Partition ratios, Pareto optimal cake division, and related notions

Abstract We consider the problem of partitioning a `cake' C among n players. Various criteria have been considered for deciding whether a partition 〈 P 1 , P 2 ,…, 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.