An Optimal Procedure to Check Pareto-Optimality in House Markets with Single-Peaked Preferences

Pareto optimal and popular house allocation with lower and upper quotas

In the house allocation problem with lower and upper quotas, we are given a set of applicants and a set of projects. Each applicant has a strictly ordered preference list over the projects, while the projects are equipped with a lower and an upper quota. A feasible matching assigns the applicants to the projects in such a way that a project is either matched to no applicant or to a number of applicants between its lower and upper quota. In this model we study two classic optimality concepts: Pareto optimality and popularity. We show that finding a popular matching is hard even if the maximum l…

Fairness and Efficiency in Cake-Cutting with Single-Peaked Preferences

We study the cake-cutting problem when agents have single-peaked preferences over the cake. We show that a recently proposed mechanism by Wang-Wu (2019) to obtain envy-free allocations can yield large welfare losses. Using a simplifying assumption, we characterize all Pareto optimal allocations, which have a simple structure: are peak-preserving and non-wasteful. Finally, we provide simple alternative mechanisms that Pareto dominate that of Wang-Wu, and which achieve envy-freeness or Pareto optimality.

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 cla…

Counting Houses of Pareto Optimal Matchings in the House Allocation Problem

Let $A,B$ with $|A| = m$ and $|B| = n\ge m$ be two sets. We assume that every element $a\in A$ has a reference list over all elements from $B$. We call an injective mapping $τ$ from $A$ to $B$ a matching. A blocking coalition of $τ$ is a subset $A'$ of $A$ such that there exists a matching $τ'$ that differs from $τ$ only on elements of $A'$, and every element of $A'$ improves in $τ'$, compared to $τ$ according to its preference list. If there exists no blocking coalition, we call the matching $τ$ an exchange stable matching (ESM). An element $b\in B$ is reachable if there exists an exchange st…

False-Name-Proof Facility Location on Discrete Structures

We consider the problem of locating a single facility on a vertex in a given graph based on agents' preferences, where the domain of the preferences is either single-peaked or single-dipped. Our main interest is the existence of deterministic social choice functions (SCFs) that are Pareto efficient and false-name-proof, i.e., resistant to fake votes. We show that regardless of whether preferences are single-peaked or single-dipped, such an SCF exists (i) for any tree graph, and (ii) for a cycle graph if and only if its length is less than six. We also show that when the preferences are single-…

公平社会：房屋分配的算法

Fair Societies: Algorithms for House Allocations

本文研究房屋分配中的公平性问题，提出两种可处理算法：通过有界重分配减少嫉妒智能体数量，以及在单峰偏好下多项式时间最小化嫉妒并满足帕累托效率，并实证分析公平与福利的权衡。