Super Envy-Free Cake Division and Independence of Measures

Abstract

We define a strengthening of the notions of fair and envy-free cake division, which we call super envy-free cake division. We establish that there exists a super envy-free partition of a "cake" among n people if and only if the n measures used by these people to evaluate sizes of pieces of cake are linearly independent.

On the possibilities for partitioning a cake

“…, m p can be expressed as a linear combination of the measures in this set. By the previously stated result from [1], if E = { m 1 (A), m 2 (A), . .…”

“…We shall also need the following result, which we established in [1]: If the measures m 1 , m 2 , . .…”

“…Corollary 1 (see [1] We note that the partition obtained above has the property that each person judges his or her own piece of cake to be the same size as everyone else judges their own piece of cake. In other words, m 1 (A 1 ) = m 2 (A 2 ) = · · · = m p (A p ).…”

“…In [1], we defined and studied the notion of super envy-freeness. Super envyfreeness is a strengthening of the notions of fairness and envy-freeness.…”

Consensus-halving via theorems of Borsuk-Ulam and Tucker

“…Alon (1987) proves i a generalization that produces k equal portions according to n probability measures; it yields our result when k 5 2, but it is also non-constructive and based on a topologicaĺŕ esult of Barany et al (1981). Another approach to produce the existence of the sets A 1 and A is to use Lyapunov's theorem (see Barbanel (1996)); however, it is even less 2 constructive because it does not even say how many cuts are required or what the sets A and A might look like.…”