How to cut a cake with a gram matrix

In this article we study the problem of fair division. In particular we study a notion introduced by J. Barbanel that generalizes super envy-free fair division. We give a new proof of his result. Our approach allows us to give an explicit bound for this kind of fair division. Furthermore, we also give a theoretical answer to an open problem posed by Barbanel in 1996. Roughly speaking, this question is: how can we decide if there exists a fair division satisfying some inequalities constraints? Furthermore, when all the measures are given with piecewise constant density functions then we show how to construct effectively such a fair division. Introduction In the followingX will be a measurable set. This set represents an heterogeneous good, e.g. a cake, that we want to divide between n players. A division of the cake is a partition X = ⊔i=1Xi, where each Xi is a measurable subset of X . After the division Xi is given to the i-th player. A natural and old problem is: how to get a fair division? This problem appears when we study division of land, time or another divisible resource between different agents with different points of view. These problems appear in the economics, mathematics, political science, artificial intelligence and computer science literature, see [Mou03, BCE16]. In order to study this problem, to each player is associated a non-atomic probability measure μi. Thus, in particular μi(X) = 1, and μi(A ⊔B) = μi(A) + μi(B), where A, and B are disjoint measurable sets. These measures represent the preference of each player. Severall notions of fair divisions exist: • Proportional division: ∀i, μi(Xi) ≥ 1/n. • Exact division: ∀i, ∀j, μi(Xj) = 1/n. • Equitable division: ∀i, ∀j, μi(Xi) = μj(Xj). • Envy-free division: ∀i, ∀j, μi(Xi) ≥ μi(Xj). All these fair divisions are possible, see e.g [Ste48, DS61, BT96, RW98, Chè17, SHS, Wel85]. Some fair divisions are possible under some conditions. Barbanel has shown in [Bar96b] that a super envy-free division is possible if and only if the measures μi are linearly independent. We recall the definition of a super envy-free fair division: • Super envy-free division: ∀i, ∀j 6= i, μi(Xi) > 1/n > μi(Xj). Actually, if the measures are linearly independent then there exists a real δ > 0 such that: μi(Xi) ≥ 1/n+ δ, and μi(Xj) ≤ 1/n− δ/(n− 1). Date: July 11, 2017. 1 2 G. CHÈZE AND L. AMODEI We can define an even more demanding fair division. For example, we can imagine that the first player would like to get a partition such that: μ1(X1) = 1/n+ 3δ, μ1(X3) = 1/n+ 2δ, μ1(X4) = μ1(X5) = 1/n+ δ, μ1(X2) = μ1(X6) = 1/n− 6δ. This means that the third, the forth and the fifth player are friends with the first player, but the second and sixth player are not friends with this player. Furthermore, the first player prefers the third to the forth and the fifth player. We can also imagine that the other players have also preferences between the other players. These kinds of conditions have also been studied by Barbanel in [Bar96a, Bar05]. This leads to a notion of fair division that we call hyper envy-free. Definition 1. Consider a matrix K = (kij) ∈ Mn(R), such that for all i = 1, . . . , n, ∑n j=1 kij = 0 and a point p = (p1, . . . , pn) such that ∑n j=1 pj = 1, with pi ≥ 0. We say that a partition X = ⊔i=1Xi is hyper envy-free relatively to K and p when there exists a real number δ > 0 such that μi(Xj) = pj + kijδ. For example, for a super envy-free division we have kij = −1/(n − 1), kii = 1 and p = (1/n, . . . , 1/n). Barbanel has given a criterion for the existence of an hyper envy-free division. Unfortunately, the proof of this result does not give an explicit bound on δ and a natural question is: How big δ can be? In this paper we give a new proof of Barbanel’s result. Furthermore, our strategy generalizes the approach for computing super envy-free fair division given by Webb in [Web99]. This allows us to give a bound on δ in terms of the measures. Another question related to fair division and asked by Barbanel is the following, see [Bar96a]: Suppose that p = (p1, . . . , pn) is a point such that p1 + · · · + pn = 1 with pi positive and rij are n 2 relations in {<,=, >}. How can we decide if there exists a partition X = ⊔i=1Xi such that μi(Xj) rij pj? This problem is also a generalization of the super envy-free fair division problem. Indeed, super envy-free fair division corresponds to the situation where p is (1/n, . . . , 1/n), rii is the relation “>” and rij is “<”. We give a theoretical answer to Barbanel’s question in the last section of this article. The organization of our article is the following. In the next section we present our toolbox. We recall the Dvoretzky, Wald, Wolfowitz’s theorem that will be the main ingredient of our proof. In Section 2, we prove the existence of an hyper envy-free division under some linear conditions on the measures μi. This gives a new proof of Barbanel’s theorem. A direct consequence of our construction gives a HOW TO CUT A CAKE WITH A GRAM MATRIX 3 bound on δ. At last, in Section 3 we solve the open question asked by Barbanel. Furthermore, when all the measures are given by piecewise constant density functions we show how to construct effectively an hyper envy-free fair division. This gives a method to construct a partition such that μi(Xj) rij pj . 1. Our toolbox Definition 2. A matrix M = (μi(Xj)) is said to be a sharing matrix when X = ⊔i=1Xi is a partition of X . A sharing matrix is a row stochastic matrix, this means that each coefficients are nonnegative and the sum of the coefficients of each row is equal to 1. In the following, we will use classical results about this kind of matrices. We recall below without proofs these results. Lemma 3. We denote by e the vector (1, . . . , 1) . If S = (sij) and T = (tij) are matrices, and s, t ∈ R are such that S e = s e and T e = t e, then: (1) N = S T is a matrix such that N e = st e. (2) If S is invertible, then s 6= 0 and S is such that S e = 1s e. In the following we define a new measure. This measure will be usefull to write each μi(Xj) in term of the same measure. Definition 4. We denote by μ the measure μ = μ1 + · · ·+ μn. The Radon-Nikodym derivative dμi/dμ is denoted by fi. The Radon-Nikodym derivative fi exists because μi is absolutely continuous relatively to μ. By definition we have for any measurable subset A ⊂ X : μi(A) = ∫ A fi(x)dμ. Definition 5. The Dvoretzky, Wald, Wolfowitz set (DWW set) is the set of all matrices