Mathematics of Social Choice: Voting, Compensation, and Division
By: Christoph Borgers (author)Paperback
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Mathematics of Social Choice is a fun and accessible book that looks at the choices made by groups of people with different preferences, needs, and interests. Divided into three parts, the text first examines voting methods for selecting or ranking candidates. A brief second part addresses compensation problems wherein an indivisible item must be assigned to one of several people who are equally entitled to ownership of the item, with monetary compensation paid to the others. The third part discusses the problem of sharing a divisible resource among several people. Mathematics of Social Choice can be used by undergraduates studying mathematics and students whose only mathematical background is elementary algebra. More advanced material can be skipped without any loss of continuity. The book can also serve as an easy introduction to topics such as the Gibbard-Satterthwaite theorem, Arrow's theorem, and fair division for readers with more mathematical background.
Christoph Borgers has been a Professor in the Department of Mathematics at Tufts University since 1994. He has also worked at the University of Michigan and at the IBM T. J. Watson Research Center. He received his PhD from New York University.
Preface; Part I. Voting: 1. Winner selection; 2. Rule of the majority; 3. Election spoilers; 4. The Smith set; 5. Smith-fairness and the no-weak-spoiler criterion; 6. Schulze's beatpath method; 7. Monotonicity; 8. Elections with many or few voters; 9. Irrelevant comparisons and the Muller-Satterthwaite theorem; 10. Strategic voting and the Gibbard-Satterthwaite theorem; 11. Winner selection versus ranking; 12. Irrelevant alternatives and Arrow's theorem; Part II. Compensation: 13. Fairness and envy-freeness; 14. Pareto-optimability and equitability; 15. Equality, equitability and Knaster's procedure; Part III. Division: 16. Envy-free, Pareto-optimal, and equitable cake cutting; 17. 'I cut, you choose' for three: Steinhaus' method; 18. Hall's marriage theorem; 19. 'I cut, you choose' for more than three: Kuhn's methods; 20. The method of Selfridge and Conway; 21. The geometry of Pareto-optimal division between two people; 22. The adjusted winner method of Brams and Taylor; 23. Conflict resolution using the adjusted winner method; 25. Proportional allocation; 26. Dividing a piecewise homogeneous cake among N>2 people; Part IV: Appendices: A. Sets; B. Logic; C. Mathematical induction; D. Solutions to selected exercises; Index.
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