Hedonic Games Research Articles

Pareto optimality in coalition formation

A minimal requirement on allocative efficiency in the social sciences is Pareto optimality. In this paper, we identify a close structural connection between Pareto optimality and perfection that has various algorithmic consequences for coalition formation. Based on this insight, we formulate the Preference Refinement Algorithm (PRA) which computes an individually rational and Pareto optimal outcome in hedonic coalition formation games. Our approach also leads to various results for specific classes of hedonic games. In particular, we show that computing and verifying Pareto optimal partitions in general hedonic games, anonymous games, three-cyclic games, room-roommate games and B-hedonic games is intractable while both problems are tractable for roommate games, W-hedonic games, and house allocation with existing tenants.

Subordinated Hedonic Games

Hedonic games are simple models of coalition formation whose main solution concept is that of core partition. Several conditions guaranteeing the existence of core partitions have been proposed so far. In this paper, we explore hedonic games where a reduced family of coalitions determines the development of the game. We allow each coalition to select a subset of it so as to act as its set of representatives (a distribution). Then, we introduce the notion of subordination of a hedonic game to a given distribution. Subordination roughly states that any player chosen as a representative for a coalition has to be comfortable with this decision. With subordination we have a tool, within hedonic games, to compare how a “convenient” agreement reached by the sets of representatives of different groups of a society is “valued” by the rest of the society. In our approach, a “convenient” agreement is a core partition, so this paper is devoted to relate the core of a hedonic game with the core of a hedonic game played by the sets of representatives. Thus we have to tackle the existence problem of core partitions in a reduced game where the only coalitions that matter are those prescribed by the distribution as a set of representatives. We also study how a distribution determines the whole set of core partitions of a hedonic game. As an interesting example, we introduce the notion of hedonic partitioning game, which resembles partitioning games studied in the case where a utility, transferable or not, is present. The existence result obtained in this new class of games is later used to provide a nonconstructive proof of the existence of a stable matching in the marriage model.

Two hardness results for core stability in hedonic coalition formation games

We establish NP-completeness of two problems on core stable coalitions in hedonic games. In the first problem every player has only two acceptable coalitions in his preference list, and in the second problem the preference structures arise from the distances in an underlying metric space.

Meritocracy, Egalitarianism and the Stability of Majoritarian Organizations

Egalitarianism and meritocracy are competing principles to distribute the joint benefits of cooperation. We examine the consequences of letting members of society vote between those two principles, in a context where individuals must joint with others into coalitions of a certain size to become productive. Our setup induces a hedonic game of coalition formation. We study the existence of core stable partitions (organizational structures) of this game. We show that the inability of voters to commit to one distributional rule or another is a potential source of instability. But we also prove that, when stable organizational structures exist, they may be rich in form, and different than those predicted by alternative models of coalition formation. Non-segregated coalitions may arise within core stable structures. Stability is also compatible with the coexistence of meritocratic and egalitarian coalitions. These phenomena are robust, and persist under alternative variants of our initial model.

Farsightedly Stable Matchings

We study the properties of von Neumann-Morgenstern farsightedly stable sets in application to matching models. We show that the result by Diamantoudi and Xue (2003) for hedonic games can be extended to a general matching with contracts framework: a collection of singleton stable sets constitutes a weak core of the matching with contracts game. We also show that singleton stable sets are invariant under diff erent contractual languages.

A hardness result for core stability in additive hedonic games

We investigate the computational complexity of a decision problem in hedonic coalition formation games. We prove that core stability in additive hedonic games is complete for the second level of the polynomial hierarchy.

Computing desirable partitions in additively separable hedonic games

An important aspect in systems of multiple autonomous agents is the exploitation of synergies via coalition formation. Additively separable hedonic games are a fundamental class of coalition formation games in which each player has a value for any other player and the value of a coalition to a particular player is simply the sum of the values he assigns to the members of his coalition. In this paper, we consider a number of solution concepts from cooperative game theory, welfare theory, and social choice theory as criteria for desirable partitions in hedonic games. We then conduct a detailed computational analysis of computing, checking the existence of, and verifying stable, fair, optimal, and popular partitions for additively separable hedonic games.

On non-trivial Nash stable partitions in additive hedonic games with symmetric 0/1-utilities

We consider the problem of computing non-trivial Nash equilibria in additive hedonic games with symmetric 0/1-utilities. Such a game can be represented by an undirected unweighted graph G(V,E) where a non-trivial Nash equilibrium corresponds to a partition of V into at least two sets such that each node has at least as many neighbours in its own set compared to any other set in the partition. We show that computing such an equilibrium is NP-complete. On the other hand, we show that such an equilibrium is computable in polynomial time if (1) G is triangle free, (2) G contains no 4-cycles sharing and edge, and (3) G is not a star. If G is not a star with girth at least 5 we show how to compute a non-trivial equilibrium in O(n)-time.

Coalition Formation in Generalized Apex Games

The class of games with one apex player is generalized to the class of games with a collection of apex sets. These simple games, together with a power index, canonically induce a hedonic coalition formation game. A monotonicity property of solutions is introduced and its meaning for the induced hedonic game is analyzed. Necessary and sufficient conditions for the existence of core stable partitions are stated and core stable partitions are characterized.

