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Stable Marriage and Roommate Problems with Individual-Based Stability

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Stable Marriage and Roommate Problems with Individual-Based Stability Stable Marriage and Roommate Problems with Individual-based Stability Haris Aziz NICTA and University of New South Wales 2033 Sydney, Australia [email protected] ABSTRACT interest in the mathematical economics, computer science Research regarding the stable marriage and roommate prob- and operations research communities (see e.g., [12, 26]). In- lem has a long and distinguished history in mathematics, formally, a matching is deemed ‘stable’ if the agents do not computer science and economics. Stability in this context is have an incentive to deviate to achieve a better matching for predominantly core stability or one of its variants in which themselves. In the matching theory literature, the predom- each deviation is by a group of players. We consider stability inant notion of stability is indeed the core in which no pair concepts such as Nash stability and individual stability in of agents prefer to be matched to each other than remain which the deviation is by a single player. Such stability con- in their current matching. Core stability (also simply called cepts are suitable especially when trust for the other party is stability) has been extensively investigated in the context of limited, complex coordination is not feasible, or when only the stable marriage (SM) problem [11] and stable roommate unmatched agents can be approached. Furthermore, weaker (SR) problem [15] which are two of the most fundamental stability notions such as individual stability may in principle settings in matching theory. A comprehensive survey of the circumvent the negative existence and computational com- stable marriage and roommate problems is present in [12]. plexity results in matching theory. We characterize the com- We formulate the stable marriage and stable roommate putational complexity of checking the existence and comput- settings as marriage games and roommate games. Both of ing individual-based stable matchings for the marriage and these games are basic subclasses of hedonic coalition forma- roommate settings. Some of our key computational results tion games in which an agent’s preference of a partition only also carry over to different classes of hedonic games and net- depends on the coalition (of arbitrary size) he is a member of work formation games for which individual-based stability and not on how the remaining agents are grouped (see e.g., has already been of much interest. [6, 13]). Of course, in the roommate and marriage games, feasible partitions simply correspond to matchings because each coalition is of size at most two. The main focus in hedo- Categories and Subject Descriptors nic games has been on different natural notions of stability of F.2 [Theory of Computation]: Analysis of Algorithms partitions. The stability concepts include individual-based and Problem Complexity; I.2.11 [Distributed Artificial stability concepts (Nash stability (NS), individual stability Intelligence]: Multiagent Systems; J.4 [Computer Ap- (IS), and contractual individual stability (CIS)) and group- plications]: Social and Behavioral Sciences - Economics based stability concepts (core (C) and strict core (SC)) (see e.g., [6]). Another individual-based stability concept is con- General Terms tractual Nash stability (CNS) which is stronger than CIS and is defined in an analogous way to IS [28]. Economics, Theory and Algorithms In this paper, we characterize the complexity of check- ing existence of and computing individual-based stable out- Keywords comes in marriage and roommate games. A number of ex- Game theory (cooperative and non-cooperative), teamwork, istence results are also presented. Our results shed further coalition formation, and coordination light on the dynamics of stability concepts like Nash stability in fundamental settings such as marriage games. 1. INTRODUCTION There are a number of reasons why individual-based sta- bility in matching and hedonic models may be of interest. In stable matching problems, the aim is to match agents Individual-based stability applies in situations when form- in a stable manner to objects or to other agents, keeping in ing arbitrary new coalitions may be ‘costly or may require view the preference of the agents involved. These problems complex coordination among the players’[24]. Furthermore, have significant applications in matching residents to hospi- ‘if information on the preferences of other players is scarce tals, students to schools, etc. and have received tremendous [...], then considering the actions of individual players only may be quite compelling’[24]. Since marriage games and Appears in: Proceedings of the 12th International Confer- roommate games may not admit a strict core stable and core ence on Autonomous Agents and Multiagent Systems (AA- stable matching respectively [12], it makes sense to exam- MAS 2013), Ito, Jonker, Gini, and Shehory (eds.), May, 6–10, 2013, Saint Paul, Minnesota, USA. ine weaker stability notions such as IS. IS may also apply to Copyright c 2013, International Foundation for Autonomous Agents and other matching models. For e.g., in hospital-resident match- Multiagent Systems (www.ifaamas.org). All rights reserved. 