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
14 - Strong Pareto Optimality
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 study a natural strengthening of Pareto maximality and Pareto minimality. After introducing this notion in Section 14A, we present various characterizations in Section 14B. In Sections 14C and 14D, we consider existence questions in the two-player context and in the general n-player context, respectively. In Sections 14A, 14B, 14C, and 14D, we assume that the measures are absolutely continuous with respect to each other. In Section 14E, we consider what happens when absolute continuity fails. In Section 14F, we also do not assume that the measures are absolutely continuous with respect to each other and we consider connections with the main theorem of Section 12E.
Introduction
One way to describe Pareto optimality is to say that a partition P is Pareto optimal if and only if no collection of transfers of cake among the players produces a partition that makes every player at least as happy and makes at least one player strictly happier. We strengthen this by insisting that any non-trivial (in a sense to be made precise) collection of transfers produces a partition that makes at least one player less happy.
Notice that if we start with a partition P and transfer various pieces of cake between various players, and each transferred piece has measure zero, then certainly the resulting partition makes no player less (or more) happy.
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- Information
- The Geometry of Efficient Fair Division , pp. 385 - 415Publisher: Cambridge University PressPrint publication year: 2005