Partition ratios, Pareto optimal cake division, and related notions

We consider the problem of partitioning a ‘cake' C among n players. Various criteria have been considered for deciding whether a partition P₁,P₂,…,P_n of C, where piece P_i goes to player i, is a ‘good' partition. See, for example, Barbanel (1996) [Barbanel, J.B., 1996. Super envy-free cake division and independence of measures. J. Math. Anal. Appl. 197, 54–60] or Brams and Taylor (1996) [Brams, S.J., Taylor, A.D., 1996. Fair Division: From Cake-Cutting To Dispute Resolution. Cambridge Univ. Press]. In this paper we study certain real numbers (the ‘partition ratios' of this paper's Section 2) which can be associated in a natural way with any partition. We show that various types of products of these numbers provide us with useful information about certain trades and transfers between players.