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
16 - Geometric Object #1d: The Multicake Individual Pieces Set (MIPS) Symmetry Restored
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
By Lemma 2.3, the IPS is always symmetric about the point (½, ½) when there are two players. In particular, given any point in the IPS, we obtain the reflection of that point about (½, ½) by simply having the two players trade pieces. This provides a one-to-one correspondence between the set of Pareto maximal points and the set of Pareto minimal points. However, we have seen that there is no analogous symmetry when there are more than two players. (See the discussion following Corollary 4.6, the concluding comments in Chapter 7, Theorem 11.5, and the discussion before and after Theorem 11.5. Corollary 4.9 revealed a type of symmetry, but not a precise symmetry about a particular point.) In this chapter, we show that the IPS can be viewed as part of a larger and more general structure, the Multicake Individual Pieces Set, or MIPS. The MIPS has nice symmetry properties that are not generally present in the IPS when there are more than two players.
In Section 16A, we consider the MIPS for three players. (At the end of that section, we comment on why the two-player situation is trivial and uninteresting.) In Section 16B, we consider the general case of n players. We make no general assumptions about absolute continuity in this chapter.
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- Information
- The Geometry of Efficient Fair Division , pp. 444 - 450Publisher: Cambridge University PressPrint publication year: 2005