Multi-Apartment Rent Division
Ariel D. Procaccia, Benjamin Schiffer, Shirley Zhang
TL;DR
This work generalizes rent division to multiple apartments by introducing negotiated envy-freeness, a fairness notion that enables a group to negotiate both the apartment choice and within-apartment rents while preserving fairness, Pareto optimality, and individual rationality. It proves the existence of negotiated envy-free solutions and shows polynomial-time optimization over such solutions for linear objectives, including maximin and equitability, with a key multi-welfare lemma ensuring cross-assignment consistency. The paper also defines universal envy-freeness, analyzes its (non-)existence, and provides probabilistic results under random valuations, along with a stronger variant called strong negotiated envy-freeness and several extensions. Together, these results offer a principled, computation-friendly framework for fair, cross-apartment rent division with potential practical applicability to group housing decisions and fair division software.
Abstract
Rent division is the well-studied problem of fairly assigning rooms and dividing rent among a set of roommates within a single apartment. A shortcoming of existing solutions is that renters are assumed to be considering apartments in isolation, whereas in reality, renters can choose among multiple apartments. In this paper, we generalize the rent division problem to the multi-apartment setting, where the goal is to both fairly choose an apartment among a set of alternatives and fairly assign rooms and rents within the chosen apartment. Our main contribution is a generalization of envy-freeness called negotiated envy-freeness. We show that a solution satisfying negotiated envy-freeness is guaranteed to exist and that it is possible to optimize over all negotiated envy-free solutions in polynomial time. We also define an even stronger fairness notion called universal envy-freeness and study its existence when values are drawn randomly.