Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015)
Simple Causes of Complexity in Hedonic Games
Dominik Peters and Edith Elkind Department of Computer Science University of Oxford, UK {dominik.peters, edith.elkind}@cs.ox.ac.uk
Abstract SNSSCRCRNSIS Hedonic games provide a natural model of coali- IRCL of length 6 9 NP-h. NP-c. NP-c. NP-c. NP-c. tion formation among self-interested agents. The Hedonic Coalition Nets NP-h. NP-h. NP-h. NP-c. NP-c. associated problem of finding stable outcomes in such games has been extensively studied. In this W-preferences (no ties) (P) (P) NP-c. NP-c. paper, we identify simple conditions on expressivity W-preferences NP-h. NP-c. NP-c. NP-c. of hedonic games that are sufficient for the problem WB-preferences (no ties) (P) (P) NP-c. NP-c. of checking whether a given game admits a stable WB-preferences NP-h. NP-c. NP-c. NP-c. outcome to be computationally hard. Somewhat B- & W-hedonic games NP-h. NP-h. NP-c. NP-c. surprisingly, these conditions are very mild and in- tuitive. Our results apply to a wide range of stability Additively separable NP-h. NP-h. NP-h. NP-c. NP-c. concepts (core stability, individual stability, Nash Fractional hedonic games NP-h. NP-h. NP-h. NP-c. NP-c. stability, etc.) and to many known formalisms for Social FHGs NP-h. NP-h. (+) (+) hedonic games (additively separable games, games Median NP-h. NP-h. with W-preferences, fractional hedonic games, etc.), Midrange ( 1 B + 1 W) NP-h. NP-h. NP-c. NP-c. and unify and extend known results for these for- 2 2 4-Approval NP-h. NP-h. NP-h. NP-c. NP-c. malisms. They also have broader applicability: for several classes of hedonic games whose computa- tional complexity has not been explored in prior Table 1: Some of the hardness results implied by our framework work, we show that our framework immediately im- for the problem of identifying hedonic games with stable outcomes. Gray entries are results that have not appeared in the literature before. plies a number of hardness results for them. (P) indicates known polynomial-time algorithms, (+) means that a stable outcome always exists. See Section 6 for details. 1 Introduction Hedonic games [Dreze` and Greenberg, 1980; Banerjee et al., formalisms for hedonic games, i.e., ones where a game de- 2001; Bogomolnaia and Jackson, 2002] provide an elegant scription size scales polynomially with the number of agents n. and versatile model of coalition formation among strategic Typically, such formalisms are not universally expressive, but agents. In such games, each agent has preferences over coali- capture important classes of hedonic games. For instance, if tions (subsets of players) that she can be a part of, and an the utility that an agent assigns to a coalition is given by the outcome of the game is a partition of agents into coalitions. sum/average/minimum/maximum of the utilities she assigns to Clearly, the quality of an outcome depends on how well it individual members of that coalition, the entire game can be de- reflects the agents’ preferences. In particular, it is desirable to scribed by n(n−1) numbers (such games are known as, respec- have outcomes that are stable, i.e., do not offer the agents an tively, additively separable games [Bogomolnaia and Jackson, opportunity to profitably deviate. Many different concepts of 2002], fractional hedonic games [Aziz et al., 2014], and games stability have been proposed in the hedonic games literature with W- and B-preferences [Cechlarov´ a´ and Hajdukova,´ 2003; (see Section 2 for a brief summary, and [Aziz and Savani, 2004b]). There are also representation formalisms that are 2015] for an in-depth discussion), and for each of them a natu- universally expressive (and hence exponentially verbose in ral computational question is whether a given game admits an the worst case), but provide succinct descriptions of hedo- outcome that is stable in that sense. nic games