+612 9045 4394
 
CHECKOUT
The Geometry of Efficient Fair Division - Julius B. Barbanel

The Geometry of Efficient Fair Division

By: Julius B. Barbanel, Alan D. Taylor (Introduction by)

Hardcover Published: 24th January 2005
ISBN: 9780521842488
Number Of Pages: 472

Share This Book:

Hardcover

RRP $373.99
$258.75
31%
OFF
or 4 easy payments of $64.69 with Learn more
Ships in 7 to 10 business days

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 (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 (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.

'In Chapters 12 and 13, he studies the relationship between the IPS and the RNS, and he provides a new presentation of the fundamental result that ensures the existence of a partition that is both Pareto optimal and envy-free.' Zentralblatt MATH
"The monograph is a clearly-written, matter-of-fact presentation of definitions, theorems, and proofs." MAA Reviews, Stephen Ahearn, Macalester College
"The main virtue of the book is the depth at which the author studies the division problem while maintaining the breadth across the mathematical sciences and the mathematical elegance with which he presents the results. The book should be of special interest not only to mathematicians and mathematical scientists but also to graduate students and researchers in management science, operations research, and system science who study resource allocation, optimization and decision making." Margaret M. Wiecek, MATHEMATICAL REVIEWS

Introductionp. 1
Notation and Preliminariesp. 7
Geometric Object #1a: The Individual Pieces Set (IPS) for Two Playersp. 16
What the IPS Tells Us About Fairness and Efficiency in the Two-Player Contextp. 25
Fairnessp. 25
Efficiencyp. 31
Fairness and Efficiency Together: Part 1ap. 37
The Situation Without Absolute Continuityp. 41
The Individual Pieces Set (IPS) and the Full Individual Pieces Set (FIPS) for the General n-Player Contextp. 56
Geometric Object #1b: The IPS for n Playersp. 56
Why the IPS Does Not Sufficep. 67
Geometric Object #1c: The FIPSp. 70
A Theorem on the Possibilities for the FIPSp. 74
What the IPS and the FIPS Tell Us About Fairness and Efficiency in the General n-Player Contextp. 82
Fairnessp. 82
Efficiencyp. 96
Fairness and Efficiency Together: Part 1bp. 107
The Situation Without Absolute Continuityp. 111
Examples and Open Questionsp. 134
Characterizing Pareto Optimality: Introduction and Preliminary Ideasp. 151
Characterizing Pareto Optimality I: The IPS and Optimization of Convex Combinations of Measuresp. 160
Introduction: The Two-Player Contextp. 160
The Characterizationp. 163
The Situation Without Absolute Continuityp. 168
Characterizing Pareto Optimality II: Partition Ratiosp. 190
Introduction: The Two-Player Contextp. 190
The Characterizationp. 192
The Situation Without Absolute Continuityp. 208
Geometric Object #2: The Radon-Nikodym Set (RNS)p. 220
The RNSp. 220
The Situation Without Absolute Continuityp. 230
Characterizing Pareto Optimality III: The RNS, Weller's Construction, and w-Associationp. 236
Introduction: The Two-Player Contextp. 236
The Characterizationp. 240
The Situation Without Absolute Continuityp. 260
The Shape of the IPSp. 286
The Two-Player Contextp. 286
The Case of Three or More Playersp. 291
The Relationship Between the IPS and the RNSp. 298
Introductionp. 298
Relating the IPS and the RNS in the Two-Player Contextp. 301
Relating the IPS and the RNS in the General n-Player Contextp. 308
The Situation Without Absolute Continuityp. 336
Fairness and Efficiency Together: Part 2p. 341
Other Issues Involving Weller's Construction, Partition Ratios, and Pareto Optimalityp. 352
The Relationship between Partition Ratios and w-Associationp. 352
Trades and Efficiencyp. 358
Classifying the Failure of Pareto Optimalityp. 369
Convexityp. 374
The Situation Without Absolute Continuityp. 376
Strong Pareto Optimalityp. 385
Introductionp. 385
The Characterizationp. 386
Existence Questions in the Two-Player Contextp. 394
Existence Questions in the General n-Player Contextp. 400
The Situation Without Absolute Continuityp. 409
Fairness and Efficiency Together: Part 3p. 414
Characterizing Pareto Optimality Using Hyperreal Numbersp. 416
Introductionp. 416
A Two-Player Examplep. 419
Three-Player Examplesp. 424
The Characterizationp. 435
Geometric Object #1d: The Multicake Individual Pieces Set (MIPS) Symmetry Restoredp. 444
The MIPS for Three Playersp. 444
The MIPS for the General n-Player Contextp. 447
Referencesp. 451
Indexp. 453
Symbol and Abbreviations Indexp. 462
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780521842488
ISBN-10: 0521842484
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 472
Published: 24th January 2005
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 23.5 x 15.7  x 3.0
Weight (kg): 0.74

This product is categorised by