Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence (AAAI-16) Complexity of Hedonic Games with Dichotomous Preferences Dominik Peters Department of Computer Science University of Oxford, UK [email protected] Abstract SW PF PO NS IS CR SCR p ∗ Hedonic games provide a model of coalition formation Boolean NP-c. NP-c. NP-h. NP-c. P FNP-h. Σ2-c. in which a set of agents is partitioned into coalitions 1-lists NP-c. P P P P P NP-c. and the agents have preferences over which set they be- long to. Recently, Aziz et. al. (2014) have initiated the 2-lists NP-c. P P P P P NP-c. study of hedonic games with dichotomous preferences, 3-lists NP-c. NP-c. NP-h. ? P P NP-c. where each agent either approves or disapproves of a 4-lists NP-c. NP-c. NP-h. NP-c. P P NP-c. given coalition. In this work, we study the computa- ∗ tional complexity of questions related to finding opti- Anonymous NP-c. NP-c. NP-h. NP-c. P P NP-c. mal and stable partitions in dichotomous hedonic games Intervals P P P ? P P ? under various ways of restricting and representing the Roommates P P P∗ NP-c. P P P∗ collection of approved coalitions. Encouragingly, many of these problems turn out to be polynomial-time solv- Majority ? P ? P P P P able. In particular, we show that an individually stable outcome always exists and can be found in polynomial Table 1: Overview of complexity results for various dichoto- time. We also provide efficient algorithms for cases in mous preference representations; results marked (∗) were which agents approve only few coalitions, in which they obtained elsewhere. The columns describe the problems of only approve intervals, and in which they only approve maximising welfare, and of finding (respectively) perfect, sets of size 2 (the roommates case). These algorithms pareto-optimal, Nash-stable, individually stable, core-stable, are complemented by NP-hardness results, especially and strict-core-stable partitions. for representations that are very expressive, such as in the case when agents’ goals are given by propositional formulas. optimal, or fair has turned out to be intractable even for a large variety of severely restricted preference structures. Introduction Recently, Peters and Elkind (2015) have shown that decid- ing whether a given hedonic game admits any stable outcome A coalition is an alliance between a group of individuals, at all is NP-hard for preference restrictions and representa- formed in order to achieve a common goal. How do such tions that allow agents to express more than 4 or 5 preference coalitions form if agents are selfish? An extensive literature ‘intensities’ (with some mild additional qualifiers). A result in economics and computer science has studied this question of Deineko and Woeginger (2013) shows this to also be the using the natural model of a hedonic game (see the survey case for a specific restriction allowing 3 intensities. These byAziz and Savani (2016)). A hedonic game consists of a set results suggest that if we want to stand a chance of finding of agents, each of which submits a preference ordering over polynomial time algorithms for a restricted class of hedonic all possible coalitions this agent could join. An outcome of games, we will need to go all the way down to dichotomous the game is a partition of the agent set into disjoint coalitions. preferences, which allow only 2 preference intensities. If agents are selfish, we want to find a stable outcome, while in other situations a welfare-optimal or fair outcome might In the context of hedonic games, studying the restriction to be desired. dichotomous preferences has recently been proposed by Aziz There have turned out to be multiple obstacles to achieving et al. (2014). They represent agents’ preferences by formulas these tasks. First, not all hedonic games admit any stable of propositional logic. In particular, we can use the names outcome, and thus the search for one may be futile. Second, of agents as propositional variables. An agent then approves the computational problem of finding a partition that is stable, a coalition if the members of that coalition satisfy her goal formula. Accordingly, they term games with this preference Copyright c 2016, Association for the Advancement of Artificial representation “boolean hedonic games”. Intelligence (www.aaai.org). All rights reserved. This logic representation is attractive in that it is univer- 579 sally expressive for dichotomous preferences and often suc- Preliminaries cinct. A further advantage is that we may use it to translate A hedonic game N,(i)i∈N is given by a finite set N of computational questions such as “find a stable partition” into