Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence (AAAI-16) Graphical Hedonic Games of Bounded Treewidth Dominik Peters Department of Computer Science University of Oxford, UK [email protected] p Abstract NP-hard to maximise social welfare; and it is often even Σ2- complete to identify hedonic games with non-empty core. Hedonic games are a well-studied model of coalition See Ballester (2004), Sung and Dimitrov (2010), Woeginger formation, in which selfish agents are partitioned into (2013), and Peters and Elkind (2015) for a selection of such disjoint sets and agents care about the make-up of the coalition they end up in. The computational problems results. of finding stable, optimal, or fair outcomes tend to be A standard criticism of hardness results such as these is that computationally intractable in even severely restricted they apply only in the worst case. Instances arising in practice instances of hedonic games. We introduce the notion of can be expected to show much more structure than the highly a graphical hedonic game and show that, in contrast, on contrived instances produced in 3SAT-reductions. In non- classes of graphical hedonic games whose underlying cooperative game theory, graphical games (Kearns, Littman, graphs are of bounded treewidth and degree, such prob- and Singh 2001) are an influential way to allow formalisation lems become easy. In particular, problems that can be of the notion of a ‘structured’ game. In a graphical game, specified through quantification over agents, coalitions, and (connected) partitions can be decided in linear time. agents form the vertices of an undirected graph, and each The proof is by reduction to monadic second order logic. agent’s utility function only depends on the actions taken by We also provide faster algorithms in special cases, and her neighbours. The underlying graph can guide algorithms show that the extra condition of the degree bound cannot in finding a stable outcome, and imposing restrictions on the be dropped. Finally, we note that the problem of allo- graph topology can yield tractability (Gottlob, Greco, and cating indivisible goods can be modelled as a hedonic Scarcello 2005). game, so that our results imply tractability of finding A particularly successful restriction on graph topology is fair and efficient allocations on appropriately restricted bounded treewidth. The treewidth of a graph (Robertson and instances. Seymour 1986) measures how ‘tree-like’ a given graph is. Many NP-hard problems become fixed-parameter tractable 1 Introduction with respect to the treewidth of some graph naturally associ- ated with the problem instances. Indeed, dynamic program- Hedonic games, first studied by Banerjee, Konishi, and ming over a given tree decomposition often yields algorithms Sonmez¨ (2001) and Bogomolnaia and Jackson (2002), pro- that are exponential in the treewidth, but linear in the prob- vide a general framework for the study of coalition formation. lem size. See Bodlaender’s (1994) ‘tourist guide through Hedonic games subsume the well-studied matching prob- treewidth’ for an introduction to this area. lems (stable marriage, stable roommates, hospital-residents), The treewidth approach was first applied to the domain but are able to express more general preference structures. of hedonic games by Elkind and Wooldridge (2009), who They have been applied to problems in public good provi- propose a representation formalism for hedonic games called sion, voting, and clustering, and, as we show below, they also hedonic coalition nets that expresses agents’ preferences by encapsulate a variety of allocation problems. weighted boolean formulas. They also introduce a notion of A hedonic game consists of a set N of agents, each of treewidth for hedonic games—very close to ours—and show whom has a preference ordering over all coalitions S ⊆ N that (when numbers in the input are polynomially bounded) containing her. The outcome of such a game is a partition of it is fixed-parameter tractable with respect to treewidth to the agent set into disjoint coalitions, with agents preferring decide whether a given partition is core-stable. those partitions in which they are in a preferred coalition. Unfortunately, it has turned out that many key questions In this paper, we study graphical hedonic games, which we about hedonic games are computationally hard to answer. define in analogy to graphical games. Here, agents are again For example, it is typically NP-complete to decide whether a arranged in an underlying graph, and need to be partitioned hedonic game admits a Nash stable outcome; it is typically into coalitions. Every agent only cares about which of her neighbours are in the same coalition as her. Every hedonic Copyright c 2016, Association for the Advancement