Social Welfare in One-sided Matching Markets without Money

We study social welfare in one-sided matching markets where the goal is to efficiently allocate n items to n agents that each have a complete, private preference list and a unit demand over the items. Our focus is on allocation mechanisms that do not involve any monetary payments. We consider two natural measures of social welfare: the ordinal welfare factor which measures the number of agents that are at least as happy as in some unknown, arbitrary benchmark allocation, and the linear welfare factor which assumes an agent's utility linearly decreases down his preference lists, and measures the total utility to that achieved by an optimal allocation. We analyze two matching mechanisms which have been extensively studied by economists. The first mechanism is the random serial dictatorship (RSD) where agents are ordered in accordance with a randomly chosen permutation, and are successively allocated their best choice among the unallocated items. The second mechanism is the probabilistic serial (PS) mechanism of Bogomolnaia and Moulin [8], which computes a fractional allocation that can be expressed as a convex combination of integral allocations. The welfare factor of a mechanism is the infimum over all instances. For RSD, we show that the ordinal welfare factor is asymptotically 1/2, while the linear welfare factor lies in the interval [.526, 2/3]. For PS, we show that the ordinal welfare factor is also 1/2 while the linear welfare factor is roughly 2/3. To our knowledge, these results are the first non-trivial performance guarantees for these natural mechanisms

Finding large stable matchings

When ties and incomplete preference lists are permitted in the stable marriage and hospitals/residents problems, stable matchings can have different sizes. The problem of finding a maximum cardinality stable matching in this context is known to be NP-hard, even under very severe restrictions on the number, size, and position of ties. In this article, we present two new heuristics for finding large stable matchings in variants of these problems in which ties are on one side only. We describe an empirical study involving these heuristics and the best existing approximation algorithm for this problem. Our results indicate that all three of these algorithms perform significantly better than naive tie-breaking algorithms when applied to real-world and randomly-generated data sets and that one of the new heuristics fares slightly better than the other algorithms, in most cases. This study, and these particular problem variants, are motivated by important applications in large-scale centralized matching schemes

Popular matchings in the marriage and roommates problems

Popular matchings have recently been a subject of study in the context of the so-called House Allocation Problem, where the objective is to match applicants to houses over which the applicants have preferences. A matching M is called popular if there is no other matching M′ with the property that more applicants prefer their allocation in M′ to their allocation in M. In this paper we study popular matchings in the context of the Roommates Problem, including its special (bipartite) case, the Marriage Problem. We investigate the relationship between popularity and stability, and describe efficient algorithms to test a matching for popularity in these settings. We also show that, when ties are permitted in the preferences, it is NP-hard to determine whether a popular matching exists in both the Roommates and Marriage cases

Matching under Preferences

Matching theory studies how agents and/or objects from different sets can be matched with each other while taking agents\u2019 preferences into account. The theory originated in 1962 with a celebrated paper by David Gale and Lloyd Shapley (1962), in which they proposed the Stable Marriage Algorithm as a solution to the problem of two-sided matching. Since then, this theory has been successfully applied to many real-world problems such as matching students to universities, doctors to hospitals, kidney transplant patients to donors, and tenants to houses. This chapter will focus on algorithmic as well as strategic issues of matching theory. Many large-scale centralized allocation processes can be modelled by matching problems where agents have preferences over one another. For example, in China, over 10 million students apply for admission to higher education annually through a centralized process. The inputs to the matching scheme include the students\u2019 preferences over universities, and vice versa, and the capacities of each university. The task is to construct a matching that is in some sense optimal with respect to these inputs. Economists have long understood the problems with decentralized matching markets, which can suffer from such undesirable properties as unravelling, congestion and exploding offers (see Roth and Xing, 1994, for details). For centralized markets, constructing allocations by hand for large problem instances is clearly infeasible. Thus centralized mechanisms are required for automating the allocation process. Given the large number of agents typically involved, the computational efficiency of a mechanism's underlying algorithm is of paramount importance. Thus we seek polynomial-time algorithms for the underlying matching problems. Equally important are considerations of strategy: an agent (or a coalition of agents) may manipulate their input to the matching scheme (e.g., by misrepresenting their true preferences or underreporting their capacity) in order to try to improve their outcome. A desirable property of a mechanism is strategyproofness, which ensures that it is in the best interests of an agent to behave truthfully

