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.

In this chapter, we introduce and study partition ratios. These ratios are numbers that can be associated with any partition. They provide us with our second approach to characterizing Pareto maximality and Pareto minimality and will be useful in future chapters. In Section 8A we consider the two-player context, and in Section 8B we establish our characterization in the general n-player context. In these sections, we assume that the measures are absolutely continuous with respect to each other. In Section 8C we consider the situation without absolute continuity. The definition of partition ratios and the corresponding characterization are very similar to a notion and result that appeared in a preliminary version of E. Akin's [1] but did not appear in the published version.

Introduction: The Two-Player Context

Suppose that P = 〈P1, P2〉 is a partition of C that is not Pareto maximal, and let us assume for simplicity that P1 and P2 are both of positive measure. (We shall drop this simplifying assumption when we consider the general n-player context in the next section.) Since P is not Pareto maximal, there is a partition Q = 〈Q1, Q2〉 that is Pareto bigger than P. We can imagine the change from partition P to partition Q as being accomplished by a trade between the two players.

In this chapter, we explore an assortment of issues that did not fit naturally into previous chapters. In Sections 13A, 13B, 13C, and 13D, we assume that the measures are absolutely continuous with respect to each other. In Section 13E, we reconsider the results of these sections without this assumption.

The Relationship between Partition Ratios and w-Association

Suppose that a partition P of the cake C is Pareto maximal. For simplicity, we also assume that P ∈ Part+. (We recall that Part+ denotes the set of all partitions of C that give each player a piece of cake of positive measure.) For distinct i, j = 1, 2, …, n, let pri j be the i j partition ratio. (These are given by Definition 8.6. By Theorem 8.9, for any associated cyclic sequence φ, CP(φ), the cyclic product of φ, is at most one.) By Theorem 10.9, P is w-associated with some ω ∈ S+. In this section, we investigate the relationship between ω and the prij.

Let us first consider the two-player context with Player 1 and Player 2 and associated measures m1 and m2, respectively. The relevant RNS is the one-simplex, which is the line segment from Player 1's vertex, (1, 0), to Player 2's vertex, (0, 1).

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.

In this chapter, we consider the general case of n players. In Section 4A we consider the IPS, and then in Section 4B we shall see that the IPS is not a sufficient structure for studying all fairness properties when there are more than two players. In Section 4C, we generalize the IPS to the FIPS, the Full Individual Pieces Set. In Section 4D, we prove a general result about the possibilities for the FIPS. This result will be a central tool in our work in Chapter 5. We make no general assumptions about absolute continuity in this chapter.

Geometric Object #1b: The IPS for n Players

Before considering fairness and efficiency issues, we consider more general geometric issues, as we did in the two-player context. More specifically, we examine Theorems 2.2, 2.4, and 2.6 and consider appropriate generalizations to the n-player context. The definition of the IPS, of s-equivalence, and of p-equivalence are the obvious generalizations of the corresponding definitions for two players.

Definition 4.1

1. a. For any partition P = 〈P1, P2, …, Pn〉 of C, let m(P) = (m1(P1), m2(P2), …, mn(Pn)). The Individual Pieces Set, or IPS, is the set {m(P) : P ∈ Part}.

2. […]

In this chapter, we shall use the IPS to provide characterizations of Pareto maximality and Pareto minimality using the notion of convex combinations of measures. In Section 7A, we begin with a purely geometric description of this notion in the two-player context, and in Section 7B, we establish our characterization in the general n-player context. In these sections, we assume that the measures are absolutely continuous with respect to each other. In Section 7C, we consider the situation without absolute continuity.

Introduction: The Two-Player Context

We shall focus only on Pareto maximality in this section. All of the ideas in this section have analogous chores versions, but we will not state these here. We shall do so in the general n-player context in the next section.

Consider Figure 7.1. In each of the figures, we see the IPS for some cake C and measures m1 and m2, with the outer boundary darkened. (The IPS in Figures 7.1b and 7.1c is the same.) As illustrated in each figure, we imagine a line with negative slope, beginning to the upper right of the IPS and moving in a parallel manner until it makes contact with the IPS. Since the IPS is a closed subset of the plane, we know that there is a line in this family of parallel lines that makes first contact with the IPS.

In Chapter 7, we studied the relationship between the IPS and maximization of convex combinations of measures. In Chapter 10 (see Theorem 10.6) we used the notion of w-association to study the relationship between the RNS and maximization of convex combinations of measures. In this chapter, we put these ideas together to enable us to understand the relationship between the IPS and the RNS. In Section 12A, we introduce a relation that will be useful in Sections 12B, 12C, and 12D. In Section 12B, we examine this relation in the two-player context. In Section 12C, we consider the general n-player context. We assume in Sections 12A, 12B, and 12B that the measures are absolutely continuous with respect to each other. In Section 12D, we consider the situation without this assumption. In Section 12E, we also do not assume that the measures are absolutely continuous with respect to each other and we use the IPS and the RNS together to show that there exists a partition that is Pareto maximal and envy-free.

Introduction

We recall that S+ denotes the interior of the simplex S. For ω ∈ S+, we let ω* denote the set of partitions that are w-associated with ω.

