In this chapter, we use the structure introduced in the previous chapter (i.e., the RNS) to develop our third approach to characterizing Pareto maximality and Pareto minimality. We begin in Section 10A by examining the two-player context. In Section 10B, we show how to use the RNS to associate one or more partitions with each point in the interior of the simplex, and then we use this idea to characterize Pareto maximality and Pareto minimality. In Sections 10A and 10B, we assume that the measures are absolutely continuous with respect to each other. In Section 10C, we consider what happens when absolute continuity fails.

Introduction: The Two-Player Context

We begin this section with a brief discussion and three examples in the two-player context. This will provide motivation for the general situation.

We assume that there are two players, Player 1 and Player 2, whom we shall refer to as “she” and “he” respectively, and we consider the RNS associated with these players' measures. Since there are two players, the setting for the RNS is the one-simplex, which is the line segment between (1, 0) and (0, 1). The closer a point of the RNS is to (1, 0), the more it is valued by Player 1 (in comparison with Player 2) and the closer a point of the RNS is to (0, 1), the more it is valued by Player 2 (in comparison with Player 1).