Locally Stable Marriage with Strict Preferences

We study stable matching problems with locality of information and control. In our model, each agent is a node in a fixed network and strives to be matched to another agent. An agent has a complete preference list over all other agents it can be matched with. Agents can match arbitrarily, and they learn about possible partners dynamically based on their current neighborhood. We consider convergence of dynamics to locally stable matchings -- states that are stable with respect to their imposed information structure in the network. In the two-sided case of stable marriage in which existence is guaranteed, we show that the existence of a path to stability becomes NP-hard to decide. This holds even when the network exists only among one partition of agents. In contrast, if one partition has no network and agents remember a previous match every round, a path to stability is guaranteed and random dynamics converge with probability 1. We characterize this positive result in various ways. For instance, it holds for random memory and for cache memory with the most recent partner, but not for cache memory with the best partner. Also, it is crucial which partition of the agents has memory. Finally, we present results for centralized computation of locally stable matchings, i.e., computing maximum locally stable matchings in the two-sided case and deciding existence in the roommates case.Comment: Conference version in ICALP 2013; to appear in SIAM J. Disc Mat

A constraint programming approach to the hospitals/residents problem

An instance I of the Hospitals/Residents problem (HR) involves a set of residents (graduating medical students) and a set of hospitals, where each hospital has a given capacity. The residents have preferences for the hospitals, as do hospitals for residents. A solution of I is a <i>stable matching</i>, which is an assignment of residents to hospitals that respects the capacity conditions and preference lists in a precise way. In this paper we present constraint encodings for HR that give rise to important structural properties. We also present a computational study using both randomly-generated and real-world instances. We provide additional motivation for our models by indicating how side constraints can be added easily in order to solve hard variants of HR

A Constraint Programming Approach to the Hospitals / Residents Problem

An instance I of the Hospitals / Residents problem (HR) involves a set of residents (graduating medical students) and a set of hospitals, where each hospital has a given capacity. The residents have preferences for the hospitals, as do hospitals for residents. A solution of I is a stable matching, which is an assignment of residents to hospitals that respects the capacity conditions and preference lists in a precise way. In this paper we present constraint encodings for HR that give rise to important structural properties. We also present a computational study using both randomly-generated and real-world instances. Our study suggests that Constraint Programming is indeed an applicable technology for solving this problem, in terms of both theory and practice

Maximum Cardinality Popular Matchings in Strict Two-sided Preference Lists

We consider the problem of computing a maximum cardinality {\em popular} matching in a bipartite graph G = (\A\cup\B, E) where each vertex u \in \A\cup\B ranks its neighbors in a strict order of preference. This is the same as an instance of the {\em stable marriage} problem with incomplete lists. A matching $M^*$ is said to be popular if there is no matching $M$ such that more vertices are better off in $M$ than in $M^*$. \smallskip Popular matchings have been extensively studied in the case of one-sided preference lists, i.e., only vertices of \A have preferences over their neighbors while vertices in \B have no preferences; polynomial time algorithms have been shown here to determine if a given instance admits a popular matching or not and if so, to compute one with maximum cardinality. It has very recently been shown that for two-sided preference lists, the problem of determining if a given instance admits a popular matching or not is NP-complete. However this hardness result assumes that preference lists have {\em ties}. When preference lists are {\em strict}, it is easy to show that popular matchings always exist since stable matchings always exist and they are popular. But the complexity of computing a maximum cardinality popular matching was unknown. In this paper we show an $O(mn)$ algorithm for this problem, where n = |\A| + |\B| and $m = |E|$