Pareto Optimal Matching Research Articles

On the Convergence of Swap Dynamics to Pareto-Optimal Matchings

We study whether Pareto-optimal stable matchings can be reached via pairwise swaps in one-to-one matching markets with initial assignments. We consider housing markets, marriage markets, and roommate markets as well as three different notions of swap rationality. Our main results are as follows. While it can be efficiently determined whether a Pareto-optimal stable matching can be reached when defining swaps via blocking pairs, checking whether this is the case for all such sequences is computationally intractable. When defining swaps such that all involved agents need to be better off, even deciding whether a Pareto-optimal stable matching can be reached via some sequence is intractable. This confirms and extends a conjecture made by Damamme, Beynier, Chevaleyre, and Maudet (2015) who have shown that convergence to a Pareto-optimal matching is guaranteed in housing markets with single-peaked preferences. We prove that in marriage and roommate markets, single-peakedness is not sufficient for this to hold, but the stronger restriction of one-dimensional Euclidean preferences is.

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. By using a characterisation result of Bade, we show that any randomised mechanism that is a symmetrisation of a truthful, non-bossy and Pareto optimal mechanism has an improved lower bound of frac{e}{e-1}. Since our new mechanism is a symmetrisation of RSDM for strict preferences, it follows that this lower bound is tight. 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.

Traffic-Oriented Resource Allocation for mmWave Multi-Hop Backhaul Networks

This letter investigates traffic-oriented resource allocation for millimeter-wave (mmW) multi-hop backhaul (BH) networks. The problem is formulated to minimize the sum of the distances between the data rate and BH traffic of all the small cell base stations (SBSs) under limited BH resources, considering concurrent transmissions to cater for the huge BH traffic occurred in the aggregated SBSs. The problem is casted as a matching game, since it is a two-sided stable and Pareto optimal matching between sub-channels and SBSs. By analyzing the preference profiles and judge functions of SBSs and sub-channels, respectively, an algorithm based on a matching theory is proposed. Simulations demonstrate that our algorithm can effectively cope with the huge aggregated traffic occurred in the mmW multi-hop BH networks with limited BH resources.

Pareto optimal matchings of students to courses in the presence of prerequisites

We consider the problem of allocating applicants to courses, where each applicant has a subset of acceptable courses that she ranks in strict order of preference. Each applicant and course has a capacity, indicating the maximum number of courses and applicants they can be assigned to, respectively. We thus essentially have a many-to-many bipartite matching problem with one-sided preferences, which has applications to the assignment of students to optional courses at a university.We consider additive preferences and lexicographic preferences as two means of extending preferences over individual courses to preferences over bundles of courses. We additionally focus on the case that courses have prerequisite constraints: we will mainly treat these constraints as compulsory, but we also allow alternative prerequisites. We further study the case where courses may be corequisites.For these extensions to the basic problem, we present the following algorithmic results, which are mainly concerned with the computation of Pareto optimal matchings (POMs). Firstly, we consider compulsory prerequisites. For additive preferences, we show that the problem of finding a POM is NP-hard. On the other hand, in the case of lexicographic preferences we give a polynomial-time algorithm for finding a POM, based on the well-known sequential mechanism. However we show that the problem of deciding whether a given matching is Pareto optimal is co-NP-complete. We further prove that finding a maximum cardinality (Pareto optimal) matching is NP-hard. Under alternative prerequisites, we show that finding a POM is NP-hard for either additive or lexicographic preferences. Finally we consider corequisites. We prove that, as in the case of compulsory prerequisites, finding a POM is NP-hard for additive preferences, though solvable in polynomial time for lexicographic preferences. In the latter case, the problem of finding a maximum cardinality POM is NP-hard and very difficult to approximate.

ACCOUNTING FOR COMPLEMENTARY SKILL SETS: EVALUATING INDIVIDUAL MARGINAL VALUE TO A TEAM IN THE NATIONAL BASKETBALL ASSOCIATION

Identifying an individual worker's contribution to firm production can be difficult in a team setting where spillovers in labor productivity exist among team members. This paper studies a model of labor productivity where workers have heterogeneous abilities, and can differently affect the productivity of their fellow teammates. Applying the model to the setting of the National Basketball Association (NBA), I can identify the marginal value that a basketball player brings to a particular team lineup, both through his own individual contributions and his complementary contribution to teammates' productivity. Estimates from the model imply that teammates have a significant impact on individual player productivity, and that taking into account spillovers across teammates is important to assessing both overall team productivity and an individual player's contribution to team productivity. I then evaluate whether player complementarities are valued in the NBA labor market in terms of higher salaries, and find that they are undervalued, and that players are instead paid mainly for their individual offensive production. This creates an asymmetry between player incentives and the team objective. To assess the size of this inefficiency, the top trading cycle algorithm of Shapley and Scarf (1974) is used to identify a Pareto optimal matching between players and teams, that accounts for the complementarities between heterogeneous players' skill sets. (JEL J30, L25, L83, M51)

