Existence of a simple and equitable fair division: A short proof

In this note we study how to share a good between n players in a simple and equitable way. We give a short proof for the existence of such fair divisions. In this note we study a fair division problem. Fair divisions problems are sometimes called cake cutting problems. These kinds of problems appear when we study division of land, time or computer memory between different agents with different point of view. This problem is old: the “cut and choose” algorithm already appears in the Bible. In a more scientific way it has been formulated by Steinhaus in 1948, see [Ste48]. Nowadays, there exists several articles, see e.g. [DS61, BT95, RW97, BJK13], and books, see e.g. [RW98, BT96, Pro16, Bar05], about this topic. These results appears in the mathematics, economics, political science, artificial intelligence and computer science literature. In this note, our heterogeneous good, e.g. a cake, will be represented by the interval [0; 1]. We consider n players and we associate to each player a non-atomic probability measure μi on the interval X = [0; 1] with density fi. These measures represent the utility functions of the player. The set X represents the cake and we want to get a partition of X = X1 ⊔ . . . ⊔Xn, where the i-th player get Xi. In this situation several notions of fair division exists: • Proportional division: ∀i, μi(Xi) ≥ 1/n. • Exact division: ∀i, ∀j, μi(Xj) = 1/n. • Envy-free division: ∀i, ∀j, μi(Xi) ≥ μi(Xj). • Equitable division: ∀i, ∀j, μi(Xi) = μj(Xj). All these fair divisions are possible, see e.g [Ste48, DS61, BT96, RW98, CDP13]. The minimal number of cuts in order to get a fair division has also been studied. For a proportional fair division n − 1 cuts are sufficient. This means that for all i there exists an index j such that Xi = [xj , xj+1], where x0 = 0, xn = 1. A fair division where each Xi is an interval is called a simple fair division. The BanachKnaster algorithm given in [Ste48] shows how to get a simple proportional fair division. Stromquist and Woodall have shown that we can always get a simple envy-free fair division, see [Str80, Str81, Woo80]. These two different proofs use the Brouwer