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
7 - Characterizing Pareto Optimality I: The IPS and Optimization of Convex Combinations of Measures
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 shall use the IPS to provide characterizations of Pareto maximality and Pareto minimality using the notion of convex combinations of measures. In Section 7A, we begin with a purely geometric description of this notion in the two-player context, and in Section 7B, we establish our characterization in the general n-player context. In these sections, we assume that the measures are absolutely continuous with respect to each other. In Section 7C, we consider the situation without absolute continuity.
Introduction: The Two-Player Context
We shall focus only on Pareto maximality in this section. All of the ideas in this section have analogous chores versions, but we will not state these here. We shall do so in the general n-player context in the next section.
Consider Figure 7.1. In each of the figures, we see the IPS for some cake C and measures m1 and m2, with the outer boundary darkened. (The IPS in Figures 7.1b and 7.1c is the same.) As illustrated in each figure, we imagine a line with negative slope, beginning to the upper right of the IPS and moving in a parallel manner until it makes contact with the IPS. Since the IPS is a closed subset of the plane, we know that there is a line in this family of parallel lines that makes first contact with the IPS.
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- The Geometry of Efficient Fair Division , pp. 160 - 189Publisher: Cambridge University PressPrint publication year: 2005