Mathematics of Social Choice
Voting, Compensation, and Division
Published on 30. December 2009
Book
Paperback/Softback
256 pages
978-0-89871-695-5 (ISBN)
Description
How do you select a winner from a field of candidates? How do you rank a field of candidates? How do you share a divisible resource like a cake, or an indivisible one like a pet or a house? These are the questions addressed in this fun and accessible book that takes an entertaining look 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 mathematics majors as well as students whose only mathematical background is elementary algebra. Material geared toward more sophisticated readers can be skipped without any loss of continuity. The book includes many elementary and usually simple, but rigorous mathematical proofs appropriate for beginning mathematics majors. Students will also find appendices with background material on set notation, logic, and mathematical induction and solutions to many of the homework exercises.
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 mathematics majors as well as students whose only mathematical background is elementary algebra. Material geared toward more sophisticated readers can be skipped without any loss of continuity. The book includes many elementary and usually simple, but rigorous mathematical proofs appropriate for beginning mathematics majors. Students will also find appendices with background material on set notation, logic, and mathematical induction and solutions to many of the homework exercises.
More details
Language
English
Place of publication
New York
United States
Target group
Professional and scholarly
Product notice
Paperback (trade)
Unsewn / adhesive bound
Dimensions
Height: 259 mm
Width: 179 mm
Thickness: 22 mm
Weight
481 gr
ISBN-13
978-0-89871-695-5 (9780898716955)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Person
Christoph Boergers 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.
Content
Preface
Part I: Voting: Chapter 1: Winner selection
Chapter 2: Rule of the majority
Chapter 3: Election spoilers
Chapter 4: The Smith set
Chapter 5: Smith-fairness and the no-weak-spoiler criterion
Chapter 6: Schulze's beatpath method
Chapter 7: Monotonicity
Chapter 8: Elections with many or few voters
Chapter 9: Irrelevant comparisons and the Muller-Satterthwaite theorem
Chapter 10: Strategic voting and the Gibbard-Satterthwaite theorem
Chapter 11: Winner selection versus ranking
Chapter 12: Irrelevant alternatives and Arrow's theorem
Part II: Compensation: Chapter 13: Fairness and envy-freeness
Chapter 14: Pareto-optimality and equitability
Chapter 15: Equality, equitability, and Knaster's procedure
Part III: Division: Chapter 16: Envy-free, Pareto-optimal, and equitable cake cutting
Chapter 17: "I cut, you choose" for three: Steinhaus's method
Chapter 18: Hall's marriage theorem
Chapter 19: "I cut, you choose" for more than three: Kuhn's methods
Chapter 20: The method of Selfridge and Conway
Chapter 21: The geometry of Pareto-optimal division between two people
Chapter 22: The adjusted winner method of Brams and Taylor
Chapter 23: Conflict resolution using the adjusted winner method
Chapter 24: The effect of dishonesty on the adjusted winner method
Chapter 25: Proportional allocation
Chapter 26: Dividing a piecewise homogeneous cake among N>2 people
Part IV: Appendices: Appendix A: Sets
Appendix B: Logic
Appendix C: Mathematical induction
Appendix D: Solutions to selected exercises
Index.
Part I: Voting: Chapter 1: Winner selection
Chapter 2: Rule of the majority
Chapter 3: Election spoilers
Chapter 4: The Smith set
Chapter 5: Smith-fairness and the no-weak-spoiler criterion
Chapter 6: Schulze's beatpath method
Chapter 7: Monotonicity
Chapter 8: Elections with many or few voters
Chapter 9: Irrelevant comparisons and the Muller-Satterthwaite theorem
Chapter 10: Strategic voting and the Gibbard-Satterthwaite theorem
Chapter 11: Winner selection versus ranking
Chapter 12: Irrelevant alternatives and Arrow's theorem
Part II: Compensation: Chapter 13: Fairness and envy-freeness
Chapter 14: Pareto-optimality and equitability
Chapter 15: Equality, equitability, and Knaster's procedure
Part III: Division: Chapter 16: Envy-free, Pareto-optimal, and equitable cake cutting
Chapter 17: "I cut, you choose" for three: Steinhaus's method
Chapter 18: Hall's marriage theorem
Chapter 19: "I cut, you choose" for more than three: Kuhn's methods
Chapter 20: The method of Selfridge and Conway
Chapter 21: The geometry of Pareto-optimal division between two people
Chapter 22: The adjusted winner method of Brams and Taylor
Chapter 23: Conflict resolution using the adjusted winner method
Chapter 24: The effect of dishonesty on the adjusted winner method
Chapter 25: Proportional allocation
Chapter 26: Dividing a piecewise homogeneous cake among N>2 people
Part IV: Appendices: Appendix A: Sets
Appendix B: Logic
Appendix C: Mathematical induction
Appendix D: Solutions to selected exercises
Index.