Fully Polynomial-Time Approximation Schemes for Fair Rent Division

Abstract

We study the problem of fair rent division that entails splitting the rent and allocating the rooms of an apartment among roommates (agents) in a fair manner. In this setup, a distribution of the rent and an accompanying allocation is said to be fair if it is envy free, i.e., under the imposed rents, no agent has a strictly stronger preference for any other agent's room. The cardinal preferences of the agents are expressed via functions which specify the utilities of the agents for the rooms for every possible room rent/price. While envy-free solutions are guaranteed to exist under reasonably general utility functions, efficient algorithms for finding them were known only for quasilinear utilities. This work addresses this notable gap and develops approximation algorithms for fair rent division with minimal assumptions on the utility functions.Specifically, we show that if the agents have continuous, monotone decreasing, and piecewiselinear utilities, then the fair rent-division problem admits a fully polynomial-time approximation scheme (FPTAS). That is, we develop algorithms that find allocations and prices of the rooms such that for each agent a the utility of the room assigned to it is within a factor of (1 + ε) of the utility of the room most preferred by a. Here, ε > 0 is an approximation parameter, and the running time of the algorithms is polynomial in 1/ε and the input size. In addition, we show that the methods developed in this work provide efficient, truthful mechanisms for special cases of the rent-division problem. Envy-free solutions correspond to equilibria of a two-sided matching market with monetary transfers; hence, this work also provides efficient algorithms for finding approximate equilibria in such markets. We complement the algorithmic results by proving that the fair rent division problem (under continuous, monotone decreasing, and piecewise-linear utilities) lies in the intersection of the complexity classes PPAD and PLS .

An algorithm for identifying least manipulable envy‐free and budget‐balanced allocations in economies with indivisibilities

“…For weaker assumptions on preferences, preference elicitation is more cumbersome and it will typically require repeated interactions with the agents and, normally, also some type of approximation. See, e.g., Andersson and Svensson (2018), Arunachaleswaran et al (2019), or Su (1999. Furthermore, quasi-linearity implicitly assumes that agents are not budget constrained.…”

Expressive mechanisms for equitable rent division on a budget

“…The so-called equitable rent allocation problem has attracted the attention of both economists and computer scientists (Gal et al, 2017;Arunachaleswaran et al, 2018;see Velez, 2018, for a survey of incentives related studies). 1 Recently, the theoretical results of both fields have been put in practice.…”

“…Most importantly, even if it were feasible to ask for this information, we would have two additional significant problems. The first is computational complexity (Arunachaleswaran et al, 2018). The second is incentives.…”

“…At a technical level, our paper is closely related with Arunachaleswaran et al (2018) who provide an algorithm that computes an envy-free allocation for piece-wise linear preferences, a domain containing our budget constrained preferences. The algorithm they construct does not allow one to choose a specific allocation within the envy-free set, as the minmax utility allocation.…”

“…The algorithm they construct does not allow one to choose a specific allocation within the envy-free set, as the minmax utility allocation. Our algorithm shares some features with Arunachaleswaran et al (2018)'s, of which we give a detailed account in Sec. 4.…”