Core Stability in Hedonic Games Among Friends and Enemies: Impact of Neutrals

Proceedings of the Twenty-Sixth International Joint Conference on Artiﬁcial Intelligence (IJCAI-17)

Core Stability in Hedonic Games among Friends and Enemies: Impact of Neutrals

Kazunori Ota1, Nathanaël Barrot1, Anisse Ismaili1, Yuko Sakurai2 and Makoto Yokoo1 1Kyushu University, Fukuoka 819-0395, Japan 2National Institute of Advanced Industrial Science and Technology [email protected], [email protected], [email protected], [email protected], [email protected]

Abstract Enemies aversion Friends appreciation May be empty (Th. 1)∗ Non-empty (Th. 9)∗ We investigate hedonic games under enemies aver- Core VERIF is coNP-c (Th. 5) CONSTRUCTION takes ∗ ∗ sion and friends appreciation, where every agent EXIST is NPNP-c (Th. 6) polynomial time (Th. 9) considers other agents as either a friend or an en- Strict May be empty (Ex. 1) May be empty (Th. 2)∗ ∗ ∗ emy. We extend these simple preferences by al- core VERIF is coNP-c (Th. 3) VERIF is coNP-c (Th. 7) ∗ ∗ lowing each agent to also consider other agents to EXIST is NPNP-c (Th. 4) EXIST is NPNP-c (Th. 8) be neutral. Neutrals have no impact on her pref- Table 1: Summary of results: new ones marked with ∗. erence, as in a graphical hedonic game. Surpris- ingly, we discover that neutral agents do not sim- of other agents; the rest are neutral. One might think that plify matters, but cause complexity. We prove that adding such a graphical assumption would simplify the com- the core can be empty under enemies aversion and putational problems or that since neutral agents do not im- [ ] the strict core can be empty under friends apprecia- pact preferences, the previous results in Dimitrov et al. 2006 tion. Furthermore, we show that under both pref- would still hold in this extended model. erences, deciding whether the strict core is non- It turns out that under enemies aversion, a core stable coali- empty, is NPNP-complete. This complexity extends tion structure might not exist, and it might not exist under to the core under enemies aversion. We also show friends appreciation for the strict core either. Then, we inves- that under friends appreciation, we can always find tigate the complexity of (VERIF) to verify whether a given a core stable coalition structure in polynomial time. coalition structure is (strict) core stable and (EXIST) whether the (strict) core is non-empty. Our findings are in Table 1. Related work Lang et al. [2015] proposed friends- 1 Introduction neutrals-enemies hedonic games using the generalized Coalitions are a central part of economics, political, and so- Bossong-Schweigert extension principle. Peters [2016] con- cial life. A natural question is whether a coalition structure sidered graphical hedonic games. If the agent graph has (which is a partition of the set of agents) exists that is stable. a bounded treewidth and a bounded degree, deciding the In coalition formation games with hedonic preferences, each core’s existence is polynomial-time tractable. Aziz and agent only cares about her own coalition. Since the number Brandl [2012] clarified the inclusions between stability con- of coalitions that she can join is exponential, various compact cepts such as core and Nash stability, and provided some ex- classes of hedonic games have been proposed. istence results. Aziz et al. [2014] proposed conditions that In particular, Dimitrov et al. [2006] developed games in guarantee a core stable outcome in fractional hedonic games. which each agent divides the other agents into friends or ene- Aziz et al. [2016] proposed Boolean hedonic games where mies. They propose two alternative preferences. One is ene- an agent partitions a set of other agents into satisfactory and mies aversion, where each agent prefers coalitions with fewer unsatisfactory groups and showed core non-emptiness. Sung enemies and in case of a tie, with more friends. The other is and Dimitrov [2007] showed that the verification problem friends appreciation, where each agent prefers coalitions with for the core is coNP-complete in additive hedonic games. more friends and in case of a tie, with fewer enemies. Under Woeginger [2013] showed that the existence problem for the enemies aversion, there always exists a core stable coalition core is NPNP-complete in additively separable hedonic games structure that is NP-hard to find; and under friends apprecia- (ASHG). Peters [2015] proved that the existence problem for tion, there always exists a strict core stable coalition structure, the strict core is NPNP-complete in ASHGs. Rey et al. [2016] which can be found in polynomial-time. show that under enemy-aversion, deciding the existence of In this paper, we examine a slight extension of this model the strict-core is DP-hard. Peters and Elkind [2015] devel- where each agent divides the others into three groups, friends, oped a framework to prove the NP-hardness of existence enemies, and neutrals, who do not impact her preference, in problems, which widely applies to various hedonic games the fashion of graphical hedonic games [Peters, 2016]. In- such as individually rational coalition lists [Ballester, 2004] deed, in practice, agents commonly only care about a subset and hedonic coalition nets [Elkind and Wooldridge, 2009].

