The Geometry of Efficient Fair Division

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Cambridge University Press, Jan 24, 2005 - Mathematics
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.
 

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Contents

Introduction by Alan D Taylor page
1
Notation and Preliminaries
7
The Individual Pieces Set IPS
16
What the IPS Tells Us About Fairness and Efficiency in
25
The Individual Pieces Set IPS and the Full Individual
56
What the IPS and the FIPS Tell Us About Fairness
82
Introduction
151
The IPS
160
The RNS Wellers
236
The Shape of the IPS
286
The Relationship Between the IPS and the RNS
298
Other Issues Involving Wellers Construction Partition
352
Strong Pareto Optimality
385
Characterizing Pareto Optimality Using Hyperreal
416
The Multicake Individual Pieces
444
References
451

Partition Ratios
190
The RadonNikodym Set RNS
220

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About the author (2005)

Julius B. Barbanel is Professor of Mathematics at Union College, where he has also served as Department Chair. He has published numerous articles in the areas of both Logic and Set Theory, and Fair Division in leading mathematical journals. He is a member of the Mathematical Association of American, the Association of Symbolic Logic, and the Game Theory Society.

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