that have certain structural properties; examples The complexity of this question depends on how the game include Individually Rational Coalition Lists [Ballester, 2004] is represented: while every hedonic game can be described and Hedonic Coalition Nets [Elkind and Wooldridge, 2009]. by explicitly listing each agent’s preference relation over all The complexity of stability-related problems under these and coalitions that may contain her, in recent years there has been other representations for hedonic games has been investigated a considerable amount of research on succinct representation by a number of researchers (see [Woeginger, 2013a] for a sur-
617 vey); with a few exceptions, checking whether a game admits disjoint coalitions. We write π(i) for the coalition of π that 0 0 a stable outcome turns out to be computationally hard. contains i. For partitions π and π , we write π solution concept for group deviations is the automatically derive new hardness results for hedonic games: core (CR). We say that a non-empty coalition S CR-blocks π instead of coming up with a hardness reduction, one can sim- if S i π(i) for all i ∈ S; it SCR-blocks π if S
618 order >i over N. We interpret >i as i’s preference order over To avoid difficulties with binary representations that are very the set of players. We write j >i k if j >i k but not k >i j, short, we will assume that the binary encoding of hN,(
619 REMARK II. The same result holds for SIS-EXISTENCEFOR C. blocks. By consistency on pairs, it can be checked that no It applies to SNS-EXISTENCEFOR C if we add {1-1}-toxicity coalition of size 2 blocks. Now let S be a coalition with |S| > 3. and weak {2-1}-toxicity. It applies to SSNS-EXISTENCEFOR C Consider the friendship graph on N with friends connected by if we add strict {0-1,1-1}-toxicity and weak {2-1}-toxicity. an edge. This graph has girth 6 and does not contain a cubic subgraph. If S contained an isolated agent or a leaf (a member 5 The Reductions with at most 1 friend in S), then S is not blocking by {0-1}- toxicity and weak {1-1}-toxicity. So S contains a cycle and The proofs of our results are by reduction from a restricted thus |S| > 6. Since S is not cubic, there is an agent, matched version of 3SAT. The reduction behind Theorem 1 is in- in π, who has 2 friends in S and so by weak {2-2}-toxicity spired by an argument of Ronn [1990] showing that STABLE- is worse off in S, so S does not block. Hence there are no ROOMMATES with ties is NP-complete. Theorem 3 introduces blocking coalitions and π is in the core. triangles into this reduction to allow strict preferences. Let π be a CR-stable partition of G. We sketch an argu- We sketch the proof of Theorem 1 but omit proofs of the ment giving a satisfying assignment of ϕ. Because the players other claims due to space constraints. The omitted arguments {c1,...,c9} of a clause are unstable on their own (toxicity are similar to the one given, but more complicated due to limits coalitions within them to size 2, no agent wants to be SNS-like stability concepts imposing little structure. Full clockwise last in a coalition, and the number of members is proofs are given in an extended version of this paper, available odd), stability of π implies that for each clause one of its on arXiv.org [Peters and Elkind, 2015]. players must be in a coalition with the literal connected to it. Define a propositional assignment A so that all literals in a PROOFOF THEOREM 1 (SKETCH). We reduce from (3,B2)- coalition with a c-player are set true, and set other variables SAT, which is 3SAT restricted to formulas in which each clause arbitrarily. This assignment is well-defined. Indeed, suppose contains exactly 3 literals, and each variable occurs exactly variable x is to be set both true and false. Then WLOG ei- twice positively and twice negatively [Berman et al., 2003]. ther both x1 and x1 or both x1 and x2 are matched with