Much of our daily lives are spent taking part in various types of social procedures. Examples range from voting in a national election to deliberating with others in small committees. Many interesting philosophical and mathematical issues arise when we carefully examine our group decision-making processes. For example, suppose that a group of friends are deciding where to go for dinner. If everyone agrees on which restaurant is best, then it is obvious where to go. But, how should the friends decide where to go if they have different opinions about which restaurant is best?

The course will examine the main topics in voting theory. Topics include a brief history of the theory of voting, a survey of different voting methods (such as plurality rule, Borda count, Approval voting, plurality with runoff), Voting paradoxes (the Condorcet Paradox, multiple election paradox, the no-show paradox), and axiomatic characterization of voting rules. The second part of the course will introduce the general theory of fair division (including cake-cutting algorithms and an introduction to the theory of social welfare).

This is an introductory course. Students will come away from this course with a working knowledge of voting theory and the theory of fair division.

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Related Courses. For more information about the topics which may be discussed during the semester, you can consult the webpages of the following related courses that I have taught:

• Rationality, Masters of philosophy course on the general theory of rational and social choice.
• Social Choice Theory for Logicians, Short course taught at the North American Summer School for Logic, Language and Information (NASSLLI 2012).

The required reading material for the course will be drawn from:

• K. Shepsle, Analyzing Politics, W. W. Norton & Company, 2010.
• S. Brams and A. Taylor, Fair Division, Cambridge University Press, 1996.
• E. Pacuit, Voting Methods, Stanford Encyclopedia of Philosophy, 2011.
• E. Pacuit, Background reading on sets and relations.

Reading will also include excerpts from the following texts

• Steven Brams, Mathematics and Democracy, Princeton University Press, 2007.
• S. Brams and P. Fishburn, Approval Voting
• I. McClean and A. Urken, Classics of Social Choice, University of Michigan Press, 1995.
• E. Pacuit, Voting Methods, Stanford Encyclopedia of Philosophy, 2011.
• W. Poundstone, Gaming the Vote: Why Elections aren't Fair (and What We Can Do About It), Hill and Wang, 2008
• M. Regenwetter, B. Grofman, A.A.J. Marley, I. Tsetlin, Behavioral Social Choice, Cambridge University Press, 2006.
• Donald Saari, Disposing Dictators, Demystifying Voting Paradoxes: Social Choice Analysis, Cambridge University Press, 2008.
• Donald Saari, Basic Geometry of Voting, Springer, 2003.
• Y. Shoham and K. Leyton-Brown, Chapter 9: Aggregation Preferences: Social Choice, in Multiagent Systems: Algorithmic, Game Theoretic and Logical Foundations, Cambridge University Press, 2009.
• Alan D. Taylor, Social Choice and the Mathematics of Manipulation, Cambridge University Press, 2005.

Please consult this schedule regularly throughout the semester as meeting times and readings may change.

