Book contents
- Frontmatter
- Contents
- Introduction by Alan D. Taylor
- 1 Notation and Preliminaries
- 2 Geometric Object #1a: The Individual Pieces Set (IPS) for Two Players
- 3 What the IPS Tells Us About Fairness and Efficiency in the Two-Player Context
- 4 The Individual Pieces Set (IPS) and the Full Individual Pieces Set (FIPS) for the General n-Player Context
- 5 What the IPS and the FIPS Tell Us About Fairness and Efficiency in the General n-Player Context
- 6 Characterizing Pareto Optimality: Introduction and Preliminary Ideas
- 7 Characterizing Pareto Optimality I: The IPS and Optimization of Convex Combinations of Measures
- 8 Characterizing Pareto Optimality II: Partition Ratios
- 9 Geometric Object #2: The Radon–Nikodym Set (RNS)
- 10 Characterizing Pareto Optimality III: The RNS, Weller's Construction, and w-Association
- 11 The Shape of the IPS
- 12 The Relationship Between the IPS and the RNS
- 13 Other Issues Involving Weller's Construction, Partition Ratios, and Pareto Optimality
- 14 Strong Pareto Optimality
- 15 Characterizing Pareto Optimality Using Hyperreal Numbers
- 16 Geometric Object #1d: The Multicake Individual Pieces Set (MIPS) Symmetry Restored
- References
- Index
- Symbol and Abbreviations Index
2 - Geometric Object #1a: The Individual Pieces Set (IPS) for Two Players
Published online by Cambridge University Press: 19 August 2009
- Frontmatter
- Contents
- Introduction by Alan D. Taylor
- 1 Notation and Preliminaries
- 2 Geometric Object #1a: The Individual Pieces Set (IPS) for Two Players
- 3 What the IPS Tells Us About Fairness and Efficiency in the Two-Player Context
- 4 The Individual Pieces Set (IPS) and the Full Individual Pieces Set (FIPS) for the General n-Player Context
- 5 What the IPS and the FIPS Tell Us About Fairness and Efficiency in the General n-Player Context
- 6 Characterizing Pareto Optimality: Introduction and Preliminary Ideas
- 7 Characterizing Pareto Optimality I: The IPS and Optimization of Convex Combinations of Measures
- 8 Characterizing Pareto Optimality II: Partition Ratios
- 9 Geometric Object #2: The Radon–Nikodym Set (RNS)
- 10 Characterizing Pareto Optimality III: The RNS, Weller's Construction, and w-Association
- 11 The Shape of the IPS
- 12 The Relationship Between the IPS and the RNS
- 13 Other Issues Involving Weller's Construction, Partition Ratios, and Pareto Optimality
- 14 Strong Pareto Optimality
- 15 Characterizing Pareto Optimality Using Hyperreal Numbers
- 16 Geometric Object #1d: The Multicake Individual Pieces Set (MIPS) Symmetry Restored
- References
- Index
- Symbol and Abbreviations Index
Summary
In this chapter, we introduce the first of two geometric objects that we associate with cake division. We also introduce various notions and questions that will be important in later chapters. We call this geometric object the Individual Pieces Set, or IPS. Our present focus is the two-player context. In Chapter 4, we consider the general case of n players, where we shall also introduce a generalized version of the IPS, called the Full Individual Pieces Set. Throughout this chapter, the measures m1 and m2 may or may not be absolutely continuous with respect to each other.
Definition 2.1 For any partition P = 〈P1, P2〉 of C, let m(P) = (m1(P1), m2(P2)). The Individual Pieces Set, or IPS, is the set {m (P) : P ∈ Part}.
Notice that IPS ⊆ R2.
Of course, the IPS depends upon C, m1, and m2, and thus we shall always need to be sure that when we write “the IPS” the corresponding cake and measures are clear by context.
We wish to understand the general shape and geometric properties of the IPS. What do we know about points in the IPS? We can imagine all of the cake being given to Player 1. The associated partition is 〈C, ø〉 and the corresponding point in the IPS is (1, 0).
- Type
- Chapter
- Information
- The Geometry of Efficient Fair Division , pp. 16 - 24Publisher: Cambridge University PressPrint publication year: 2005