their Given an instance formula ϕ with variable set X and clause c1. Either {xa,x1} or {x1,x2} will then end up blocking, a set C, we construct the following agent set N: contradiction. Clearly, A satisfies ϕ. S 0 00 0 00 S x∈X {x1,x1,x2,x2,xa,xa,xa,xb,xb,xb} ∪ c∈C{c1, . .. ,c9}. The four occurrences (two positive ones and two negative 6 Applications ones) of a variable x ∈ X are called x1,x2,x1,x2. respectively. Our NP-hardness results have implications for many well- For a clause c = `1 ∨`2 ∨`3, we write c(`1) := c1, c(`2) := c4, known classes of hedonic games. In this section we briefly c(`3) = c7. Construct orderings (>i)i∈N as follows: describe some of these (see [Aziz and Savani, 2015] for details) 0 x1 : xa > x2 > c(x1) xa : x1 ∼ x1 > xa c1 : `1 > c2 > c9 and check which of our conditions they satisfy. In this way, 0 we recover—and sometimes strengthen—a number of known x2 : xb > x1 > c(x2) xb : x2 ∼ x2 > xb c4 : `2 > c5 > c3 0 00 hardness results for these games. We also introduce five new x1 : xa > x2 > c(x1) xa : xa > xa c7 : `3 > c8 > c6 x : x > x > c(x ) x0 : x > x00 c : c > c classes of games, and show how our framework allows us to 2 b 1 2 b b b i i+1 i−1 deduce hardness results for them with ease. x00 : x0 x00 : x0 a a b b The extended version [Peters and Elkind, 2015] of this paper For each agent i we have only listed i’s friends Fi, each friend gives further details on constructions outlined here. being strictly better than i. Any agent not mentioned in i’s list Individually Rational Coalition Lists (IRCL). Balles- is an enemy, i.e., an element of Ei. Figure 1(a) illustrates the ter [2004] proposes to represent a hedonic game by listing orderings (>i)i∈N. Note that no agent has more than 3 friends, and that these orderings are mutual. the agent preferences i)i∈N given above. We show that ϕ is satisfiable if and only if G admits a CR-stable partition. rems 1 and 2 apply even if each agent has only 3 friends, we Let A be a satisfying assignment of ϕ. Take the partition therefore have a hardness result for α ∈ {SNS,SIS,CR,NS,IS} even if the list of each agent includes at most 3 entries, each π = {{`,c(`)} : ` a true variable occurrence} of which is a pair. A similar result is shown by Deineko ∪ {{xa,x1},{xb,x2} : x ∈ X set true in A} and Woeginger [2013]. They prove that CR-EXISTENCEFOR IRCL is hard even for lists of length 2, with entries being ∪ {{xa,x1},{xb,x2} : x ∈ X set false in A} 0 00 0 00 coalitions of size 3. Theorem 3 applies if we allow lists up ∪ {{xa,xa},{xb,xb} : x ∈ X} to length 9, which can encode a triangle-appreciating game ∪ {{ci,ci+1},...,{c j} : c ∈ C}. where agents have up to 4 friends. In the last line we partition clause players that are not matched Hedonic Coalition Nets. Elkind and Wooldridge [2009] to true variables into pairs and singletons in some stable way study a rule-based representation for hedonic games in which as in Figure 1(a), see full proof for details. agents’ preferences are described by weighted boolean for- We show that π is CR-stable in G. Since π is IR, no singleton mulas. It can be shown that polynomial size nets are suf-
620 (a) (b) (c)
3 3 3 3
x1 x2 x1 x2 c x1 x1 c 5 5 6 5 6 5 4 4 4 4 xa xb xa 7 xb xc 7 xd 5 5 5 6 5 6 x x1 x2 x1 x2 c 2 x2 c 3 3 3 3
Figure 1: Graphical presentations of the 3SAT reductions used. Figure (a) is used in Theorem 1, (b) in Theorem 3, and (c) in Theorem 2. Agents not connected by an edge are enemies. The gray sets of agents indicate a stable partition π in the hedonic game. The 9-gons form clause gadgets in (a) and (b) and a variable gadget in (c). On their own, these groups of 9 players do not admit a stable outcome. So in (a) and (b), stability can only occur if for each clause one of its agents can be connected to one of its (true) literals, i.e. if the underlying formula is satisfiable. In (c), each variable must be set true or false for stability to occur.