Date	Lecture	Notes/Readings
8/30	Introductory Remarks (initial quiz)
9/4	Introductory remarks (slides)	Section 1 of [Voting Methods]
9/6	The rational choice model (slides)	Chapter 2 of [AP] Problem Set 1: exercises 2, 4 and 6, Chapter 2 of [AP], pgs. 35- 37
9/11	Paradoxes of Voting, I (slides)	Section 3.1 of [Voting Methods] and Chapter 4, pgs. 53 - 67 of [AP]
9/13	Paradoxes of Voting, II (slides)	Chapter 4, pgs. 53 - 67 of [AP] and Chapter 3 of [AP] Problem Set 1 Due G. Asan and R. Sanver, Another characterization of the majority rule, Economics Letters, 75:3, 2002
9/18	Paradoxes of Voting, III (slides)	Sections 3.3 and 4.2 of [Voting Methods]
9/20	Arrow's Theorem (handout) J. Geanakoplos, Three Brief Proofs of Arrow's Theorem (we will discuss Proof 1), Economic Theory, 26, pgs. 211 - 215, 2005.	Chapter 4, pgs. 67 -89 of [AP] Chapter 5 of Riker's Liberalism without Populism
9/25	Arrow's Theorem, continued (handout) Domain Restrictions: Single-Peakedness and Value Restrictions (slides)	pgs. 81- 85 of [AP] Problem Set 2: exercises 2, 3 and 4, Chapter 4 of [AP], pgs. 86, 87
9/27	Spatial Models of Majority Rule (slides)	Chapter 5 of [AP], pgs. 90 - 123
10/2	Condorcet Jury Theorem and Paradoxes of Judgement Aggregation (slides)	Chapter 11 of S. Nitzan, Collective Preference and Choice, Cambridge University Press, 2010 (available in class)
10/4	Doctrinal Paradox, Sen's Liberal Paradox (slides)	C. List and P. Pettit, Aggregating Sets of Judgments: An Impossibility Result, Economics and Philosophy 18: 89-110, 2002 A. Sen, Impossibility of the Paretian Liberal, Journal of Political Economy, 78:1, pgs. 152 - 157, 1970 Problem Set 2 Due
10/9	Strategizing, I (slides)	Chapter 2 of A. Taylor, Social Choice and the Mathematics of Manipulation, Cambridge University Press, 2005.
10/11	No Class (instructor away at conference)
10/16	Strategizing, II (slides)	Chapter 2 of A. Taylor, Social Choice and the Mathematics of Manipulation, Cambridge University Press, 2005.
10/18	Strategizing, III	Section 3.1 of A. Taylor, Social Choice and the Mathematics of Manipulation, Cambridge University Press, 2005. K. Dowding and M. van Hees, In Praise of Manipulation, British Journal of Political Science, 38:1, pgs. 1 - 15, 2008.
10/23	Distance Based Voting Methods, I (slides)	H. P. Young. Optimal Voting Rules. The Journal of Economic Perspectives, 9:1, pgs. 51 - 64, 1995.
10/25	Distance Based Voting Methods, II and Approval Voting (slides)	S. Brams, Chapter 2 in Mathematics and Democracy
10/30	University Closed
11/1	Approval Voting (slides)	E. Aragones, I Gilboa, and A. Weiss, Making statements and approval voting,Theory and Decision, 71:4, pp. 461 - 472, 2011
11/6	Majoritarian Judgement, I (slides)	Problem Set 3: exercise 7 on pg. 190, 1 on pg. 222, 3a and 3b. on pg. 223,224 Midterm Papers Due
11/8	Majoritarian Judgement, II (slides)
11/13	Combining Approval and Preference (slides)	Problem Set 3 Due
11/15	Introduction to Fair Division (slides)
11/20	Adjusted Winner (slides)	http://www.nyu.edu/projects/adjustedwinner/
11/22	Thanksgiving break
11/27	Cake-Cutting Algorithms, I	Chapter 1 of J. Robertson and W. Web. Cake-Cutting Algorithms: Be Fair If You Can. A K Peters/CRC Press, 1998.
11/29	Cake-Cutting Algorithms, II	Chapter 1 of J. Robertson and W. Web. Cake-Cutting Algorithms: Be Fair If You Can. A K Peters/CRC Press, 1998.
12/4	Cake-Cutting Algorithms, III (slides)	Problem Set 4
12/6	Cake-Cutting Algorithms, IV (slides)
12/11	Concluding Remarks and Review (slides)	Final Exam Material Problem Set 4 Due
12/18	Final Exam, 4-6 PM, SKN 1112

Your final grade will be based on problem sets (approximately 4-6 assigned throughout the semester), a short midterm paper (5-10 pages on a paradox of voting, due on October 23), and an in-class final exam. Your final grade will be calculated as follows: The problem sets count for 30% of your grade, the midterm paper counts for 35%, and the final exam counts for 35%.

For the problem sets, you are encouraged to work in small groups. You may discuss the problems with one another or with me as much as you want. But you must always do the final write-up completely on you own. A good strategy when working together is to use a blackboard and erase it completely before writing up your (separate) answers. Please write the name of your discussion partner(s) on the front page of your assignments.