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
1 - Notation and Preliminaries
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
Our “cake” C is some set. We wish to partition C among n players, whom we shall refer to as Player 1, Player 2, …, Player n. For each i = 1, 2, …, n, Player i uses a measure mi to evaluate the size of pieces of cake (i.e., subsets of C). Unless otherwise noted, we shall always assume that C is non-empty.
Definition 1.1 A σ- algebra on C is a collection of subsets W of C satisfying that
a. C ∈ W,
b. if A ∈ W then C\A ∈ W, and
c. if Ai ∈ W for every i ∈ N, then (∪i∈NAi) ∈ W (where N denotes the set of natural numbers).
Definition 1.2 Assume that some σ-algebra W has been defined on C. A countably additive measure on W is a function µ : W → R (where R denotes the set of real numbers) satisfying that
a. µ(A) ≥ 0 for every A ∈ W,
b. µ(ø) = 0, and
c. if A1, A2, … is a countable collection of elements of W and this collection is pairwise disjoint, then µ(∪i∈NAi) = ∑i∈N µ(Ai).
In addition, µ is
d. non-atomic if and only if, for any A ∈ W, if µ(A) > 0 then for some B ⊆ A, B ∈ W and 0 < µ(B) < µ(A) and
e. a probability measure if and only if µ(C) = 1.
- Type
- Chapter
- Information
- The Geometry of Efficient Fair Division , pp. 7 - 15Publisher: Cambridge University PressPrint publication year: 2005