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
10 - Characterizing Pareto Optimality III: The RNS, Weller's Construction, and w-Association
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 use the structure introduced in the previous chapter (i.e., the RNS) to develop our third approach to characterizing Pareto maximality and Pareto minimality. We begin in Section 10A by examining the two-player context. In Section 10B, we show how to use the RNS to associate one or more partitions with each point in the interior of the simplex, and then we use this idea to characterize Pareto maximality and Pareto minimality. In Sections 10A and 10B, we assume that the measures are absolutely continuous with respect to each other. In Section 10C, we consider what happens when absolute continuity fails.
Introduction: The Two-Player Context
We begin this section with a brief discussion and three examples in the two-player context. This will provide motivation for the general situation.
We assume that there are two players, Player 1 and Player 2, whom we shall refer to as “she” and “he” respectively, and we consider the RNS associated with these players' measures. Since there are two players, the setting for the RNS is the one-simplex, which is the line segment between (1, 0) and (0, 1). The closer a point of the RNS is to (1, 0), the more it is valued by Player 1 (in comparison with Player 2) and the closer a point of the RNS is to (0, 1), the more it is valued by Player 2 (in comparison with Player 1).
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- The Geometry of Efficient Fair Division , pp. 236 - 285Publisher: Cambridge University PressPrint publication year: 2005