Rental harmony with roommates

Abstract

Abstract. 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 then show how our theorem may be applied in two related problems: Existence of envy-free allocations in a version of the cake-cutting problem, and existence of equilibrium in an exchange economy with indivisible goods and money.

KKM type theorems with boundary conditions

“…where p * (respectively, n * ) is the number of (ordering) pairs (p i , p i+1 ) such that L(p i ) = 1 and L(p i+1 ) = 2 (respectively, L(p i ) = 2 and L(p i+1 ) = 1). It is clear, that instead of [1,2] we can take [2,3] or [3,1].…”

“…Following Su [21], we call such a situation rental harmony. In [1] consideration is given to different aspects of this model.…”

Competitive Division of a Mixed Manna

“…See the precise definition in Section 5 3. Selecting the competitive utility profiles maximizing the product of disutilities on this part of the efficiency frontier almost surely gives a unique utility profile (Lemmas 3 and 4), but does not eliminate the computational and continuity issues, as explained by Proposition 3.…”

Fairness and externalities

“…This is very restrictive in our environment. In a contemporary paper, Azrieli and Shmaya (2014) generalize the simplicial covering methods in Stromquist (1980) and Woodall (1980) to accommodate 11 Our theorem states that the requirement of the compensation assumption can be waived for one agent.…”

“…(2014) for the allocation of indivisible goods that can be shared by several agents. We discuss in detail the relevant papers to us, i.e., Gale (1984), Dufwenberg et al (2011), and Azrieli and Shmaya (2014), in Sections 3 and 5.…”