Fully Polynomial-Time Approximation Schemes for Fair Rent Division

A polynomial algorithm for maxmin and minmax envy-free rent division on a soft budget

“…We construct a family of polynomial algorithms, with the same fixed parameters of Arunachaleswaran et al (2018), that compute allocations selected by the maxmin and minmax utility and maxmin and minmax money linear selections of the envy-free set (Theorem 1). Sec.…”

“…Second, at a technical level, we bridge two independent branches of the literature on envy-free rent division. On the one hand, we work on the algorithmic framework developed by Gal et al (2017) and Arunachaleswaran et al (2018). On the other hand, we manage to exploit from a computational perspective the topological properties of the envy-free set that economists have used in unrestricted continuous economies to understand the normative and structural properties of these allocations (c.f., Alkan et al, 1991;Velez, , 2018.…”

“…Furthermore, our restriction is necessary to leverage a topological property of these selections, rent monotonicity, that was never taken advantage of before from an algorithmic perspective. 5 At a high level, we borrow from Gal et al (2017) the representation of our problems as linear programs, and from Arunachaleswaran et al (2018) the strategy of first solving the problem for a rent high enough in which budget constraints are all violated and then proceeding recursively rebating amounts of rent. Our algorithm significantly differs in that we are able to maintain control of the location of our iterations with respect to the solution we want to calculate.…”

“…Starting from an envy-free allocation, decrease the rent of each room and reshuffle rooms so no-envy is preserved. In what follows we introduce notation and provide the intuition why this is always possible for a small enough rebate, the so called Perturbation Lemma (Alkan et al, 1991;Arunachaleswaran et al, 2018). This notation and intuition is essential to later introduce our algorithms for the more general problem.…”

“…This recommendation is not informed by the agents' preferences to go over their budget. The second approach, advanced by Arunachaleswaran et al (2018), addresses this issue. They propose to enlarge the space of admissible preferences to include piece-wise linear preferences as follows: Each agent has a reference rent (a soft budget) that determines a breaking point in the marginal disutility of paying rent (see functional form in Sec.…”