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).