of Artificial game can be made graphical by introducing edges whenever Intelligence (www.aaai.org). All rights reserved. one agent’s utility depends on the other’s presence. We will 586 then consider graphical hedonic games whose agent graphs into disjoint coalitions. We write π(i) for the coalition S ∈ π have bounded treewidth and bounded degrees. that contains i ∈ N. We are interested in finding partitions In the context of hedonic games, restricting treewidth and that are stable, optimal, and/or fair. There are multiple ways also degrees in the underlying social network seems particu- of formalising these goals. For example, a partition π is Nash larly natural. Consider for example Dunbar’s number (Dun- stable if no agent wants to join another (possibly empty) bar 1992), a suggested limit on the number of individuals that coalition of π, that is π(i) i S ∪{i} for all S ∈ π ∪{∅} a single human being can maintain stable social relationships and all i ∈ N.Itisindividually stable if there is no agent i with. This number has been suggested to lie between 100 and coalition S ∈ π ∪{∅} such that S ∪{i}i π(i) and and 250, which gives us a natural bound on the degree of any S ∪{i} j S for all j ∈ S.Wesayπ is core-stable if there is social network. Intuitively, it also seems sensible to suppose no non-empty blocking coalition S ⊆ N such that S i π(i) that social networks have relatively small treewidth, though for each i ∈ S.Wesayπ is strict-core-stable if there is no see Adcock, Sullivan, and Mahoney (2013) who find mixed non-empty coalition S ⊆ N such that S i π(i) for each empirical support for this proposition. i ∈ S, with at least one preference strict. For fairness, we We show that when restricted to a class of graphical hedo- say that π is envy-free if no agent prefers taking another nic games whose agent graphs have bounded treewidth and agent’s place: π(i) i π(j) \{j}∪{i} for all i, j ∈ N bounded degrees, many standard problems related to these with π(i) = π(j). We will only consider the concept of hedonic games become linear-time solvable. More precisely, envy-freeness for hedonic games where every coalition is by a somewhat involved translation to monadic second-order acceptably to every player, i.e., if S i {i} for all S ∈Ni, logic and by appealing to Courcelle’s theorem, it follows that so that a player never envies a player being alone. we can decide in linear time whether a given such hedonic Note that the preference relations i have an exponentially game satisfies any logical sentence of what we call HG-logic, sized domain of 2n−1 coalitions. For computational purposes, which allows quantification over partitions, coalitions, and we need to use a language that represents such preferences agents. Using this approach and on this restricted domain, we succinctly, so that the representation preferably uses only can efficiently find stable or fair outcomes of a hedonic game poly(n) symbols, where n is the number of agents. An attrac- for all notions of stability that are commonly discussed in the tive such representation is given by additively separable hedo- literature. nic games, in which each agent specifies a valuation function We also show that HG-logic is expressive enough to cap- vi : N → R assigning each agent a numeric value. We then ture problems that would at first appear to lie outside the say that S i T if and only if j∈S vi(j) j∈T vi(j). domain of hedonic games, such as the problem of fair and An additively separable game is thus given by n2 numbers. efficient allocation of indivisible goods. This implies that A more expressive representation is proposed by Elkind questions regarding those problems can also be answered and Wooldridge (2009). They define hedonic coalition nets efficiently when we restrict treewidth and degree. (or HC-nets) in which each agent specifies a set of weighted Our appeal to Courcelle’s meta-algorithmic result, while propositional formulas, called rules, with propositional atoms powerful, comes at the cost of the hidden ‘constant factor’ given by the agents. For example, the rule i2 ∧i3 ∧¬i4 →i 5 growing dramatically as the treewidth and degree of the hedo- 1 means that agent i1 derives utility 5 when i1 is together nic game increase (indeed, this growth cannot be bounded by with i2 and i3 but not together with i4. If an agent specifies an elementary function unless P = NP). To show that despite multiple rules, the agent obtains the sum of the weights of this the restriction to bounded treewidth is useful in practice, those formulas that are satisfied in the given coalition. By we present a variety of more specific problems that can be taking rules of form j →i vi(j), we see that HC-nets can ˜ kd2 solved in the more manageable runtime O(2 n), where encode additively separable games.