359 Proceedings of the Twenty-Sixth International Joint Conference on Artiﬁcial Intelligence (IJCAI-17)

2 Model Depending on the context, we sometimes omit one of the Let N = {1, . . . , n} denote the set of agents.A coalition three sets AE, AF or A⊥, since it can be deduced from the C ⊆ N is a subset of agents. A coalition structure π is a par- two others. For instance, omitting set A⊥, graph GEF = tition of set N. Let π(i) denote the coalition to which agent (N,AE ∪ AF ) shows a graphical hedonic game [Peters, i belongs in π. Let CN denote the set of coalition structures. 2016]. The following remark is also useful: For every agent i, her preference %i is based on the coali- Remark 1. Under enemies aversion, consider an HG/E and tions to which she belongs; let i (respectively ∼i) denote graph GF ⊥ = (N,AF ∪A⊥), where only friendly and neutral the preference’s asymmetric (respectively symmetric) part. A arcs are represented, and enemy arcs omitted. Given coalition hedonic game (N,P ) is defined by set of agents N and pref- C, if arc (i, j) is missing in subgraph GF ⊥[C], then agent j erence profile P = (%i)i∈N . is an enemy of i, and C is unacceptable to her. Thus, (i) if Coalition structure π ∈ CN admits blocking coalition π is a member of the (strict) core, for each C ∈ π, subgraph 4 ∅ 6= X ⊆ N iff for every i ∈ X, X i π(i) holds. The GF ⊥[C] is necessarily a clique , and (ii) given π where every core is the set of coalition structures that do not admit any coalition C ∈ π induces clique GF ⊥[C], a (weakly) blocking N blocking coalition. Similarly, coalition structure π ∈ C ad- coalition X ⊆ N necessarily induces clique GF ⊥[X]. mits weakly blocking coalition X ⊆ N iff for every i ∈ X, % X i π(i), and there exists j ∈ X such that X j π(j). The 3 Stable Coalitions May Not Exist. strict core is the set of coalition structures that do not admit any weakly blocking coalition.1 Coalition C is acceptable for In this section, we discuss the existence of a (strict) core sta- 2 ble coalition structure. Under enemies aversion, consider- agent i if and only if C %i {i} holds. In this paper, we consider the following simple (and com- ing only friends and enemies (no neutrals), the possible non- pact) preferences. For each agent i, set N is partitioned into existence of a strict core stable coalition structure was shown [ ] {F ,E , ⊥ }. Agents F are her friends, E are her enemies, in Example 1 Dimitrov et al., 2006 and implies that the strict i i i i i core can be empty, when neutral agents are also allowed. and ⊥i are neutral agents (i ∈⊥i). Under enemies aversion, she first compares the number of This example does not extend to the (non strict) core in enemies. Hence, without loss of generality, any coalition that the friends-and-enemies model and indeed a core stable coali- [ contains an enemy is unacceptable. Within acceptable coalition structure always exists under enemies aversion Dimitrov ] tions, she prefers coalitions with more friends, i.e., for two et al., 2006 . However, when we add neutral agents to the friends-and-enemies model, the following theorem holds. acceptable coalitions, C i D holds iff |C ∩ Fi| > |D ∩ Fi|. Under friends appreciation, she prefers