Generalized Rental Harmony

Rental harmony with roommates

We prove existence of envy-free allocations in markets with heterogenous indivisible goods and money, when a given quantity is supplied from each of the goods and agents have unit demands. We depart from most of the previous literature by allowing agents’ preferences over the goods to depend on the entire vector of prices. Our proof uses Shapley’s K-K-M-S theorem and Hall’s marriage lemma. We t...

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 di...

Harmony Search for Generalized Orienteering Problem: Best Touring in China

In order to overcome the drawbacks of mathematical optimization techniques, soft computing algorithms have been vigorously introduced during the past decade. However, there are still some possibilities of devising new algorithms based on analogies with natural phenomena. A nature-inspired algorithm, mimicking the improvisation process of music players, has been recently developed and named Harm...

Sperner’s Lemma and Fair Division Problems

This paper is a seminar paper that summarizes Fran­ cis Edward Su’s paper “Rental harmony: Sperner’s lemma in fair division” [1] in 1999. In this paper we introduce and prove Sperner’s Lemma. Then, we introduce the fair division problem and an al­ gorithm based on Sperner’s lemma that can solve some variants of the fair division problem.

Rental Properties