ficient to describe, for any collection of orderings (>i)i∈N, Aziz also notes that it is possible to embed SMTI into other a game satisfying all our conditions, implying hardness of classes C of hedonic games, and thus to deduce hardness of α-EXISTENCE for all α considered in this work. This is NS-EXISTENCEFOR C; this observation provides an (alterna- perhaps not surprising: while Elkind and Wooldridge only tive) method of deriving hardness results for several classes establish the hardness of CR-EXISTENCE in their work, they of hedonic games. show that one can compile an IRCL representation into a W-preferences. Cechlarov´ a´ and Hajdukova´ [2004b] consider hedonic coalition net representation with at most polynomial hedonic games where each agent first ranks all other agents overhead. Because our hardness results hold even if each and then compares coalitions based on their worst member player is only allowed 3 or 4 friends, we can say in addition under this ranking. Clearly, the game so obtained is con- that -EXISTENCE for hedonic coalition nets remains hard α sistent on pairs and strictly {k-1}-toxic for all k. It follows even if we restrict each player’s preferences to be described that (with ties allowed) CR-EXISTENCE is NP-hard by Theo- by at most 4 or 5 formulas, and even if the weights of these rem 1 (a result first obtained by [Cechlarov´ a´ and Hajdukova,´ formulas are given in unary. 2004b]), and that NS- and IS-EXISTENCE are NP-complete Stable Roommates. The reduction behind Theorem 1 is a by Theorem 2, first shown by Aziz et al. [2012]. NS- and modified version of Ronn’s construction showing that CR- IS-EXISTENCE are hard even if preferences are strict; the EXISTENCE FOR SRT, the stable roommate problem with latter result was previously unknown. ties, is NP-complete [Ronn, 1990]. It is thus no surprise that the class of stable roommate problems, considered as WB-preferences. Noting that agents in W-hedonic games are hedonic games in which sets with 3 or more members are extremely pessimistic, Cechlarov´ a´ and Hajdukova´ [2004a] unacceptable, fulfills the conditions of Theorem 1 (but note propose a compromise: Agents still rank coalitions according that this formulation corresponds to SRTI, not SRT). Indeed, to their worst member, but break ties in favor of the coalition CR-EXISTENCE FOR SRTI remains hard even if the preference with better best member. Again the game obtained is consis- list of each agent has length at most 3, and by Theorem 2 tent on pairs and strictly {k-1}-toxic for all k, so CR-, NS-, IS EXISTENCE this is also true of NS- and IS-EXISTENCE. Now, consider a and - are hard. generalization of STABLE-ROOMMATES where rooms have W- and B-hedonic games. In these two classes of games capacity 1, 2, or 3, and rooms with capacity 3 are generally [Aziz et al., 2012; 2013], agents rank coalitions according preferred because they are cheaper per person. Then it can be to their worst or best member, but coalitions containing an checked that the conditions of Theorems 2 and 3 are satisfied, enemy are not individually rational. As for W-preferences, giving hardness of α-EXISTENCE for all α for this model. we see that CR-, NS-, and IS-EXISTENCE are hard. Stable Marriages. A version of Theorem 2 implies that NS- EXISTENCEFORSMI, the stable marriage problem with In all of the following classes of games, agents first assign incomplete lists, is NP-complete. This extends the NP- cardinal utilities vi( j) ∈ R to all agents in N = {1,...,n}, and completeness result for SMTI obtained by [Aziz, 2013]. then lift these utilities to coalitions (e.g., by computing the
621 sum or average of the utilities of coalition members). if agents take either the ordinal or cardinal median of the The following method of constructing integer-valued func- coalition S \{i} then both Theorems 1 and 2 apply. tions vi : N → R from orderings (>i)i∈N will be used repeat- Geometric Mean Games. In these games agents evaluate |pS| edly: given x,y ∈ Z, we set vi(i) = 0, vi( j) = x for j ∈ Ei and coalitions according to the geometric mean ∏vi( j) of let y 6 vi( j) 6 y + n for j ∈ Fi so that for each j,k ∈ Fi we member utilities. We obtain the same hardness results as have vi( j) > vi(k) iff j >i k (this is accomplished by assigning for FHGs by taking logs. utility y + k + 1 to friends at the k-th ‘preference level’). We Nash Product Games. This is the class of games that are refer to such utilities as x,y -utilities. J K ‘multiplicatively separable’; agents evaluate coalitions ac- Additively Separable Games (ASGs). In these games, pref- cording to ∏ j∈S vi( j). As far as hardness is concerned erences are given by S ∑ j∈T vi( j). This these games behave identically to additively separable games, class of games satisfies all our theorems, so α-EXISTENCE is again by taking logs. hard for all α we consider. Indeed, given N = {1,...,n} and 1 1 2 Midrange ( B + W). In this case, agents evaluate a coali- (>i)i∈N, we consider the ASG with −(n +2n),4 -utilities. 