The geometry of efficient fair division / Julius B. Barbanel ; with an introduction by Alan D. Taylor.

Format:

Book

Author/Creator:

Barbanel, Julius B., 1951-

Contributor:

Hazel M. Hussong Fund.

Language:

English

Subjects (All):

Partitions (Mathematics).

Physical Description:

ix, 462 pages : illustrations ; 24 cm

Place of Publication:

Cambridge, UK ; New York : Cambridge University Press, 2005.

Summary:

What is the best way to divide a "cake" and allocate the pieces among some finite collection of players? In this book, the cake is a measure space, and each player uses a countably additive, non-atomic probability measure to evaluate the size of the pieces of cake, with different players generally using different measures. The author investigates efficiency properties (such as Pareto maximality: is there another partition that would make everyone at least as happy, and would make at least one player happier, than the present partition?) and fairness properties (such as envy - freeness: do all players think that their piece is at least as large as every other player's piece?). He focuses exclusively on abstract existence results rather than algorithms, and on the geometric objects that arise naturally in this context. By examining the shape of these objects and the relationship between them, he demonstrates results concerning the existence of efficient and fair partitions.

This is a work of mathematics that will be of interest to both mathematicians and economists.

Contents:

2 Geometric Object #1a: The Individual Pieces Set (IPS) for Two Players 16

3 What the IPS Tells Us About Fairness and Efficiency in the Two-Player Context 25

3A Fairness 25

3B Efficiency 31

3C Fairness and Efficiency Together: Part 1a 37

3D The Situation Without Absolute Continuity 41

4 The Individual Pieces Set (IPS) and the Full Individual Pieces Set (FIPS) for the General n-Player Context 56

4A Geometric Object #1b: The IPS for n Players 56

4B Why the IPS Does Not Suffice 67

4C Geometric Object #1c: The FIPS 70

4D A Theorem on the Possibilities for the FIPS 74

5 What the IPS and the FIPS Tell Us About Fairness and Efficiency in the General n-Player Context 82

5A Fairness 82

5B Efficiency 96

5C Fairness and Efficiency Together: Part 1b 107

5D The Situation Without Absolute Continuity 111

5E Examples and Open Questions 134

6 Characterizing Pareto Optimality: Introduction and Preliminary Ideas 151

7 Characterizing Pareto Optimality I: The IPS and Optimization of Convex Combinations of Measures 160

7A Introduction: The Two-Player Context 160

7B The Characterization 163

7C The Situation Without Absolute Continuity 168

8 Characterizing Pareto Optimality II: Partition Ratios 190

8A Introduction: The Two-Player Context 190

8B The Characterization 192

8C The Situation Without Absolute Continuity 208

9 Geometric Object #2: The Radon-Nikodym Set (RNS) 220

9A The RNS 220

9B The Situation Without Absolute Continuity 230

10 Characterizing Pareto Optimality III: The RNS, Weller's Construction, and w-Association 236

10A Introduction: The Two-Player Context 236

10B The Characterization 240

10C The Situation Without Absolute Continuity 260

11 The Shape of the IPS 286

11A The Two-Player Context 286

11B The Case of Three or More Players 291

12 The Relationship Between the IPS and the RNS 298

12B Relating the IPS and the RNS in the Two-Player Context 301

12C Relating the IPS and the RNS in the General n-Player Context 308

12D The Situation Without Absolute Continuity 336

12E Fairness and Efficiency Together: Part 2 341

13 Other Issues Involving Weller's Construction, Partition Ratios, and Pareto Optimality 352

13A The Relationship between Partition Ratios and w-Association 352

13B Trades and Efficiency 358

13C Classifying the Failure of Pareto Optimality 369

13D Convexity 374

13E The Situation Without Absolute Continuity 376

14 Strong Pareto Optimality 385

14B The Characterization 386

14C Existence Questions in the Two-Player Context 394

14D Existence Questions in the General n-Player Context 400

14E The Situation Without Absolute Continuity 409

14F Fairness and Efficiency Together: Part 3 414

15 Characterizing Pareto Optimality Using Hyperreal Numbers 416

15B A Two-Player Example 419

15C Three-Player Examples 424

15D The Characterization 435

16 Geometric Object #1d: The Multicake Individual Pieces Set (MIPS) Symmetry Restored 444

16A The MIPS for Three Players 444

16B The MIPS for the General n-Player Context 447.

Notes:

Includes bibliographical references (pages 451-452) and indexes.

Local Notes:

Acquired for the Penn Libraries with assistance from the Hazel M. Hussong Fund.

ISBN:

0521842484

OCLC:

54972740

Online:

Publisher description