# Logical Methods in the Humanities

## The Logic and Mathematics of Voting Procedures

Weekly seminar using logic and mathematics to study voting procedures.

1. Wulf Gaertner, A Primer in Social Choice Theory, Oxford University Press, pgs. 1 - 53
This selection contains a general introduction to Social Choice Theory, a statement and proof of Arrow's Theorem (actually three different proofs), a characterization of majority rule, a discussion of domain restrictions (eg., single-peaked preferences) and a statement and proof of Sen's Theorem about individual rights.
2. Alan D. Taylor, Social Choice and the Mathematics of Manipulation, pgs. 3 - 77
This selection contains a different perspective on Arrow's Theorem, discusses the issue of manipulability and the Gibbard-Satterthwaite Theorem (including a proof). It also contains a very nice discussion of different voting methods (plurality, borda, hare system, etc.) and some voting paradoxes.
3. Donald Saari, Decisions and Elections, Cambridge University Press, pgs. 69 - 101
Chapter 3 of this book (the first 2 chapters contain material we cover above) discusses Saari's perspective on some classic results in Social Choice Theory including his "explanation" of Arrow's Theorem (he also discusses Sen's result).
4. Steven Brams, Mathematics and Democracy, Princeton University Press, pgs. 23 - 45
Chapter 2 (Electing a Single Winner: Approval Voting in Theory) surveys a number of theoretical results about Approval Voting.
5. Steven Brams, Mathematics and Democracy, Princeton University Press, pgs. 3 - 21
6. This chapter discusses the use of Approval voting in actual elections.
7. Donald Saari, Basic Geometry of Voting, Springer, pgs. 1 - 27
This selection introduces Saari's geometric framework for studying voting procedures. Saari also argues that the Borda count is the "best" voting procedure.