The Shape of the IPS
Julius B. Barbanel · Cambridge University Press eBooks · 2005
In this chapter, we investigate the possible shapes of the IPS. In Section 11A, we consider the two-player context, where we shall provide a complete answer. In Section 11B, we consider the n -player context for n > 2, where we are able to provide only a partial answer. For any cake C and corresponding measures m 1 , m 2 , …, m n , let IPS( C ; m 1 , m 2 , …, m n ) denote the IPS corresponding to cake C and measures m 1 , m 2 , …, m n on C . We make no general assumptions about absolute continuity in this chapter. The Two-Player Context In Chapter 2, we considered various properties of the IPS for the case of two players. In particular, we established Theorem 2.4, which told us that the IPS a. is a subset of [0, 1] 2 , b. contains the points (1, 0) and (0, 1), c. is closed, d. is convex, and e. is symmetric about the point (½, ½). In this section, we show that these five properties completely characterize the possible shapes of the IPS. In other words, for any G ⊆ R 2 that satisfies these five conditions, there is a cake C and measures m 1 and m 2 on C so that G = IPS( C ; m 1 , m 2 ).