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.