Pressing Toward the Prize

Posts Tagged ‘Saari

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.

In creating my outline last week, I began to suspect that my initial goals for Capstone were a bit too ambitious. There is so much interesting material that could be covered, but in order to include everything I wanted to, I would practically be re-writing Mr. Saari’s book. Since that is not an option – for a multitude of reasons – I realize now that I need to narrow my focus in order to do justice to the topic. I will be meeting with my professor this coming week to get some direction on refining my goals, so that I will know how to proceed when I begin writing my rough draft during spring break.

Although I did not get to read Chaotic Elections as much as I had hoped (I did read a little), I have begun to formulate ideas for my Capstone outline. These are my first thoughts:

1. Define voting paradox and explain election conditions that may be problematic, i.e., there are three or more candidates or issues on the ballot, and the winning candidate or issue does not receive a majority of the votes.

2. Define various voting methods: plurality, Borda Count, Condorcet.

3. Offer brief historical background: Borda, de Condorcet, Arrow, Saari.

4. Give simple example to demonstrate the three voting methods and how different methods can produce different results when applied to the same election.

5. Explain some of the voting biases inherent to each of the methods, with examples, and the kinds of fallacies that can result.

6. Discuss how Donald Saari used mathematics to identify specific kinds of voting paradoxes, to analyze each for the causes of the various voting biases, and to determine the most reasonable voting method, i.e., the one most immune from voting method biases.

7. Offer possible reasons why, in light of existing evidence, wholesale changes have not yet been made to elections to insure that outcomes accurately reflect the will of the people

I will need much more research before I am ready to finalize my outline, and continuing research will be my plan for the week. I will refine the outline as I go, using this rough draft as a  starting place.

It’s hard to believe that I just finished my first week of spring semester at Pacific Lutheran University. When fall semester ended in December, I had such high hopes for working on my Capstone project during the break, but life took an unexpected turn. My house sold after being on the market for only a week, so I spent Christmas break moving, spending time with my dad, and getting settled in my new home as best I could. Then J-term started, and that was a wild four-week ride through Tudor England, and a noble attempt at learning ballroom dance!! To top it off, I had a family crisis during the first week of J-term, and it is continuing to impact my life. The week between J-term and spring semester was filled with doctor appointments, the application process for grad school, the GRE, and more unpacking and organizing. I only had time to read a little in Donald Saari’s book, Chaotic Elections, and that was a far cry from what I had hoped to accomplish.

So here I am, back at it. This is my final semester at PLU, at least that’s the plan, and my Capstone project is due in May. Although I find myself a bit under the gun and nowhere near where I had hoped to be at this point, I will continue moving forward. My immediate goal is to refresh my LaTex skills by completing the assignment we were given this week. I hope to also continue reading Chaotic Elections and begin formulating my project outline. I’ll check in next week and report how it goes… wish me luck!!

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.



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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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