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
3 - What the IPS Tells Us About Fairness and Efficiency in the Two-Player Context
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 continue to restrict our attention to the two-player context and we consider how the fairness or efficiency of partitions is reflected in the IPS. In other words, if a partition P has some fairness property or some efficiency property, what can be said about the location of m(P) in the IPS? In Section 3A, we consider fairness; in Section 3B, we consider efficiency; and in Section 3C, we consider fairness and efficiency together. In these sections, we assume that measures m1 and m2 on some cake C are absolutely continuous with respect to each other. In Section 3D, we consider the situation when absolute continuity fails.
Fairness
We begin by noting that when there are only two players, proportionality and envy-freeness correspond:
〈P1, P2〉 is a proportional partition if and only if
〈P1, P2〉 is an envy-free partition if and only if
m1(P1) ≥ ½ and m2(p2) ≥ ½
Similarly, strong proportionality, strong envy-freeness, and super envy-freeness correspond:
〈P1, P2〉 is a strongly proportional partition if and only if
〈P1, P2〉 is a strongly envy-free partition if and only if
〈P1, P2〉 is a super envy-free partition if and only if
m1(P1) ≥ ½ and m2(p2) ≥ ½
In Chapter 4, we shall see that these notions are all distinct if there are more than two players.
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- Information
- The Geometry of Efficient Fair Division , pp. 25 - 55Publisher: Cambridge University PressPrint publication year: 2005