In this chapter, we introduce the first of two geometric objects that we associate with cake division. We also introduce various notions and questions that will be important in later chapters. We call this geometric object the Individual Pieces Set, or IPS. Our present focus is the two-player context. In Chapter 4, we consider the general case of n players, where we shall also introduce a generalized version of the IPS, called the Full Individual Pieces Set. Throughout this chapter, the measures m1 and m2 may or may not be absolutely continuous with respect to each other.

Definition 2.1 For any partition P = 〈P1, P2〉 of C, let m(P) = (m1(P1), m2(P2)). The Individual Pieces Set, or IPS, is the set {m (P) : P ∈ Part}.

Notice that IPS ⊆ R2.

Of course, the IPS depends upon C, m1, and m2, and thus we shall always need to be sure that when we write “the IPS” the corresponding cake and measures are clear by context.

We wish to understand the general shape and geometric properties of the IPS. What do we know about points in the IPS? We can imagine all of the cake being given to Player 1. The associated partition is 〈C, ø〉 and the corresponding point in the IPS is (1, 0).

In this chapter, we investigate the possible shapes of the IPS. In Section 11A, we consider the two-player context, where we shall provide a complete answer. In Section 11B, we consider the n-player context for n > 2, where we are able to provide only a partial answer. For any cake C and corresponding measures m1, m2, …, mn, let IPS(C; m1, m2, …, mn) denote the IPS corresponding to cake C and measures m1, m2, …, mn on C. We make no general assumptions about absolute continuity in this chapter.

The Two-Player Context

In Chapter 2, we considered various properties of the IPS for the case of two players. In particular, we established Theorem 2.4, which told us that the IPS

1. a. is a subset of [0, 1]2,

2. b. contains the points (1, 0) and (0, 1),

3. c. is closed,

4. d. is convex, and

5. e. is symmetric about the point (½, ½).

In this section, we show that these five properties completely characterize the possible shapes of the IPS. In other words, for any G ⊆ R2 that satisfies these five conditions, there is a cake C and measures m1 and m2 on C so that G = IPS(C; m1, m2).

In this chapter, we consider generalizations of our fairness and efficiency results from Chapter 3 to the general n-player context. In Section 5A, we consider fairness; in Section 5B, we consider efficiency; and in Section 5C, we consider fairness and efficiency together. In these sections, we assume that the measures are absolutely continuous with respect to each other. In Section 5D, we consider the situation when absolute continuity fails. In Section 5E, where we consider examples and open questions, absolute continuity will sometimes hold and sometimes fail.

Fairness

We begin by recalling that, when there are two players, proportionality and envy-freeness correspond, as do strong proportionality, strong envy-freeness, and super envy-freeness. The following two facts are implied by Corollary 3.3 for the two-player context.

1. Fact 1: There exist infinitely many mutually non-s-equivalent partitions that are proportional and envy-free.

2. Fact 2: There exist partitions that are strongly proportional, strongly envy-free, and super envy-free if and only if the measures are not equal (and, in this case, there are infinitely many mutually non-p-equivalent such partitions).

Analogous facts for chores fairness properties are implied by Corollary 3.5.

The natural guess for the generalization of Fact 1 when there are more than two players is obvious and turns out to be true.

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

1. a. C ∈ W,

2. b. if A ∈ W then C\A ∈ W, and

3. 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

1. a. µ(A) ≥ 0 for every A ∈ W,

2. b. µ(ø) = 0, and

3. 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).

4. In addition, µ is

5. 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

6. e. a probability measure if and only if µ(C) = 1.

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.

In this chapter, we introduce the second of the two geometric objects that we associate with cake division. We call this object the Radon—Nikodym Set, or RNS. For our first geometric object, the IPS (or, more generally, the FIPS), we were interested in a geometric perspective on the set of all partitions of C. Our present goal is quite different. When we introduced the IPS, we started with the cake C, we considered the set of all partitions of C, and then we formed a geometric object, the IPS (or the FIPS), that contains useful information about this set. Now, we start with the cake C, we form a geometric object, the RNS, and then we use this new geometric object to construct partitions having desired properties. In the next chapter, we will use the RNS to obtain a new characterization of Pareto maximality and Pareto minimality. In Chapter 12, we will study the relationship between the IPS and the RNS.

In Section 9A, we assume that the measures are absolutely continuous with respect to each other. In Section 9B, we consider what happens when absolute continuity fails. Much of the material in this chapter is attributable to D. Weller ([43]).

In this chapter, we show how the use of hyperreal numbers can simplify our previous characterizations of Pareto optimality. In particular, we shall see that this approach will allow us to avoid the iterative procedures using partition sequence pairs, which involved a-maximization and b-maximization of convex combinations of measures in Chapter 7, and w-association in Chapter 10. Our new approach will also allow us to avoid the assumption (which we needed at times in Chapters 7 and 10) that each player receives a piece of cake that he or she believes to be of positive measure.

In Section 15A, we give the necessary background on hyperreals. In Section 15B, we illustrate the use of hyperreals by considering a two-player example. In Section 15C, we do the same for a three-player example. In Section 15D, we state and prove our new characterization. We make no general assumptions about absolute continuity in this chapter (although our examples in Sections 15B and 15C involve the failure of absolute continuity).

Introduction

Although the foundations of calculus evolved over many years, Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716) are generally considered to be the inventors of modern calculus. Their work involved two different but related notions, limits and infinitesimals.