Status-seeking in hedonic games with heterogeneous players

We study group formation by heterogeneous players when players’ preferences are dictated by status-seeking. Status can be intrinsic or associative. Considering these two types of status as either complements or substitutes results in non-emptiness of the core of the corresponding games and allows for a full characterization of the core stable outcomes.

Hedonic games related to many-to-one matching problems

We consider the existence problem of stable matchings in many-to-one matching problems. Unlike other approaches which use algorithmic techniques to give necessary and sufficient conditions, we adopt a game theoretic point of view. We first associate, with each many-to-one matching problem, a hedonic game to take advantage of recent results guaranteeing the existence of core-partitions for that class of games, to build up our conditions. The main result states that a many-to-one matching problem, with no restrictions on individual preferences, has stable* matchings if and only if a related hedonic game is pivotally balanced. In the case that the preferences in the matching problem are substitutable, the notions of stability and stability* coincide.

An axiomatic analysis of ranking sets under simple categorization

This paper contributes to the axiomatization of additive rules for ranking sets of objects under the psychological principle of categorization. Firstly we proceed with the case where the elements in the sets are categorized into at most three groups, namely good (with value 1), neutral (with value 0), and bad (with value −1). Secondly, we solve the case where there are only good and neutral elements. In both instances the evaluation of the sets is purely additive. Lastly, we show that dropping one of the axioms in our general characterization produces an axiomatization of the more general class of evaluations where good and bad elements are weighted differently. Areas of research in Economics such as committee selection problems, hedonic games and matching are among the ranking sets models where our results could potentially be applied.

Size Monotonicity and Stability of the Core in Hedonic Games

We show that the core of each strongly size monotonic hedonic game is not empty and is externally stable. This is in sharp contrast to other sufficient conditions for core non-emptiness which do not even guarantee the existence of a stable set in such games.

Computational complexity in additive hedonic games

We investigate the computational complexity of several decision problems in hedonic coalition formation games and demonstrate that attaining stability in such games remains NP-hard even when they are additive. Precisely, we prove that when either core stability or strict core stability is under consideration, the existence problem of a stable coalition structure is NP-hard in the strong sense. Furthermore, the corresponding decision problems with respect to the existence of a Nash stable coalition structure and of an individually stable coalition structure turn out to be NP-complete in the strong sense.

Noncooperative formation of coalitions in hedonic games

We study a bargaining procedure of coalition formation in the class of hedonic games, where players’ preferences depend solely on the coalition they belong to. We provide an example of nonexistence of a pure strategy stationary perfect equilibrium, and a necessary and sufficient condition for existence. We show that when the game is totally stable (the game and all its restrictions have a nonempty core), there always exists a no-delay equilibrium generating core outcomes. Other equilibria exhibiting delay or resulting in unstable outcomes can also exist. If the core of the hedonic game and its restrictions always consist of a single point, we show that the bargaining game admits a unique stationary perfect equilibrium, resulting in the immediate formation of the core coalition structure.

Status-Seeking in Hedonic Games with Heterogeneous Players

We study hedonic games with heterogeneous player types that reflect her nationality, ethnic background, or skill type. Agents' preferences are dictated by status-seeking where status can be either local or global. The two dimensions of status define the two components of a generalized constant elasticity of substitution utility function. In this setting, we characterize the core as a function of the utility's parameter values and show that in all cases the corresponding cores are non-empty. We further discuss the core stable outcomes in terms of their segregating versus integrating properties.

Hedonic Coalition Formation in Conservative Societies

Top responsiveness is a concept which provides a model of progressive society in wihch each player pays more attention to the best aspect of each coalition, and is shown to be a sufficient conditaion for the existence of core stable coalition structures. In this paper, we propose a new concept, called bottom refuseness, for modeling societies with conservative players, i.e., each player pays more attention to the worst aspect of each coalition, and has tendency to choose among options a safer one. We show that all hedonic games satisfying bottom refuseness have core stable coalition structures. Moreover, we show that some previous results can be shown when additivity is imposed on top responsiveness and bottom refuseness.

A GENERIC APPROACH TO COALITION FORMATION

We propose an abstract approach to coalition formation that focuses on simple merge and split rules transforming partitions of a group of players. We identify conditions under which every iteration of these rules yields a unique partition. The main conceptual tool is a specific notion of a stable partition. The results are parametrized by a preference relation between partitions of a group of players and naturally apply to coalitional TU-games, hedonic games and exchange economy games.

Nash Stability in Additively Separable Hedonic Games and Community Structures

We prove that the problem of deciding whether a Nash stable partition exists in an Additively Separable Hedonic Game is NP-complete. We also show that the problem of deciding whether a non trivial Nash stable partition exists in an Additively Separable Hedonic Game with non-negative and symmetric preferences is NP-complete. We motivate our study of the computational complexity by linking Nash stable partitions in Additively Separable Hedonic Games to community structures in networks. Our results formally justify that computing community structures in general is hard.

Computational Complexity in Additive Hedonic Games

We investigate the computational complexity of several decision problems in hedonic coalition formation games and demonstrate that attaining stability in such games remains NP-hard even when they are additive. Precisely, we prove that when either core stability or strict core stability is under consideration, the existence problem of a stable coalition structure is NP-hard in the strong sense. Furthermore, the corresponding decision problems with respect to the existence of a Nash stable coalition structure and of an individually stable coalition structure turn out to be NP-complete in the strong sense.