Pareto optimal matchings with lower quotas

We consider the problem of allocating applicants to courses, where each applicant has a capacity, possibly greater than 1, and a subset of acceptable courses that she ranks in a strict order of preference. Each course has a lower and an upper quota, indicating that if it is assigned some applicants then their number has to be between these two bounds. We further suppose that applicants extend their preferences over courses to preferences over bundles of courses lexicographically.In this setting we present several algorithmic results concerned with the computation of Pareto optimal matchings (POMs). Firstly, we extend the Serial Dictatorship with Project Closures mechanism to the case when an applicant can be assigned more than one course. We show that unlike in the one-to-many case no mechanism is strategy-proof against dropping manipulations and that this mechanism is strategy-proof against reordering strategies only for some picking sequences. We further show the intractability of the following problems: deciding about the Pareto optimality of a given matching, computation of a POM with maximum cardinality and computation of a POM in case of indifferences.

QoE-Aware Power Management in Vehicle-to-Grid Networks: A Matching-Theoretic Approach

Frequency, time, and places of charging and discharging have critical impact on the quality of experience (QoE) of using electric vehicles (EVs). EV charging and discharging scheduling schemes should consider both the QoE of using EV and the load capacity of the power grid. In this paper, we design a traveling plan-aware scheduling scheme for EV charging in driving pattern and a cooperative EV charging and discharging scheme in parking pattern to improve the QoE of using EV and enhance the reliability of the power grid. For traveling plan-aware scheduling, the assignment of EVs to charging stations is modeled as a many-to-one matching game and the stable matching algorithm (SMA) is proposed. For cooperative EV charging and discharging in parking pattern, the electricity exchange between charging EVs and discharging EVs in the same parking lot is formulated as a many-to-many matching model with ties, and we develop the Pareto optimal matching algorithm (POMA). Simulation results indicates that the SMA can significantly improve the average system utility for EV charging in driving pattern, and the POMA can increase the amount of electricity offloaded from the grid which is helpful to enhance the reliability of the power grid.

Counting houses of Pareto optimal matchings in the house allocation problem

In an instance of the house allocation problem, two sets A and B are given. The set A is referred to as applicants and the set B is referred to as houses. We denote by m and n the size of A and B, respectively. In the house allocation problem, we assume that every applicanta∈A has a preference list over the set of houses B. We call an injective mapping τ from A to B a matching. A blocking coalition of τ is a non-empty 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 a Pareto optimal matching (POM).A house b∈B is reachable if there exists a Pareto optimal matching using b. The set of all reachable houses is denoted by E∗. We show |E∗|≤∑i=1,…,m⌊mi⌋=Θ(mlogm). This is asymptotically tight. A set E⊆B is reachable (respectively exactly reachable) if there exists a Pareto optimal matching τ whose image contains E as a subset (respectively equals E). We give bounds for the number of exactly reachable sets. We find that our results hold in the more general setting of multi-matchings, when each applicant a of A is matched with ℓa elements of B instead of just one. Furthermore, we give complexity results and algorithms for corresponding algorithmic questions. Finally, we characterize unavoidable houses, i.e., houses that are used by all POMs. We obtain efficient algorithms to determine all unavoidable elements.

Pareto Optimal Matchings in Many-to-Many Markets with Ties

We consider Pareto optimal matchings (POMs) in a many-to-many market of applicants and courses where applicants have preferences, which may include ties, over individual courses and lexicographic preferences over sets of courses. Since this is the most general setting examined so far in the literature, our work unifies and generalizes several known results. Specifically, we characterize POMs and introduce the Generalized Serial Dictatorship Mechanism with Ties (GSDT) that effectively handles ties via properties of network flows. We show that GSDT can generate all POMs using different priority orderings over the applicants, but it satisfies truthfulness only for certain such orderings. This shortcoming is not specific to our mechanism; we show that any mechanism generating all POMs in our setting is prone to strategic manipulation. This is in contrast to the one-to-one case (with or without ties), for which truthful mechanisms generating all POMs do exist.

