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. […]