Rental Harmony: Sperner's Lemma in Fair Division

My friend’s dilemma was a practical question that mathematics could answer, both elegantly and constructively. He and his housemates were moving to a house with rooms of various sizes and features, and were having trouble deciding who should get which room and for what part of the total rent. He asked, “Do you think there’s always a way to partition the rent so that each person will prefer a different room?” As we shall see, with mild assumptions, the answer is yes. This rent-partitioning problem is really a kind of fair-division question. It can be viewed as a generalization of the age-old cake-cutting problem, in which one seeks to divide a cake fairly among several people, and the chore-division problem, posed by Martin Gardner in [6, p. 124], in which one seeks to fairly divide an undesirable entity, such as a list of chores. Lately, there has been much interest in fair division (see, for example, the recent books [3] and [11]), and each of the related problems has been treated before (see [1], [4], [10]). We wish to explain a powerful approach to fair-division questions that unifies these problems and provides new methods for achieving approximate envy-free divisions, in which each person feels she received the “best” share. This approach was carried out by Forest Simmons [13] for cake-cutting and depends on a simple combinatorial result known as Sperner’s lemma. We show that the Sperner’s lemma approach can be adapted to treat chore division and rent-partitioning as well, and it generalizes easily to any number of players. From a pedagogical perspective, this approach provides a nice, elementary demonstration of how ideas from many pure disciplines—combinatorics, topology, and analysis—can combine to address a real-world problem. Better yet, the proofs can be converted into constructive fair-division procedures.