Transfers and exchange-stability in two-sided matching problems

In this paper we consider one-to-many matching problems where the preferences of the agents involved are represented by monetary reward functions. We characterize Pareto optimal matchings by means of contractual exchange stability and matchings of maximum total reward by means of compensational exchange stability. To conclude, we show that in going from an initial matching to a matching of maximum total reward, one can always provide a compensation schedule that will be ex-post stable in the sense that there will be no subset of agents who can all by deviation obtain a higher reward. The proof of this result uses the fact that the core of an associated compensation matching game with constraints is nonempty.

The Complexity of Computing the Random Priority Allocation Matrix

The random priority (RP) mechanism is a popular way to allocate n objects to n agents with strict ordinal preferences over the objects. In the RP mechanism, an ordering over the agents is selected uniformly at random; the first agent is then allocated his most-preferred object, the second agent is allocated his most-preferred object among the remaining ones, and so on. The outcome of the mechanism is a bi-stochastic matrix in which entry (i, a) represents the probability that agent i is given object a. It is shown that the problem of computing the RP allocation matrix is #P-complete. Furthermore, it is NP-complete to decide if a given agent i receives a given object a with positive probability under the RP mechanism, whereas it is possible to decide in polynomial time whether or not agent i receives object a with probability 1. The implications of these results for approximating the RP allocation matrix as well as on finding constrained Pareto optimal matchings are discussed.

A Simple Mechanism for the Roommate Problem

Gale and Shapley (1962) proposed that there is a similar game to the marriage problem called "the roommate problem". And, they showed that unlike the marriage problem, the roommate problem may have unstable solutions. In other words, the stability theorem fails for the roommate problem. In this paper, we propose a new mechanism for the roommate problem. The mechanism is successful in determining the reason of instability in our game scenario. And, we show that our mechanism implements the full set of stable matchings in the existence of stability, and it ends up with Pareto Optimal matching in the instance of instability.

Pareto optimality in many-to-many matching problems

Consider a many-to-many matching market that involves two finite disjoint sets, a set A of applicants and a set C of courses. Each applicant has preferences on the different sets of courses she can attend, while each course has a quota of applicants that it can admit. In this paper, we examine Pareto optimal matchings (briefly POM) in the context of such markets, that can also incorporate additional constraints, e.g., each course bearing some cost and each applicant having a limited budget available. We provide necessary and sufficient conditions for a many-to-many matching to be Pareto optimal and show that checking whether a given matching is Pareto optimal can be accomplished in O(∣A∣2⋅∣C∣2) time. Moreover, we provide a generalized version of serial dictatorship, which can be used to obtain any many-to-many POM. We also study some structural questions related to POM. We show that, unlike in the one-to-one case, finding a maximum cardinality POM is NP-hard for many-to-many markets.

On the Existence and Stability of Pareto Optimal Endogenous Matching with Fairness

In the current paper, we study the asymmetric normal-form game between two heterogeneous groups of populations by employing the stochastic replicator dynamics driven by Levy process. A new game equilibrium, i.e., the game equilibrium of a stochastic differential cooperative game on time, is derived by introducing optimal-stopping technique into evolutionary game theory, which combines with the Pareto optimal standard leads us to the existence of Pareto optimal endogenous matching. Moreover, stability of the Pareto optimal endogenous matching is confirmed by essentially using the well-known Girsanov Theorem.

Matching as a cure for underprovision of voluntary public good supply

Matching mechanisms are widely regarded as providing a promising way to cure the public good underprovision problem, but the Pareto optimal interior matching equilibrium emerges only under very special conditions. However, we show that public good underprovision is normally avoided.

Pareto-Optimal Matching Allocation Mechanisms for Boundedly Rational Agents

This article is concerned with the welfare properties of trade when the behavior of agents cannot be rationalized by preferences. I investigate this question in an environment of matching allocation problems. There are two reasons for doing so: firstly, the finiteness of such problems entails that the domain of the agents' choice behavior does not need to be restricted in any which way to obtain results on the welfare properties of trade. Secondly, some matching allocation mechanisms have been designed for non-market environments in which we would typically expect boundedly rational behavior. I find qualified support for the statements that all outcomes of trade are Pareto-optimal and all Pareto optima are reachable through trade. Contrary to the standard case, different trading mechanisms lead to different outcome sets when the agents' behavior is not rationalizable. These results remain valid when restricting attention to minimally irrational behavior.

Matching of Buyers and Sellers by Brokers: A Comparison of Alternative Commission Structures

This note characterizes the set of Pareto optimal and stable matchings among buyers and sellers and examines the optimality of matching strategies employed by brokers under different commission structures. It is shown that the profit‐maximizing matching strategy for the broker under percentage commission and flat‐fee systems also maximizes the number of houses sold, but it minimizes the buyers' and sellers' surplus. On the other hand, net listing results in the sale of fewer houses, but it yields larger surplus for buyers and sellers.