## MATH 342: The Mathematics of Social Choice

These are lecture notes for an upper-level undergraduate mathematics course, MATH 342: The Mathematics of Social Choice. I created this course and have taught it four times, most recently in Fall 2017. Because I couldn't find any good textbooks for this audience, I created these notes, which I provide to the students each day.

On the politics side, this course covers voting (Lectures 2–5), apportionment (30–32), and fair division (33–36). On the economics side, it mainly covers auction theory, including strategy-proof auctions (6–11), matching markets (12–19), and strategizing (20–29). It seems fairly unusual to teach this at the upper-level (the politics topics tend to be covered in “math for the liberal arts“ type courses, and the economics topics are more likely to be taught in a graduate microeconomics course). But the students enjoy the mix of math, economics, politics, and computer science.

## Files

 Lecture Notes These are the complete lecture notes for the class. Lecture Notes, with blanks For the handouts to students, the notes would have had blank spots, to fill in during class with answers, steps, or explanations. Try to fill in the boxes! Problem Sets These problems comprised the weekly problem sets in Fall 2017. Syllabus From Fall 2017.

## Outline

Each lecture is designed to be worked through in a 50 minute class period. The main prerequisite is the mathematical maturity to tackle an upper-level math class. Lectures 20–29 also require the basics of integration at the level of a Calculus I course.

 Lecture(s) Description 1 An introduction and overview. 2–5 Voting methods, culminating in Arrow's Impossibility Theorem and the Gibbard-Satterthwaite Theorem (on strategizing in elections). 6–11 Theory of strategy-proof auctions, such as the second price auction and the Vickrey-Clarke-Groves mechanism. 12–16 The special case of auctions in a matching market (there are multiple objects for sale, but each buyer only wants one of them). 17–19 Matching markets, if money is not allowed to change hands. 20–23 Background in utility theory, game theory, and continuous probability so that we can talk about strategizing (in particular, Nash Equilibria and Bayesian Nash Equilibria). 24–29 Strategizing in various types of auctions, culminating in the Revenue Equivalence Theorem. 30–32 Apportionment (like in Congressional seats). This doesn't require any of the previous material 33–36 Various solutions to the fair division problem (in particular, examining what “fairness” even means). 37 Conclusion.

## Say Hi!

If you're interested in using this material to teach a class, if you're interested in using it to learn something about the subject, or you simply want to make me feel good, then please do contact me!

## Other Resources

After I created these notes, the textbook Game Theory, Alive (by Anna Karlin and Yuval Peres) came out; it covers similar topics and more, and it might be appropriate for upper-level undergraduates. The Institute for Mathematics and Democracy, which I'm an affiliate of, convinced me to post these notes online. Their website has many other great resources!

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