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Chad Watson: Is Our Voting System the Best Mathematics Can Offer?

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Chad Watson: Is Our Voting System the Best Mathematics Can Offer? Chad Watson Mathematics 482 May 8, 2013 Is Our Voting System the Best Mathematics Can Offer? In the year 2000, my friend and I were both proponents of Al Gore who supported Ralph Nader. We were committed to encouraging everyone to vote for Nader to help the Green Party gain the 5% of the popular vote that would provide public funding and a place in the debates for 2004, but we were aware he had less than a 1% chance of victory. I lived in California, my friend leaved in Florida. I was able to convince a handful of folks, but it was easy; Al Gore was going to win the California Electoral College by a landslide. The race in Florida was a much different. Bush and Gore were a toss up. I suggested to my friend he vote for Nadar, but cease recruiting voters. He shrugged off my suggestion and he continued to campaign, even convincing his blue dog Democrat mother to vote green. Gore won California in a landslide but the recount in Florida went on for over a month. While am not how many people my friend persuaded, I can safely say that my friend did not convince 537 people to vote for Nader. Twelve states were decided by a margin of less than 5%, which is less than the acceptable margin of error in surveys seen on the news. Two states threw out more ballots than the margin of victory. Nine states threw out over 2% of their vote while other states threw out no votes (Leip, http://uselectionatlas.org/RESULTS/index.html). In a contest to pick out the leader of our this is all a little too close for comfort. While the 2000 election may be the most contested election in recent memory, it is not the first in the United States’ history. William Poundstone argues since the advent of the current voting system, upwards of 7 presidential elections were won by the candidate who did not win a majority of the popular vote (Poundstone, 91). How can someone win without winning a majority, by winning with a plurality. If Millard Fillmore and Franklin Pierce are both running for comptroller of the senior class and Millard Fillmore receives 51% of votes cast while Pierce receives 49% of votes cast then Millard Fillmore wins with a majority. Let’s says George McGovern and Spiro Agnew get into the race and the vote cast is now 28% for Millard Fillmore, 27% for Pierce, 26% for McGovern and 19% for Agnew. Millard Fillmore wins with a plurality; he received the most votes, but not a more than 50%. How can this happen in a country that is a standard barer for democracy? How can someone be elected with more than half the registered voters going against him or her? Vote splitting is one of the problems with our current system. Vote splitting is when candidates have similar viewpoints on an issue or issues and voters divide their choices between the two. This often results in neither candidate winning. Most voters do not fall perfectly on one side of the binary Republican/Democrat system. When a third party candidate enters the race they tend to take votes away from the Republican or Democrat. For instance, in the original iteration of our comptroller election I supported Milard Filmore and his Whig values because I detest a Jacksonion Democrat like Pierce. The electorate is either in agreement with or stands ins complete opposition to me. Votes change when 20th century Republican and Democrat Agnew and McGovern, respectively, get into the race. Now there exists a broad spectrum of beliefs within the election and the vote is now distributed among the new candidates. My vote goes to Agnew in hopes he would better manage the finances of the class. While the class is a small part of my life what about broader issues? Suppose we had hypothetical gubernatorial candidates with their platform in their name, Stay the Course Sara with 51% of the vote and Free Tuition by Eliminating Football Fred with 49% of the vote, then Four Day Weekend William enters race. 2 5% of the votes are transferred from Sara votes to William and 2% of the voters who were with Fred are now voting for William, this Gives Free Tuition Fred a narrow victory. This is not a silly and purely hypothetical case. Take a look at the 1998 Minnesota gubernatorial race which pitted a Democratic Attorney General and Republican St. Paul mayor with over two decades of combined experience against one another. Well into the race they were forced to face off against noted World Wrestling Federation heel and conspiracy theorist Jesse “the Body” Ventura as the Reform Party candidate. Even though Ventura admitted he was not sure how he could get things done in the governor’s mansion he managed to appeal to 37% of the voters and won the plurality (Poundstone, 213). In 1992 George H. W. Bush, Bill Clinton, and Ross Perot faced off for the White House. Many analysts believe that a portion of Perot’s 19% could have turned the election in favor of Bush. Bill Clinton won this election with over 56% of the electorate voting against him. It is difficult to say if Perot or Ventura took votes from the frontrunners. Perhaps they brought people to the polls who would not have otherwise voted. Regardless, when more than two people are running, the plurality system falls apart because it is difficult to receive a majority of the vote. Mathematician and economist Kenneth Arrow won a Nobel Prize in economics for bringing to the forefront a problem called Arrow’s paradox. The main problem of ranked voting is the intransitivity of voting choices (Poundstone 38-41). We learned in elementary school that if A > B and B>C then A>C. This not the case with ranked voting. Take four candidates, the front runners President Obama and former Governor Romney, along with the two major third party candidates, former Governor Gary Johnson from the Libertarian party and Dr. Jill Stein from the Green Party. A voter may favor Johnson over Obama, Romney over Johnson, Obama over Romney, and Stein and Obama. How does this person vote? They may ideologically 3 believe in Johnson, but they do not want Romney to be in the White House, so they vote for Obama as the lesser of two evils. Another voter may favor Stein over others, but vote Romney in order to change the status quo. Vote splitting is evident to some extent in all elections when a third party enters the race. People vote strategically. Sometimes they will vote not for their favorite candidate but of the front runner they favor most. Voters may often support a weak candidate in the primaries to give their candidate an easier opponent. Often they vote their conscience and the vote is split. Are there better systems? Most are susceptible to “gaming” or strategic voting that clouds the results. But many have strong arguments that they are better methods than the current ranked plurality system. Kenneth Arrow went into detail in his 1950 paper “A Difficulty in the concept of Social Welfare.” Arrow was a practitioner in the field of social choice theory. Social choice theory uses elements of formal logic to construct axioms and social welfare functions with elements of game theory mixed in. Suppose A is a set of outcomes and N is the number of voters. Let F: L(A)N L(A) where L(A)N is the list of all profiles of voter and L(A) is the final outcome is the list of society’s profile and a ordering is listed a voter’s ranking of candidates: (R1,R2,…,Rm). “Arrow’s Paradox considers the following properties:” (http://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem) 1.) Pareto Efficiency is unanimity or the fact that no one can be made better off without making someone else worse. If a>b for all orderings (R1,R2,…,Rm) then a>b for F[(R1,R2,…,Rm)]. In short, common rankings should be the outcome. 2.) Non-Dictatorship states there is no individual whose preference always prevails 4 3.) The Independence of Irrelevant Alternatives: Two candidates ranking should only be motivated by preference of one to the other. If I have B>C>D>Z B>C should hold if I change my preference to Z>D. Arrow’s Paradox stated that if society fields over two candidates than one of the above properties is violated resulting in the best candidate not always having the best chance of winning. In 1972 Arrow was awarded the Nobel Prize in Economics which brought his idea to the masses and resulted in a boarder discussion on the validity of voting systems. The Marquis de Condorcet published one of the first alternatives to plurality voting first in a 1785 paper (Poundstone, 141). He claimed the best system was one in which each candidate goes up against the other in a two-way contest. One who wins in all head to head match ups is the Condorcet winner (Poundstone, 142). Here again we run into the problem of intransitivity. What if there is no Condorcet winner, then we have what analysts call a Condorcet cycle. If in the current contest Romney>Obama, Johnson>Romney, Stein>Romney, then we have a Condorcet cycle and there is no clear winner. Another system to consider is the Borda count system. It tries to counteract vote splitting by giving voters more than one vote via a weighted point system.
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