By Lemma 2.3, the IPS is always symmetric about the point (½, ½) when there are two players. In particular, given any point in the IPS, we obtain the reflection of that point about (½, ½) by simply having the two players trade pieces. This provides a one-to-one correspondence between the set of Pareto maximal points and the set of Pareto minimal points. However, we have seen that there is no analogous symmetry when there are more than two players. (See the discussion following Corollary 4.6, the concluding comments in Chapter 7, Theorem 11.5, and the discussion before and after Theorem 11.5. Corollary 4.9 revealed a type of symmetry, but not a precise symmetry about a particular point.) In this chapter, we show that the IPS can be viewed as part of a larger and more general structure, the Multicake Individual Pieces Set, or MIPS. The MIPS has nice symmetry properties that are not generally present in the IPS when there are more than two players.

In Section 16A, we consider the MIPS for three players. (At the end of that section, we comment on why the two-player situation is trivial and uninteresting.) In Section 16B, we consider the general case of n players. We make no general assumptions about absolute continuity in this chapter.