Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence Optimal Partitions in Additively Separable Hedonic Games∗ Haris Aziz Felix Brandt Hans Georg Seedig Department of Informatics Technische Universitat¨ Munchen¨ 85748 Garching bei Munchen,¨ Germany {aziz,brandtf,seedigh}@in.tum.de Abstract form just as much as the formation of coalitions depends on how the payoffs are distributed. We conduct a computational analysis of fair and Coalition formation games, as introduced by Dreze` and optimal partitions in additively separable hedonic Greenberg [1980], provide a simple but versatile formal games. We show that, for strict preferences, a model that allows one to focus on coalition formation as such. Pareto optimal partition can be found in polynomial In many situations it is natural to assume that a player’s appre- time while verifying whether a given partition is ciation of a coalition structure only depends on the coalition Pareto optimal is coNP-complete, even when pref- he is a member of and not on how the remaining players are erences are symmetric and strict. Moreover, com- grouped. Initiated by Banerjee et al. [2001] and Bogomolnaia puting a partition with maximum egalitarian or util- and Jackson [2002], much of the work on coalition formation itarian social welfare or one which is both Pareto now concentrates on these so-called hedonic games. optimal and individually rational is NP-hard. We The main focus in hedonic games has been on notions of also prove that checking whether there exists a par- stability for coalition structures such as Nash stability, indi- tition which is both Pareto optimal and envy-free vidual stability, contractual individual stability, or core stabil- is Σp-complete. Even though an envy-free partition 2 ity and characterizing conditions under which they are guar- and a Nash stable partition are both guaranteed to anteed to be non-empty [see, e.g., Bogomolnaia and Jackson, exist for symmetric preferences, checking whether 2002]. The most prominent examples of hedonic games are there exists a partition which is both envy-free and two-sided matching games in which only coalitions of size Nash stable is NP-complete. two are admissible [Roth and Sotomayor, 1990]. General coalition formation games have also received at- 1 Introduction tention from the artificial intelligence community, where the focus has generally been on computing partitions that give Ever since the publication of von Neumann and Morgen- rise to the greatest social welfare [see, e.g., Sandholm et stern’s Theory of Games and Economic Behavior in 1944, al., 1999]. The computational complexity of hedonic games coalitions have played a central role within game theory. The has been investigated with a focus on the complexity of crucial questions in coalitional game theory are which coali- computing stable partitions for different models of hedonic tions can be expected to form and how the members of coali- games [Ballester, 2004; Dimitrov et al., 2006; Cechlarov´ a,´ tions should divide the proceeds of their cooperation. Tra- 2008]. We refer to Hajdukova´ [2006] for a critical overview. ditionally the focus has been on the latter issue, which led to Among hedonic games, additively separable hedonic the formulation and analysis of concepts such as Gillie’s core, games (ASHGs) are a particularly natural and succinct rep- the Shapley value, or the bargaining set. Which coalitions resentation in which each player has a value for every other are likely to form is commonly assumed to be settled exoge- player and the value of a coalition to a particular player is nously, either by explicitly specifying the coalition structure, computed by simply adding his values of the players in his a partition of the players in disjoint coalitions, or, implicitly, coalition. by assuming that larger coalitions can invariably guarantee Additive separability satisfies a number of desirable ax- better outcomes to its members than smaller ones and that, as iomatic properties [Barbera` et al., 2004]. ASHGs are the a consequence, the grand coalition of all players will eventu- non-transferable utility generalization of graph games stud- ally form. ied by Deng and Papadimitriou [1994]. Sung and Dimitrov The two questions, however, are clearly interdependent: ff [2010] showed that for ASHGs, checking whether a core sta- the individual players’ payo s depend on the coalitions that ble, strict-core stable, Nash stable, or individually stable par- ∗This material is based on work supported by the Deutsche tition exists is NP-hard. Dimitrov et al. [2006] obtained posi- Forschungsgemeinschaft under grants BR-2312/6-1 (within the Eu- tive algorithmic results for subclasses of additively separable ropean Science Foundation’s EUROCORES program LogICCC) hedonic games in which each player divides