In this chapter, we consider generalizations of our fairness and efficiency results from Chapter 3 to the general n-player context. In Section 5A, we consider fairness; in Section 5B, we consider efficiency; and in Section 5C, we consider fairness and efficiency together. In these sections, we assume that the measures are absolutely continuous with respect to each other. In Section 5D, we consider the situation when absolute continuity fails. In Section 5E, where we consider examples and open questions, absolute continuity will sometimes hold and sometimes fail.

Fairness

We begin by recalling that, when there are two players, proportionality and envy-freeness correspond, as do strong proportionality, strong envy-freeness, and super envy-freeness. The following two facts are implied by Corollary 3.3 for the two-player context.

1. Fact 1: There exist infinitely many mutually non-s-equivalent partitions that are proportional and envy-free.

2. Fact 2: There exist partitions that are strongly proportional, strongly envy-free, and super envy-free if and only if the measures are not equal (and, in this case, there are infinitely many mutually non-p-equivalent such partitions).

Analogous facts for chores fairness properties are implied by Corollary 3.5.

The natural guess for the generalization of Fact 1 when there are more than two players is obvious and turns out to be true.