Tufts University Summer of 2019 in Talloires Math 19, Mathematics of Social Choice

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1 Tufts University Summer of 2019 in Talloires Math 19, Mathematics of Social Choice Instructor: Prof. Christoph Börgers Office: Bromfield-Pearson, Rm. 215 Office hours (Fall 2018): Tu, We 10:30 12:00 and by appointment Phone: Text: Christoph Börgers, Mathematics of Social Choice, SIAM Syllabus: Part I: Electing a president (a single winner) 1. What is wrong with the usual way of voting? For one thing, there is no usual way of voting! Different countries elect their presidents or parliaments in different methods. We will study several voting methods plurality voting, the runoff method, the instant runoff method, Borda count, and the Copeland method (the simplest Condorcet-fair method). We will give examples, both fictitious and real, showing that different voting methods can very well lead to different outcomes. Chapter 1 of the text How Majority Rule Might Have Stopped Donald Trump by Eric Maskin and Amartya Sen, New York Times, April 28, Both authors are Nobel Prize winning economists at Harvard University. 2. Presidential elections in the United States, France, Russia, and India The presidential election system in the United States involves the electoral college. State-by-state, however, we use plurality voting in most states. France and Russia use the runoff method. In India, the President is not elected by the people directly, but by Parliament, and the method used is instant runoff. We will study the details of that, and think about advantages and disadvantages. D. Madore, U.S. Presidential Electoral System, M. Alvarez-Rivera, Election Resources on the Internet: Presidential and Legislative Elections in France, 1 For every copy bought, I get some royalty income. It s about enough for a coffee. If you show me a new copy of the book that you bought, it will be my pleasure to take you out for a coffee.

2 R. Dubbudu, A Dummy s Guide to the President of India s Election, (a good article in spite of the title) 3. Election spoilers as an unavoidable fact of life In 2000, George W. Bush won the Presidential election in Florida and, as a result, the U.S. Presidency because Ralph Nader was on the Florida ballot. (I know that you can reasonably dispute that, but in any case it s possible that I am right. I would argue it s actually probably that I am right.) In the 2002 French Presidential election, the runoff was between a center-right candidate (Jacques Chirac) and a far-right one (Jean-Marie LePen). In the first round, the total vote for left-wing candidates far exceeded that for either Chirac or LePen, but there were so many left-wing candidates that none of them advanced into the runoff. Both of these incidents illustrate something called the spoiler effect. We will make mathematically precise what this effect is, and prove that there is no election method that can avoid it completely. Chapter 3 of the text 4. The definition of majority rule proposed by the Marquis de Condorcet The Marquis de Condorcet was a French nobleman of the 1700s 2 who made the single most famous mathematical contribution to the theory of election systems: He formalized what we should mean by rule of the majority. (If you think about it for a little while, it will become clear that a priori, the meaning of that phrase is unclear.) Methods that satisfy Condorcet s criterion tend to favor compromise candidates over ones who have an enthusiastic base, but are strongly disliked by many other voters. Chapter 2 of the text 5. John Smith s refinement of Condorcet s notion of majority rule Not all elections generate a Condorcet candidate, a majority candidate according to Condorcet s proposed definition. John Howard Smith, a now-retired professor of mathematics at Boston College, proposed a generalized definition, which we will call the Smith candidates. When there is a Condorcet candidate, that candidate is the only Smith candidate. Otherwise, there are always several Smith candidates. John Smith s idea was that an election winner should always be a Smith candidate. When you understand it, you will probably agree. Chapter 4 of the text 2 His death in 1794 was a result of the French Revolution.