other players into and BR 2312/7-1. friends and enemies. Branzei and Larson [2009] examined 43 ff ∈N the tradeo between stability and social welfare in ASHGs. coalition as i and if i is in coalition S i, then i gets utility , ∈N j∈S \{i} vi( j). For coalitions S T i, S i T if and only if ≥ j∈S \{i} vi( j) j∈T\{i} vi( j). Contribution In this paper, we analyze concepts from fair = division in the context of coalition formation games. We A preference profile is symmetric if vi( j) v j(i) for any two players i, j ∈ N and is strict if vi( j) 0 for all i, j ∈ N present the first systematic examination of the complexity of computing and verifying optimal partitions of hedonic games, such that i j. We consider ASHGs (additively separable specifically ASHGs. We examine various standard criteria hedonic games) in this paper. Unless mentioned otherwise, from the social sciences: Pareto optimality, utilitarian social all our results are for ASHGs. welfare, egalitarian social welfare, and envy-freeness [see, e.g., Moulin, 1988]. 2.2 Fair and optimal partitions In Section 3, we show that computing a partition with max- In this section, we formulate concepts from the social sci- imum egalitarian social welfare is NP-hard. Similarly, com- ences, especially the literature on fair division, for the context puting a partition with maximum utilitarian social welfare is of hedonic games. A partition π satisfies individual rational- NP-hard in the strong sense even when preferences are sym- ity if each player does as well as by being alone, i.e., for all metric and strict. i ∈ N, π(i) i {i}. For a utility-based hedonic game (N, P) and In Section 4, the complexity of Pareto optimality is studied. partition π, we will denote the utility of player i ∈ N by uπ(i). We prove that checking whether a given partition is Pareto op- The different notions of fair or optimal partitions are defined timal is coNP-complete in the strong sense, even when pref- as follows.2 erences are strict and symmetric. By contrast, we present a 1. The utilitarian social welfare of a partition is defined as polynomial-time algorithm for computing a Pareto optimal the sum of individual utilities of the players: u (π) = partition when preferences are strict.1 Interestingly, comput- ut uπ(i). A maximum utilitarian partition maximizes ing an individually rational and Pareto optimal partition is i∈N the utilitarian social welfare. NP-hard in general. In Section 5, we consider complexity questions regarding 2. The elitist social welfare is given by the utility of the envy-free partitions. Checking whether there exists a parti- player that is best off: uel(π) = max{uπ(i) | i ∈ N}.A tion which is both Pareto optimal and envy-free is shown to maximum elitist partition maximizes the utilitarian so- be Σp-complete. We present an example which exemplifies cial welfare. 2 ff the tradeo between satisfying stability (such as Nash sta- 3. The egalitarian social welfare is given by the utility of bility) and envy-freeness and use the example to prove that the agent that is worst off: ueg(π) = min{uπ(i) | i ∈ N}. checking whether there exists a partition which is both envy- A maximum egalitarian partition maximizes the egali- free and Nash stable is NP-complete even when preferences tarian social welfare. are symmetric. 4. A partition π of N is Pareto optimal if there exists no partition π of N which Pareto dominates π, that is for 2 Preliminaries all i ∈ N, π (i) i π(i) and there exists at least one player In this section, we provide the terminology and notation re- j ∈ N such that j ∈ N, π ( j) j π( j). quired for our results. 5. Envy-freeness is a notion of fairness. In an envy-free 2.1 Hedonic games partition, no player has an incentive to replace another player. A hedonic coalition formation game is a pair (N, P) where N is a set of players and P is a preference profile which specifies For the sake of brevity, we will call all the notions de- for each player i ∈ N the preference relation i, a reflexive, scribed above “optimality criteria” although envy-freeness is complete and transitive binary relation on set Ni = {S ⊆ N | rather concerned with fairness than optimality. We consider i ∈ S }. the following computational problems with respect to the S i T denotes that i strictly prefers S over T and S ∼i T optimality criteria defined above. that i is indifferent between coalitions S and T.Apartition π is a partition of players N into disjoint coalitions. By π(i), we Optimality: Given (N, P) and a partition π of N,isπ optimal? denote the coalition in π which includes player i. Existence: Does an optimal partition for a given (N, P) exist? A game (N, P)isseparable if for any player i ∈ N and any Search: If an optimal partition for a given (N, P) exists, find coalition S ∈Ni and for any player j not in S we have the one.