Logical Methods in the Humanities

The Logic and Mathematics of Voting Procedures

Weekly seminar using logic and mathematics to study voting procedures.

Readings

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
This chapter discusses the use of Approval voting in actual elections.
6. 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.

Additional Material

• Michel Balinski and Rida Laraki, A theory of measuring, electing and ranking, Proceeding of the National Academy of Sciences USA, vol. 104, no. 21, pp. 8720-8725 May 22, 2007.
• Steven Brams, Marc Kilgour and William Zwicker, The Paradox of Multiple Elections, Social Choice and Welfare, Volume 15, Number 2, pgs. 211 - 236, 1998
• K. Dowding and M. Van Hees, In Praise of Manipulation, British Journal of Political Science, Volume 38, pgs. 1 - 15, 2007.
• Peter C. Fishburn and Steven J. Brams, Paradoxes of Preferential Voting, Mathematics Magazine, Vol. 56, No. 4, 1983, pp. 207-214.
• Stephan Hartmann, Gabriella Pigozzi and Jan Sprenger, Reliable Methods of Judgement Aggregation, 2008
• M. Van Hees, The limits of epistemic democracy, Social Choice and Welfare, Volume 28, Number 4, 2007.
• Lingfang Li and Donald Saari, Sen's Theorem: Geometric Proof and New Interpretations, Social Choice and Welfare, Volume 31, Number 3, 2008
• David Makinson, Combinatorial versus decision-theoretic components of impossibility theorems, Theory and Decision, Volume 40, pgs. 181 - 190, 1996
• Amartya Sen, The Possibility of Social Choice, Nobel Prize Lecture, 1998
• Patrick Suppes, The pre-history of Kenneth Arrow's social choice and individual values, Social Choice and Welfare, Volume 25, pgs. 319 - 329, 2005

• Papers on Judgement Aggregation
• Three videos created by Donald Saari for the 2008 Mathematics Awareness Month which had the theme "mathematics of voting":
  1. Mathematics of Voting
  2. Creating Voting Paradoxes
  3. What Causes Voting Paradoxes
• Description of a very interesting new approach to Social Choice theory avoiding many of the classic "paradoxes" (Enlish follows French)

• Y. Shoham and K. Leyton-Brown, Chapter 9: Aggregation Preferences: Social Choice, in Multiagent Systems: Algorithmic, Game Theoretic and Logical Foundations.
• K. Arrow, A.K. Sen, and K. Suzumura, Handbook of Social Choice and Welfare
• K. Arrow, Social Choice and Individual Values
• W. Poundstone, Gaming the Vote: Why Elections aren't Fair (and What We Can Do About It)
• S. Brams and P. Fishburn, Approval Voting
• M. Regenwetter, B. Grofman, A.A.J. Marley, I. Tsetlin, Behavioral Social Choice
• I. McClean and A. Urken, Classics of Social Choice