” as they were presented, as well as Saari’s analysis that leads to a reasonable resolution. As has been suggested, since Saari uses linear algebra in his analysis, it would be interesting to run a few elections with fellow students and manipulate the outcomes using linear spaces. I would also like to investigate the reasons why plurality voting is still widely used, even though its flaws are fairly obvious.

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  • gramsonjanessa: I can't wait to listen to your capstone presentation in the spring! Your proposal was really interesting and I'm interested to see how the linear alge
  • dewittda: This is impressive! I thought I was good because I solved a rubik’s cube once in an hour. I served with a guy in the Air Force who could solve a r
  • ZeroSum Ruler: The Euclidean algorithm should me the mainstream way we teach students how to find the GCF. Why isn't it? A mystery.