3 Getting rid of election spoilers after all (almost) Let us say that Smith candidates are strong, and others are weak. Well, then do this: First, compute which are the Smith candidates, and remove the others from the ballots. Then, do whatever you like. I call such a method a priori Smith-fair. In some sense, it gets rid of the spoiler problem: Weak candidates no longer affect the outcome, if you accept that weak candidates are those that are not Smith candidates. Ralph Nader could never have determined the outcome of the 2000 Presidential election in the U.S. if the Florida election method had been a priori Smith-fair; he wasn t a Smith candidate. Chapter 5 of the text 7. Very good but complicated: Schulze s beatpath method Schulze s beatpath method has many good properties, and it is explained and analyzed extensively in Chapter 6 of the text. I view it as an example of a method that illustrates that not all that can be done mathematically in this field is politically relevant. The method is so complicated that there does not seem to be the slightest chance of adopting it politically. However, it has been used by a large number of organizations the Free Software Foundation of Europe, the Pirate Party in many different countries, the European Democratic Education Community, and the Tufts University Mathematics Department, to name just a few. Chapter 6 of the text (This site has a rather complete list of good and bad properties of the method.) (This site allows you to organize elections based on the beatpath method online.) 8. Simple and still good: Condorcet + Borda Copeland s method (already discussed in the first lecture) is the simplest election method that respects Condorcet s criterion, and in fact even John Smith s refinement of the criterion. But it has a terrible disadvantage: Ties become uncomfortably likely even with a very large number of voters. I propose to break the ties by combining Copeland s method with another very simple method, Borda count, named after another French nobleman of the 1700s. 3 C. Börgers, Beyond instant runoff: A better way to conduct multi-candidate elections, The Conversation, April 2017 ( 3 Jean-Charles de Borda survived the French Revolution just fine, and died of natural causes in

4 C. Börgers, Simple methods for single winner elections, Lecture at SUnMaRC 2018, University of New Mexico Betsy Kaplan, Is There A Better Way To Vote In Connecticut s Primary?, Click on the audio and you will hear the June 28, 2018 Colin McEnroe Show. One of the segments is an interview with me, in which I argue against instant runoff. (I now believe that for political reasons, not for mathematical ones, what I said there is wrong.) 9. Politics vs. mathematics: the later-no-harm criterion and instant runoff At this point in the course, you will know what instant runoff means. It is often called ranked choice voting, and it has been used in various places in the United States. The state of Maine approved a ballot initiative in 2016 making instant runoff the method of choice in Maine elections, but implementation of this reform has been blocked by the Maine Supreme Court. At this point in the course, you will also understand the main objection to instant runoff: It violates Condorcet s criterion. In practical terms, this means that it will not usually allow the election of a compromise canddiate who is widely liked, but few people s favorite. Activists who advocate instant runoff argue that its principal advantage is the later-no-harm property: You cannot worsen your favorite candidate s chances by revealing your second choice. We will prove that insisting on this property leaves us with almost no choice other than instant runoff. A Better Electoral System in Maine by Eric Maskin and Amartya Sen, New York Times, June 10, In this article, Maskin and Sen, two Nobel-Prize-winning economists at Harvard University, support instant runoff, even though their preference is for a method that satisfies Condorcet s criterion. Later-no-harm criterion, Center for Election Science, C. Börgers, On the later-no-harm criterion for single-winner election methods, lecture notes (to be distributed) 10. No reasonable voting method rules out strategic voting Strategic voting means casting a vote that does not reflect your honest opinion, and thereby affecting the outcome in a way that you like. For instance, a Libertarian who votes Republican and thereby causes the victory of the Republican is said to have voted strategically if the voter prefers the Republican to the Democrat who would otherwise have won. Nothing is unethical about strategic voting. If you don t like it, you should design your election method such that it becomes impossible. However, a famous theorem, the Gibbard-Satterthwaite