i ∈ N propositional logic, and then use an off-the-shelf SAT solver agents, and for each agent a complete and transitive preference relation over Ni = {S ⊆ N : i ∈ S}.We to answer it. Such a translation is presented in detail by Aziz ∼ et al. (2014). Note that for some solution concepts, they write i and i for the strict and indifference parts of i. A hedonic game has dichotomous preferences, and is called have not found polynomial-size expressions. Still, given the i ∈ N impressive performance of modern SAT solvers on instances a dichotomous hedonic game, if for each agent , the coalitions Ni = {S ⊆ N : i ∈ S} can be partitioned arising in practice, we might hope that this approach allows + into approved coalitions Ni and non-approved coalitions hedonic games to be applicable in practice. − Ni such that i strictly prefers approved coalitions to non- Aziz et al. (2014) argue that their specific SAT encoding approved coalitions, but is indifferent within the two groups: cannot be improved by an efficient algorithm if and only if the + − so S i T iff S ∈N and T ∈N . corresponding computational problems are NP-hard. They i i The outcome of a hedonic game is a partition π of the write that “identifying the complexity of finding partitions agent set into disjoint coalitions. We write π(i) for the coali- satisfying solution concepts for Boolean hedonic games is tion S ∈ π that contains i ∈ N. We are interested in finding therefore the most immediate direction of further research.” partitions that are stable, optimal, and/or fair. In a dichoto- In this work, we study a variety of different restrictions mous hedonic game, a partition π maximises social welfare of agents’ dichotomous preferences, and use the term “di- if it has the maximum number of agents who are in approved chotomous hedonic game” for any hedonic game in which coalitions among all partitions of N.Ifevery agent is in all agents have dichotomous preferences. an approved coalition in π, then π is called perfect (some- π Theoretically speaking, the dichotomous case is nice be- times known as wonderfully stable). A partition is Pareto- π π(i) π(i) cause every dichotomous hedonic game admits an outcome optimal if there is no partition such that i for i ∈ N π(i) π(i) i ∈ N that is simultaneously core-stable (resistant to group devia- all and i for some . Fairness can be π tions) and individually stable (resistant to deviations by any formalised using the notion of envy-freeness: a partition is i single player). This is a refinement of an observation by Aziz envy-free if there is no agent who would prefer to be in the j π(j) \{j}∪{i} π(i) et al. (2014). While the argument establishing the existence position of agent , i.e. i . of such a partition does not yield a polynomial-time algo- There are many notions of stability for a partition π in a rithm, we identify many appealing special cases in which hedonic game. We will mainly use four such concepts. A there is one. In the general case, we always can in polyno- partition π is core-stable if there is no non-empty coalition mial time find a partition that is individually stable (but not S ⊆ N with S i π(i) for all i ∈ S. Thus, every member necessarily core-stable) when given an oracle that decides of S would strictly prefer being in S to being where they whether a given coalition is approved by a given agent. have been put under π. A partition π is strict-core-stable if there is no non-empty coalition S ⊆ N with S i π(i) for We further study the computational complexity of finding all i ∈ S and S i π(i) for some i ∈ S. In both of these a partition that maximises the number of players who ap- notions, a group of agent deviates. If we restrict our attention prove it, of finding a Pareto optimal partition, and of deciding to the possibility of just a single agent deviating, we obtain whether a strict-core-stable or a Nash-stable partition exists the notion of Nash-stability. Here, no agent i prefers to join (these concepts are strengthenings of core- and individual another coalition in π, that is π(i) i π(j) ∪{i} for all j, stability). We find that these problems are all NP-hard under and also π(i) i {i}.Inanindividually stable (IS) partition, the logic representation (which is not too surprising given its no agent prefers to deviate in this way and is welcomed by expressive power) and welfare-maximisation in particular is his new coalition. Formally, an agent i IS-deviates into a inapproximable and fixed-parameter intractable. In contrast, coalition S ∈ π ∪{∅} if S i π(i) and for each j ∈ S,we we find that these problems become easy when requiring have π(j) ∪{i} j π(j).