Size versus truthfulness in the House Allocation problem

We study the House Allocation problem (also known as the Assignment problem), i.e., the problem of allocating a set of objects among a set of agents, where each agent has ordinal preferences (possibly involving ties) over a subset of the objects. We focus on truthful mechanisms without monetary transfers for finding large Pareto optimal matchings. It is straightforward to show that no deterministic truthful mechanism can approximate a maximum cardinality Pareto optimal matching with ratio better than 2. We thus consider randomised mechanisms. We give a natural and explicit extension of the classical Random Serial Dictatorship Mechanism (RSDM) specifically for the House Allocation problem where preference lists can include ties. We thus obtain a universally truthful randomised mechanism for finding a Pareto optimal matching and show that it achieves an approximation ratio of $\frac{e}{e-1}$. The same bound holds even when agents have priorities (weights) and our goal is to find a maximum weight (as opposed to maximum cardinality) Pareto optimal matching. On the other hand we give a lower bound of $\frac{18}{13}$ on the approximation ratio of any universally truthful Pareto optimal mechanism in settings with strict preferences. In the case that the mechanism must additionally be non-bossy with an additional technical assumption, we show by utilising a result of Bade that an improved lower bound of $\frac{e}{e-1}$ holds. This lower bound is tight since RSDM for strict preference lists is non-bossy. We moreover interpret our problem in terms of the classical secretary problem and prove that our mechanism provides the best randomised strategy of the administrator who interviews the applicants.Comment: To appear in Algorithmica (preliminary version appeared in the Proceedings of EC 2014

Auction Design for Value Maximizers with Budget and Return-on-spend Constraints

The paper designs revenue-maximizing auction mechanisms for agents who aim to maximize their total obtained values rather than the classical quasi-linear utilities. Several models have been proposed to capture the behaviors of such agents in the literature. In the paper, we consider the model where agents are subject to budget and return-on-spend constraints. The budget constraint of an agent limits the maximum payment she can afford, while the return-on-spend constraint means that the ratio of the total obtained value (return) to the total payment (spend) cannot be lower than the targeted bar set by the agent. The problem was first coined by [Balseiro et al., EC 2022]. In their work, only Bayesian mechanisms were considered. We initiate the study of the problem in the worst-case model and compare the revenue of our mechanisms to an offline optimal solution, the most ambitious benchmark. The paper distinguishes two main auction settings based on the accessibility of agents' information: fully private and partially private. In the fully private setting, an agent's valuation, budget, and target bar are all private. We show that if agents are unit-demand, constant approximation mechanisms can be obtained; while for additive agents, there exists a mechanism that achieves a constant approximation ratio under a large market assumption. The partially private setting is the setting considered in the previous work [Balseiro et al., EC 2022] where only the agents' target bars are private. We show that in this setting, the approximation ratio of the single-item auction can be further improved, and a $\Omega(1/\sqrt{n})$-approximation mechanism can be derived for additive agents.Comment: 29 page