The mathematics of democracy

With only a few days left until the U.S. presidential elections, I thought it would be appropriate to write about some of the mathematical and teaching opportunities an event like this provides. Especially because this has such a high profile in the general public, talking about election-related mathematics is much easier than just talking about math on any other day. Here are a few of the things that I have been talking about with my family during dinner, sharing with friends on facebook, and if I were teaching classes this semester I would like my students to read and discuss.

  1. Nate Silver and 538: One of my favorite things to read these days (and for the last month or so) is Nate Silver‘s blog, 538, on the New York Times site. Silver is a statistician who became famous for his predictive analysis of the performance of baseball players, and later on for his predictions on the 2008 elections based on polling data. I think this blog is filled with fabulous reading for a stats class. In particular, Silver is very careful to keep a very mathematical and level-headed point of view. He describes where he obtains all his data, where potential noise or inaccuracies may be coming from, and updates his predictions as more polling data gets published. In fact, he writes a post almost every day. I hope someone is having their students read this. A new, surprising development (at least, surprising to me), is all the backlash that Silver has been getting from some pundits in the media. Part of what Silver does is predict who will be the winner of the election, and given the polling data from each state and the general trends, he computes a probability of winning for each candidate. Conservatives seemed to be very angry that Obama gets (from Silver) a 77.4% chance of winning the election, and claim that his analysis is biased and inaccurate. The main problem seems to be that many people don’t seem to grasp that a probability, no matter how high, is not a certainty. Also, Romney is hardly playing the lottery here, many people would bet on those odds. A few friends of mine shared some good replies to the backlash, and in defense of Silver’s methods, like the Salon post, Jordan Ellenberg‘s blog post, and this response on the Washington Post. I think having the students read the criticism of Nate Silver’s predictions on Politico and then some of these responses would make for an excellent class discussion. Boy I wish I were teaching stats this semester. That is really the fun of it: seeing probability and statistics being used in a real-life situation and understanding when they are being used correctly or incorrectly.
  2. Arrow’s Theorem and other voting paradoxes: I recently gave a talk about this during a Sonia Kovalevsky day we recently had at Bates (read my blog post about it if you’re interested!). It is very fun to talk about this with students and very elementary (unless you get into the proof of the theorem, which requires some experience with set theory and functions on sets). It is a bit disheartening to them that there is no perfect way to collect individual choices into a societal choice. Every once in a while people get very angry that someone got kicked off American Idol (or Dancing with the Stars, or whatever people watch these days), and I think it’s because the voting system somehow eliminated a pretty good candidate (i.e. someone who most people liked). But the only alternative to these flawed systems is a dictatorship, and everyone agrees that that’s not something we want (at least the kids I give these talks to). One of my favorite examples is the 1860 Presidential Election, in which there were four candidates: Abraham Lincoln, John Bell, John C. Breckinridge, and Stephen A. Douglas. The Electoral College and plurality voting (i.e. the person with the most total votes wins) would have won the election for Lincoln, but for example a run-off vote or the Borda count method (in which points are given to each individual’s ranking of the candidates) would have given the election to Douglas. Many people complain about the Electoral College, but thanks to it Lincoln won, and we all know that turned out pretty well for the country (and I don’t mean that he became an important vampire hunter). Anyway, many fun discussions can be had with this material, I highly recommend it for almost any class, especially with the election just around the corner.
  3. Computational approaches to election prediction: In this really cool article, a much more computational and combinatorial approach is taken to understand the election and predict the result. This is similar in spirit to what Nate Silver is doing with data and statistics, but a bit more abstract. In the article, Robert St. Amant explores the electoral college votes that are “up for grabs” and how many different combinations of the swing state votes can give each candidate the win. This is based on an article by Gabriel Snyder in the Atlantic Wire, in which he goes through the combinatorial computations. St. Amant goes through the computation carefully (and clearly) and gets that of “512 possible combinations, 431 result in Obama winning the election and 76 give Romney the win (five additional ties would go to Romney)”. This means that the probability (as a purely “favorable outcomes/possible outcomes” formula) that Obama will win is about 84%. Now, again, since this is based on no data and rather on a combinatorial and abstract formula, it is even less of a predictor, but clearly indicates that the odds are in Obama’s favor (but again, odds are not certainties). I think this is a pretty nifty example of how computational and combinatorial approaches can give a better idea of what’s going on. For example, from this analysis, St. Amant points out, it’s clear why Florida is a “must-win” state for Romney.
  4. Benford’s Law and detecting voting fraud: This seems slightly less relevant in the context of this election (because we are pretty certain that there will be no serious voting fraud in the U.S. election). Briefly, Benford’s law states that for numbers that come up in real life contexts, the number 1 appears as the first digit about 30% of the time. This has very much to do with logarithmic scales and in fact was discovered by Newcomb while looking a logarithmic tables in a book, and noticing that the first pages (with numbers with leading digits of 1) were much more worn than the later pages. Since fabricated data doesn’t follow Benford’s Law, it is thought to be a good indicator of fraud. Supposedly it was used to detect fraud in the 2009 Iranian election. (Election results, since they are numbers that arise from counting people with certain votes, are thought to follow Benford’s law). It is now believed that it is actually a very weak way to detect fraud, at least in elections. In a very good article from 2009,Carl Bialik explains that since precincts are more or less the same size, if you for example have 1000 people in each precinct then you would not expect Benford’s law to hold if you have only two candidates (in fact, it doesn’t hold when there are cut-offs for the numbers). Even though it’s not entirely appropriate for election time, I still think it’s interesting that you could try to detect financial fraud and election fraud with such a simple idea, and why it doesn’t always work.

That’s it for me, dear readers. Do you have any election-related math articles you would like to share? Any examples of your own that you use in class? Who are you voting for? Just kidding! Now seriously, though, let’s keep the comments focused on the math and not on the politics. We look forward to hearing from you, as always.

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