Pressing Toward the Prize

Posts Tagged ‘voting

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.

Here we are at the end of the fall semester, and that means we are halfway through the Capstone process. This semester has been quite a learning experience for me. I remember being terrified at the prospect of doing a blog for all the world to see, because I have never done anything like that before. I’m not a texter, and I don’t follow Twitter or FaceBook, so this is a whole new realm for me. Why would anyone care what I have to say? I still don’t know the answer to that, but the good news is that I survived, and I found it wasn’t as bad as I thought it would be.

This semester I also learned how to use the mathematical typesetting program, LaTex, as well as the Beamer package that creates presentation slides. Our professor gave us awesome templates to use, so that was very helpful. We also practiced reading and “understanding” mathematics articles, and it was no surprise that I’m not very good at it. It takes a while to digest the information, if one can even follow what is being said in the first place. Those are the times when a translator would come in handy!! I have enjoyed being exposed to some of the math that is being studied by way of faculty, students, and literature, and if I had time, there are many topics I would love to pursue. But for now, I will be focusing my attention on my own Capstone project, The Complexities of Voting, and the linear algebra I will need in order to understand the articles and books on the subject!!

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.

I recently ran across an article that Donald G. Saari (author of Basic Geometry of Voting) wrote in 1996 about that year’s elections, highlighting the ease with which an unwanted outcome can occur. As he explains, when an election result is not what one thinks it should be, that is, the preferred candidate or issue does not win, it is called a voting paradox. This is generally not due to the voters, but rather the voting procedure used. In plurality voting, the process we use in which each voter votes for one candidate and the candidate with the most votes wins, multiple candidates can “split the vote” causing an “inferior” (at least according to the will of the people) candidate to be elected.

Saari demonstrates his point with a very entertaining story about his encounter with a group of precocious 4th graders in 1991. He was attempting to present a counting problem caused by a hypothetical voting example, when the students recognized a flaw in his assessment of the winner in a three-way vote. According to plurality voting, candidate A was preferred to B who was preferred to C in a 6:5:4 vote, when considering only the first choice of each voter. But the students immediately protested, because he did not factor in the “next best” preference, which alters the outcome. In fact, when all rankings were considered, not just the top-ranked candidate, the winner A under plurality voting was actually the least preferred by the majority of the voters, and C was in fact the favorite.

When he asked the students what they thought was the “correct” voting procedure to use, one of the students suggested voters assign 3 points to their favorite candidate, 2 points to the next best, and 1 point to their least desirable candidate. What this student described is the Borda Count method, named for the French mathematician JC Borda, who developed this method in 1770. When the Borda Count method was applied to the voting example, it showed that C was preferred to B who was preferred to A by a 34:29:27 vote – consistent with the students’ earlier assessment. Then Saari presented a version of Marquis de Condorcet’s puzzling example from the 1780’s that shows it is possible to have no winner, because there is a way to count the votes so that every candidate has the same number of votes. The students saw right through this example, as well.

The amazing thing about these students is how quickly they recognized a flaw and were able to suggest reasonable solutions, based solely on their examination of the problem. As Saari points out, they have not yet been conditioned to blindly accept the way things are, but rather used critical thinking skills coupled with their value of fair play. He is concerned that too often our educational system stifles the creativity of our students rather than nurture their inventiveness and innate desire to explore the world around them. He suggests that educators consider changing their classroom approach in order to foster creativity and develop problem-solving skills in their students. As these 4th graders demonstrate, children can accomplish amazing things when given the right environment.

The time has come to declare the topic for my capstone project, and I still find myself somewhat undecided. The two areas that interest me most are voting paradoxes and working with complex numbers. I like the practical application of studying voting processes, the problems that can arise when trying to determine the will of the people, and the manipulations (inadvertent or otherwise) that can greatly affect the results. Fairness in voting is a concept that should be of interest to us all, since the outcomes dictate many aspects of our lives.

A few weeks ago, my math professor suggested considering for our capstone project topics that may have been introduced in past math classes, but not studied in depth. My interest in complex numbers is a result of this suggestion. My exposure to the complex number system has mainly consisted of performing algebraic manipulations, and I would like to know more. Analysis of the complex numbers is such a broad topic that, should I decide to go this route, I will need some direction in choosing a focus. An intriguing application that my professor mentioned is one in which complex numbers are used to solve integration problems that would be quite difficult in the real number system. Both of these are worthy topics, but our formal proposal is due next week, so I will need to hop off the fence soon!

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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