5 Theorem, states that you cannot do that. We will take a look at what, exactly, the theorem says in a technical sense. Its proof is not easy, and we won t study it in this course. However, I will give a very simple mathematical argument that shows that any election method that rules out strategic voting would have to be very strange in some sense. Chapter 10 of the text 11. Ranking candidates, and Arrow s famous theorem The single most famous theorem in the theory of election methods is Arrow s theorem. It is an impossibility theorem, much like the Gibbard-Satterthwaite theorem mentioned in the preceding section. However, it refers to methods for ranking a field of candidates, not just selecting a single winner. We will study what the theorem says, and I will give a very simple mathematical argument that makes the theorem sound not very surprising. Chapter 12 of the text Part II: Electing a parliament (a group of winners) 12. Plurality voting in single-member districts, and gerrymandering A simple way of electing a parliament is to divide the country into districts, which each district electing one representative using one of the single winner methods described earlier. This approach is vulnerable to gerrymandering, the practice of drawing district boundaries to achieve outcomes desired by those who draw the boundaries. Because we read the news, we know there is gerrymandering in the United States. Is there gerrymandering in Europe? If not (or if less), what explains the difference? Mira Bernstein and Moon Duchin, A Formula Goes to Court: Partisan Gerrymandering and the Efficiency Gap, Notices of the American Mathematical Society, October Both authors are faculty at Tufts University. We will not be able to read the entire article in the time we will have, but we will read part of it. 13. Proportional representation, and the single transferable vote Proportional representation systems aim at giving a party (roughly) x percent of the seats in parliament if the party wins x percent of the vote. A simple voting system compatible with this idea is the single transferable vote system, a relative of the instant runoff method, but suitable for parliamentary elections.

6 Nicolaus Tideman, The Single Transferable Vote, J. Economic Perspectives, Eivind Stensholt, Single Transferable Votes with Tax Cuts, SIAM Review (Society for Industrial and Applied Mathematics), vol. 46, no. 3, pp (2004). This paper goes far beyond the scope of this class; we will nonetheless browse through it, and understand how the tax cut algorithm discussed in it works. 14. The French, German, UK, Italian, and European parliaments The details of the parliamentary election systems used in different countries can be complicated. Germany, for example, uses a system in which each voter has two votes one for a candidate, and one for a party list. Half the seats in parliament are filled using the first vote, but then an attempt is made at achieving proportional representation by adding candidates from the party lists. For the elections to the European Parliament, rules are somewhat different in different European countries, although all use proportional representation. We will study these and other examples and look for aspects that can be understood better by using mathematics. Internet resources such as: the-european-parliament-electoral-procedures 15. Democracy in Switzerland Less than 40 miles north of Talloires is Switzerland, which has rather unique democratic traditions, including a component of direct democracy. We will study how the direct democracy aspect of the Swiss system works, as well as the peculiarities of its proportional representation aspect, thinking about the latter from the point of view of the mathematics studied in this course particularities-switzerlands-proportional-election-system.html 16. Haute-Savoie department and Auvergne-Rhône-Alpes region We will try to learn as much as we can about voting in the Haute-Savoie department, and in the Auvergne-Rhône-Alpes region, of which Talloires is a part. We will invite local politicians from Annecy (the nearby capital of the Haute-Savoie

7 department) to speak to us about how voting is done locally. We will explore the possibility of excursions to Annecy, or even Grenoble (70 miles away, capital of the department of Isère, also part of the Auvergne-Rhône-Alpes region) or Lyon (100 miles away, capital of the Auvergne-Rhône-Alpes region). We will think about election methods in the region through the lens of the mathematics studied in this course, and in comparison with United States elections. We will be flexible about the timing of this section of the course, making it dependent on availability of guest speakers. Any excursion, of course, will in any case take place outside class time. Part III: Fair resource allocation 17. Notions of fairness in resource allocation Social choice theory is about more than voting, and the bulletin description of Math 19 has a fair resource allocation component. Here we have deliberately de-emphasized that aspect to make space for a more extensive treatment of voting, including examinations of voting methods in France and in Europe. However, we conclude the course with two sections on fair resource allocation. In the first, we define three desirable properties of a resource division: envyfreeness, Pareto-efficiency, and equitability. We give examples illustrating these ideas. Chapter 16 of the text 18. Fair resource allocation algorithms Many resource division algorithms have been proposed and analyzed by mathematicians. We give four examples: I cut, you choose (proposed for instance in the Bible, but also, in writing, long before that), Hugo Steinhaus extension of I cut, you choose to three people, Brams and Taylor s adjusted winner method (the objectively best way of dividing a resource among two people), and a brief sketch of fair resource allocation as a linear programming problem. (I will tell you what that means. The recognition that a problem can be phrased as a linear programming problem allows solving it with readily available software.) We will cut and share a real cake in class for illustration. Parts of Chapters 17, 21, 22, and 25 of the text. I will be more specific when we get here.

