The Stable Matching Problem with Couples (SMP-C) is a ubiquitous real-world extension of the stable matching problem (SMP) involving complementarities. Although SMP can be solved in polynomial time, SMP-C is NP-Complete. Hence, it is not clear which, if any, of the theoretical results surrounding the canonical SMP problem apply in this setting. In this paper, we use a recently-developed SAT encoding to solve SMP-C exactly. This allows us to enumerate all stable matchings for any given instance of SMP-C. With this tool, we empirically evaluate some of the properties that have been hypothesized to hold for SMP-C.
We take particular interest in investigating if, as the size of the market grows, the percentage of instances with unique stable matchings also grows. While we did not find this trend among the random problem instances we sampled, we did find that the percentage of instances with an resident optimal matching seems to more closely follow the trends predicted by previous conjectures. We also define and investigate resident Pareto optimal stable matchings, finding that, even though this is important desideratum for the deferred acceptance style algorithms previously designed to solve SMP-C, they do not always find one.
We also investigate strategy-proofness for SMP-C, showing that even if only one stable matching exists, residents still have incentive to misreport their preferences. However, if a problem has a resident optimal stable matching, we show that residents cannot manipulate via truncation.
We consider fair division problems where indivisible items arrive one-by-one in an online fashion and are allocated immediately to agents who have additive utilities over these items. Many existing offline mechanisms do not work in this online setting. In addition, many existing axiomatic results often do not transfer from the offline to the online setting. For this reason, we propose here three new online mechanisms, as well as consider the axiomatic properties of three previously proposed online mechanisms. In this paper, we use these mechanisms and characterize classes of online mechanisms …
LIP6, Sorbonne Université（巴黎索邦大学LIP6实验室）；ILLC, University of Amsterdam（阿姆斯特丹大学ILLC研究所）
Recently, the problem of allocating one resource per agent with initial endowments (house markets) has seen a renewed interest: indeed, while in the domain of strict preferences the Top Trading Cycle algorithm is known to be the only procedure guaranteeing Pareto-optimality, individual rationality, and strategy proofness. However, the situation differs in the single-peaked domain. Indeed, Bade presented the Crawler, an alternative procedure enjoying the same properties, with the additional advantage of being implementable in obviously dominant strategies. In this paper we further investigate t…
We consider fair allocation of indivisible items under additive utilities. When the utilities can be negative, the existence and complexity of an allocation that satisfies Pareto optimality and proportionality up to one item (PROP1) is an open problem. We show that there exists a strongly polynomial-time algorithm that always computes an allocation satisfying Pareto optimality and proportionality up to one item even if the utilities are mixed and the agents have asymmetric weights. We point out that the result does not hold if either of Pareto optimality or PROP1 is replaced with slightly stro…
Coventry University（考文垂大学）；Özyeğin University（厄兹耶金大学）；Centrum Wiskunde & Informatica（荷兰数学与计算机科学研究中心）；Technical University of Delft（代尔夫特理工大学）
Traditionally, researchers in decision making have focused on attempting to reach Pareto Optimality using horizontal approaches, where optimality is calculated taking into account every participant at the same time. Sometimes, this may prove to be a difficult task (e.g., conflict, mistrust, no information sharing, etc.). In this paper, we explore the possibility of achieving Pareto Optimal outcomes in a group by using a bottom-up approach: discovering Pareto optimal outcomes by interacting in subgroups. We analytically show that Pareto optimal outcomes in a subgroup are also Pareto optimal in …
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…
In fair division, equitability dictates that each participant receives the same level of utility. In this work, we study equitable allocations of indivisible goods among agents with additive valuations. While prior work has studied (approximate) equitability in isolation, we consider equitability in conjunction with other well-studied notions of fairness and economic efficiency. We show that the Leximin algorithm produces an allocation that satisfies equitability up to any good and Pareto optimality. We also give a novel algorithm that guarantees Pareto optimality and equitability up to one go…
Reallocating resources to get mutually beneficial outcomes is a fundamental problem in various multi-agent settings. While finding an arbitrary Pareto optimal allocation is generally easy, checking whether a particular allocation is Pareto optimal can be much more difficult. This problem is equivalent to checking that the allocated objects cannot be reallocated in such a way that at least one agent prefers her new share to his old one, and no agent prefers her old share to her new one. We consider the problem for two related types of preference relations over sets of objects. In the first part…
We consider multi-item exchange markets in which agents want to receive one of their target bundles of resources. The model encompasses well-studied markets for kidney exchange, lung exchange, and multi-organ exchange. We identify a general and sufficient condition called weak consistency for the exchange mechanisms to be strategyproof even if we impose any kind of distributional, diversity, or exchange cycle constraints. Within the class of weakly consistent and strategyproof mechanisms, we highlight two important ones that satisfy constrained Pareto optimality and strong individual rationali…
Non-obvious Manipulability with Groups in Shapley-Scarf Housing Markets
CREST, Ecole Polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院）；University of Barcelona（巴塞罗那大学）；Université Jean Monnet, Saint-Étienne, GATE-LSE（让·莫内大学）；University of Padova（帕多瓦大学）；Kyushu University（九州大学）
In a multiple partners matching problem the agents can have multiple partners up to their capacities. In this paper we consider both the two-sided many-to-many stable matching problem and the one-sided stable fixtures problem under lexicographic preferences. We study strong core and Pareto-optimal solutions for this setting from a computational point of view. First we provide an example to show that the strong core can be empty even under these severe restrictions for many-to-many problems, and that deciding the non-emptiness of the strong core is NP-hard. We also show that for a given matchin…
