In this chapter, we study a natural strengthening of Pareto maximality and Pareto minimality. After introducing this notion in Section 14A, we present various characterizations in Section 14B. In Sections 14C and 14D, we consider existence questions in the two-player context and in the general n-player context, respectively. In Sections 14A, 14B, 14C, and 14D, we assume that the measures are absolutely continuous with respect to each other. In Section 14E, we consider what happens when absolute continuity fails. In Section 14F, we also do not assume that the measures are absolutely continuous with respect to each other and we consider connections with the main theorem of Section 12E.

Introduction

One way to describe Pareto optimality is to say that a partition P is Pareto optimal if and only if no collection of transfers of cake among the players produces a partition that makes every player at least as happy and makes at least one player strictly happier. We strengthen this by insisting that any non-trivial (in a sense to be made precise) collection of transfers produces a partition that makes at least one player less happy.

Notice that if we start with a partition P and transfer various pieces of cake between various players, and each transferred piece has measure zero, then certainly the resulting partition makes no player less (or more) happy.