2 2 Then a coalition containing an enemyJ of i is not individuallyK tion by averaging the maximum and minimum utility in it. With −3n,1 -utilities, these games are strictly {k-1}-toxic rational for i, so this game is strictly {k-1}-toxic for all k, and J K it is obviously consistent on pairs, triangle-appreciating and for all k and consistent on pairs, so Theorems 1 and 2 apply. monotone on triangles. α-EXISTENCE remains hard even r-Approval. Starting with cardinal utilities, sum the (up to) if players are allowed at most 3 or 4 friends (depending on r highest elements of a coalition. If r > 3, then games α), so for ASGs, it remains hard even if vi( j) is positive with −6rn,4 -utilities satisfy the conditions of Theorems 1 for at most 3 or 4 agents j. This improves on the reduction and 2.J If r > 4,K they satisfy the conditions of Theorem 3. in [Sung and Dimitrov, 2010], where agents have up to 11 friends. 7 Conclusions Fractional Hedonic Games (FHGs). This class of games We have developed a framework that enables us to prove was recently proposed by Aziz et al. [2014]. Preferences NP-hardness of α-EXISTENCEFOR C for many choices of α are given by S 1/|T|∑ j∈T vi( j). and C. Our results show that problems in this family tend to Brandl et al. [2015] have shown hardness of CR-, NS-, be hard even for representation formalisms with very limited and IS-EXISTENCE. We recover these results and comple- expressivity, and, moreover, are unlikely to admit an efficient ment them by showing hardness of SSNS-, SNS-, SIS- and parametrized algorithm for many natural choices of parameter SCR-EXISTENCE; all these results hold even if the underly- (such as length and coalition size in the IRCL representation ing preferences are strict. FHGs with −(n2+5n),5 -utilities or number of formulas per agent in the hedonic coalition nets satisfy all of our properties; choosingJy = 5 ensuresK triangle- representation). However, they also indicate which features appreciation. of hedonic games may lead to tractability of stability-related Social FHGs. An FHG is social if agents’ utilities for each problems. In particular, restricting the number of different other are non-negative. Theorem 3 applies to the class of ‘preference intensities’ (e.g., the range of vi( j) in ASGs, FHGs, social FHGs. Indeed, given (>i)i∈N we can construct a social and median games) rules out consistency on pairs, so one may FHG with 0,7n -utilities. Toxicity follows from vi( j) > hope for easiness results when this number is small. 7n for j ∈ JFi, andK other properties can be checked as for While we focused on the problem of checking whether FHGs. To ensure that our framework applies to social FHGs, a stable partition exists, another important stability-related we carefully crafted our constructions to only require weak problem is checking whether a specific partition is stable. toxicity whenever possible. This problem is in P for IS and NS for all classes of hedonic games considered here, simply because the number of possible The next five classes of hedonic games are based on fairly deviations is polynomially bounded; however, for notions of intuitive ways of deriving utilities for coalitions from utilities stability that are based on group deviations it is often coNP- for individual players; however, to the best of our knowledge complete. It would be interesting to extend our framework to we are the first to consider the computational complexity of handle this problem as well. stability-related probems for these games (median games have Since verifying stability is often hard, α-EXISTENCEFOR C been suggested by [Hajdukova,´ 2006] as an interesting topic; is usually not known to be in NP for stability notions based the other four classes appear to be entirely new). on group deviations. Thus most of our hardness results do not Median Games. Agents evaluate coalitions according to their have a tight complexity upper bound. For all representation median value, which in odd-size coalitions is the middle p formalisms we consider, these problems are in Σ2 , and CR- element, and in even-size coalitions is the mean of the middle EXISTENCEFOR ASGS is known to be complete for this two elements. Median games with 0,5 -utilities satisfy complexity class [Woeginger, 2013b]. A natural open question Theorem 3. Notice that in this constructionJ K vi( j) are non- is whether our framework can be extended from NP-hardness negative, so hardness holds even for ‘social median games’ p proofs to Σ2 -hardness proofs. with non-negative underlying utilities. There are various other ways of defining median games. In particular, we can Acknowledgements use a purely ordinal version by taking the worse of the middle We thank the anonymous reviewers from IJCAI and CoopMAS two players in even-sized coalitions, satisfying Theorem 1; for their helpful comments and pointers to the literature.
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