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.