8 Learning objectives: Although this is a course about Social Choice Theory, there are more general learning objectives: 1. Learn to recognize when an argument (about anything that you encounter in your life, not necessarily about mathematics) is too imprecisely stated to decide whether it is correct or not, most typically because terms being used have not been defined with enough clarity. 2. Learn how to create simplified, abstracted mathematical versions of questions that arise outside mathematics, by giving your terms precise definitions, and using elementary mathematical concepts such as function, graph, set, and so on. 3. Learn or review elementary mathematical techniques to analyze the mathematical questions referred to in point 2: logic, functions, graphs, sets, mathematical induction, manipulating equations and inequalities, and others. To give you the practice needed to achieve these objectives, we primarily think about issues of fair social decision making. Without question these are important issues, but the objectives of the course are broader, and I will ask some homework and exam questions that support objectives 1 3 using other contexts as well. In-class exams: There will be a midterm exam and a final exam, dates to be announced. Online quizzes: Often (not always) you will be asked to do some reading for a class in advance, and complete an online multiple choice quiz about the material prior to coming to class. Homework: Homework problems will be assigned but not collected. You are invited to ask questions about the homework in class, during office hours, on the Piazza site for the course, or by . Eighty percent of the exam question will verbatim be taken from the homework questions. Class format and participation. The class meetings are longer than in Medford, and we will fill them with various activities, not by any means just lectures voting on various issues (to test election methods), discussions in smaller groups, discussion in the whole class, and of course the time-honored Math 19 tradition of cutting and sharing a real cake to test resource allocation methods. Thoughtful and engaged participation during class is obligatory. You are expected to ask questions, answer questions, contribute to the conversation. I understand that this is harder for some people than for others, but it is part of what is expected of you in this class. You will be assigned a class participation score out of 100. It will not be automatic, but also not difficult to get the full 100 points for class participation. You may assume that you are on track to get the full 100 points unless you hear from me about that.

9 Grades: You will be given scores out of 100 for the class participation (P), online quizzes (Q), the midterm exam (M), and the final exam (F). Then your total score will be the greater of the following numbers: 0.05P Q M F, 0.10P Q M + 0.5F. I use these weights to give you a chance to compensate for a weak midterm exam by strong performance on the final, and generally enthusiastic class participation. Course scores will be translated into grades as follows. Compute your final course score, between 0 and 100, as described above. Don t round yet. If the score (before rounding) is 98 or above, you get an A+. Otherwise, round to the nearest integer, and apply the following rules. 93 and above: A (unless you get an A+ by the rule stated earlier) 90-92: A 87-89: B : B 80-82: B 77-79: C : C If your score is below 73, but you made an honest effort 4, then you still get a C. Otherwise: 70-72: C 67-69: D : D 60-62: D 0-59: F Student Accessibility Services: If you are requesting an accommodation due to a documented disability, you must register with the Student Accessibility Services Office. To do so, call the Student Accessibility Services office at to arrange an appointment with Kirsten Behling, Director of Student Accessibility Services. 4 If for instance you skip class often, or come late often, or rarely participate in the conversation in class, or text on your phone in class, you aren t making an honest effort. Of course you aren t going to do any of those things! You may assume that I consider your effort sufficient for the guaranteed minimum grade of C if you don t hear otherwise from me.