Pressing Toward the Prize

Posts Tagged ‘elections

Spring Break has come and gone, and I spent a good part of it doing an assignment for one class (that was due the Friday of Break, if you can believe that!!!), studying for the mid-term in another class, and writing the rough draft for my Capstone. I got the assignment done on time, I did well on the mid-term exam on Tuesday, but I only managed to complete about 75% of the Capstone paper by Wednesday’s deadline. I am really grateful for the time I had to work on it, but I was quite disappointed that I didn’t manage to get it finished. I tried hard, but I just wasn’t able to pull it off.

It is critical that I complete the final sections of the paper soon, because I must begin focusing my attention on preparing for the presentation I will be giving in early May. To that end, I will be working to gather as much information as I can about Donald Saari’s analysis of election outcomes. I’ve laid the foundation in the paper, now I need to try to bring it to its logical conclusion… as soon as I determine what that is! So here we go, more reading and (hopefully!) more writing.

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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.

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.

In a recent conversation with my math professor regarding a possible topic for my Capstone, I commented that I really enjoyed studying linear algebra. I told him I might be interested in doing some type of real-world application involving linear algebra for my project, and he began to tell me about Donald Saari. Mr. Saari is a mathematician who studies what he calls “the paradoxes and problems of voting procedures,” and analyzes voting methods using linear algebra. I am currently looking at two books he wrote: “Chaotic Elections!” and “Basic Geometry of Voting.”

In “Basic Geometry” Mr. Saari makes it clear that when there are only two choices up for a vote, there is generally no difficulty in determining the winner. But when there are more than two choices, things can get rather interesting. In order to demonstrate some of the challenges that can arise,  Mr. Saari opens the book with the story of a fictional academic department chair who finds himself in hot water as a result of a departmental vote. The problem is that the same vote can be interpreted different ways, depending on which voting method is used. Even if a method ranking preferences is used, various outcomes can result depending on which way one counts the rankings. In fact, each of the options can be deemed the winner, depending on the method used.

Now things can become even more complicated if a group of voters attempts to manipulate the process. For example, assume Al, Bob, and Chuck are candidates for the new Chair position. Of the 15 people voting, 7 are for Al, 7 are for Bob, and the only one pulling for Chuck is, well, Chuck. A ranking system is used that gives 2 points to one’s first choice, 1 point for one’s second choice, and 0 points for the third. Al receives 14 points from his group, 7 points from Bob’s, and 1 from Chuck. Bob receives 14 points from his group, 7 points from Al’s, and 0 from Chuck. So Al wins, 22-21-2, with Chuck’s vote determining the winner.

Let’s assume now that Bob’s supporters predict this outcome and decide to ensure Bob’s success. They each vote for Bob as their first choice, Chuck for their second, and Al for their third. Now Bob wins 21-15-9. Al’s supporters see this coming, so they decide to be “strategic” and vote for Al as their first choice, Chuck as their second, and Bob as their third. With this turn of events, that is, both groups voting for Chuck as their second choice, even though he was the first choice for only 1 out of the 15 voters, Chuck wins 16-15-14.

With this simple story, Mr. Saari demonstrates that it is not difficult for a candidate to win an election, even though that person was not the first choice of the majority of voters. As it turns out, this is not an anomaly. The outcome of a vote does not necessarily reflect the will of the people. As Mr. Saari states, his intent is simply to share what can go wrong in elections and why, in the hopes that voting errors can be prevented in the future. All in all, I found this to be a very intriguing topic, and with a little more research, it could turn into my Capstone project.



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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.

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