287 ing, the hospital may not deviate with a resident and may with strict preferences via a normal form game in which the accept any acceptable candidate (with a minimum level of pure Nash equilibria of the normal form game coincide with competency). core stable matchings. Since marriage and roommate games are two of the most As mentioned in the introduction, our results can also be fundamental classes of hedonic games, our results have bear- viewed as results on network formation. Computation of ing on the coalition formation literature. In fact, one of Nash stable networks has been shown to be computation- our key computational results for marriage games also car- ally demanding for much more elaborate models of network ries over to different classes of hedonic games for which formation [5]. individual-based stability has already been of much inter- est. 3. PRELIMINARIES Finally, we point out the relation between marriage and roommate games and network formation games. In network formation [17], agents establish links between each other and Hedonic games. they have preferences over the links or even the whole net- We review the terminology and notation used in this pa- work structure. Therefore, marriage and roommate settings per. Let N be a set of n players. A coalition is any non- are also one of the most basic models in network forma- empty subset of N. By Ni we denote the set of all coalitions tion [18] which further motivates our study. player i may belong to, that is, Ni = {S ⊆ N : i ∈ S}. A coalition structure, or simply a partition, is a partition π 2. RELATED LITERATURE of the players N into coalitions, where π(i) is the coalition player i belongs to. The complexity of computing partitions which are Nash A hedonic game is a pair (N, %), where %= (%1,..., %n) is stable or individual stable has previously been examined a preference profile specifying the preferences of each player i for some classes of hedonic games such as additively sep- as a binary, complete, reflexive, and transitive preference arable hedonic games (see e.g., [23, 29]), and hedonic relation %i over Ni. If %i is also anti-symmetric we say games represented by individually rational lists of coali- that i’s preferences are strict. Note that S ≻i T if S %i T tions (RIRLC) [4]. Sung and Dimitrov [28] introduced CNS but not T %i S—that is, if i strictly prefers S to T —and and showed that a CNS partition is guaranteed to exist for S ∼i T if both S %i T and T %i S—that is, if i is indifferent separable hedonic games satisfying weak mutuality. Papai between S and T . [24] used restrictions on acceptable coalitions to characterize For a player i, a coalition S in Ni is acceptable if for i classes of hedonic coalition formation games with strict pref- being in S is at least as preferable as being alone—that is, if erence for which Nash stable and individual stable partitions S %i {i}—and unacceptable otherwise. If {i, j} ≻i {i}, then ∗ are guaranteed to exist. we say that i likes j. We also say that partition π is accept- For the roommate and marriage settings, there has been able or unacceptable to a player i according to whether π(i) considerable work on the stable marriage (SM) problem and is acceptable or unacceptable to i, respectively. Moreover, π stable roommate (SR) problem. Stability in this regard is is individually rational (IR) if π is acceptable to all players. mostly core stability (also simply called stability). In the stable marriage (SM) problem , the set of agents is parti- Roommate & marriage games. tioned into men and women; men and women express strict A roommate game (RG) is a hedonic game (N, %) in which preferences over all their counterparts; and the aim is to find for each i ∈ N, coalitions of size three or more are unaccept- a stable matching in which men and women are matched to able and preferences % over other players are extended nat- each other. The stable roommate problem (SR) is the uni- urally over preferences over coalitions in the following way: sex generalization of the stable roommate problem in which {i} ∪ {j} %i {i} ∪ {k} if and only if j %i k for all i, j, k ∈ N. roommates are paired with each other in a stable match- In the matching theory literature, preferences %i of player ing [15]. i over other players are represented via preferences list so Subsequently, variants of the problems SM and SR have that if j 6= i is not on the preference list of i, then j is been examined: i) SMI and RMI — stable marriage and unacceptable to i.A marriage game (MG) is a roommate stable roommate problems with incomplete preference lists game in which N is partitioned into two sets M (men) and thereby signifying that the agents not in a preference list of W (women) such that each agent considers a member of his an agent are unacceptable to the agent; ii) SMT and RMT own sex unacceptable.
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