coalitions with Theorem 1. In an HG/E, the core can be empty. more friends, and in case of a tie, she prefers the one with To prove this theorem, we utilize the following example. fewer enemies: C D holds iff (a) |C ∩ F | > |D ∩ F |, or i i i Example 2. Assume 15 agents who are divided into a cycle of (b) |C ∩ F | = |D ∩ F | and |C ∩ E | < |D ∩ E |. i i i i five groups, C ,...,C , each of which contains three agents. In both preferences, C ∼ D holds iff |C ∩ E | = |D ∩ E | 0 4 i i i C (resp. C ) denotes C (resp. C ). and |C ∩ F | = |D ∩ F |. The set of preference profiles i+k i−k i+k mod 5 i−k mod 5 i i The sets of friends, neutrals and enemies are as follows: under enemies aversion (respectively friends appreciation) is • Agents in the same group are friends. denoted by PE (respectively by PF ). In PE there is additive • Agents in C consider agents in C enemies. separability on weights {1, 0, −n}, and in PF on {n, 0, −1}. j j2 • Agents in C consider agents in C friends. Definition 1 (HG/E and HG/F). An HG/E is a hedonic game j j+1 • Agents in C consider one agent in C a neutral and % % E HG/F j j−1 (N, ( i)i∈N ) such that each i is in P . Similarly, an the other two agents to be friends. (N, (% ) ) % PF is a hedonic game i i∈N such that each i is in . We illustrate these preferences with a partial representation Such hedonic games can be represented by a labeled di- of graph GF ⊥ by focusing on group C0: graph GEF ⊥ = (N,AE ∪ AF ∪ A⊥) where each vertex rep- resents an agent, and arc (i, j) in set AE labeled by E (re- C C C spectively by F, ⊥) indicates that for agent i, agent j is an 4 0 1 3 enemy (respectively friend, neutral). 3 Friends 3 Friends 1 Example 1. [Dimitrov et al., 2006] Under enemies aversion, F F in this three-agent HG/E, the following preferences hold: F F 2 3 {1, 2} 1 {1} F 1 2 3 {1, 2, 3} 2 {1, 2} ∼2 {2, 3} 2 {2} 2 Friends, 1 Neutral 2 Friends, 1 Neutral E {2, 3} 3 {3}. The coalition structures that do not admit a blocking coalition, are set { {{1, 2}, {3}} , {{1}, {2, 3}} }. However, Proof of Theorem 1. Assume π is core stable in Example 2. there are weak deviations from these coalition structures, re- From Remark 1, only the agents in adjoining groups can form spectively {2, 3} and {1, 2}, and thus, the strict core is empty. a coalition. Also, note that if coalition Ci ∪ Ci+1 is formed, each member in Ci is with 5 friends and each member in Ci+1 1 The strict core is more demanding, thus contained in the core. is with 4 friends. First, we show that the agents in a same 2If π(i) is unacceptable, π cannot be a member of the core. 3The size of this representation is Θ(n2). 4A digraph G = (W, B) is a clique iff ∀(i, j) ∈ W 2, (i, j) ∈ B.

360 Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI-17) group cannot be separated. For the sake of contradiction, as- HG/E: (i) HG/E/SC/EXIST, (ii) HG/E/C/EXIST, and (iii) sume agent i ∈ C0 is separated from the other members in HG/F/SC/EXIST. To examine whether they belong to class C0. There are three cases to study: NP, we also study the problem of verifying whether a given coalition structure π is in the core or the strict core for a given • When π(i) ∩ C1 6= ∅: i is with at most 3 friends, and game: i.e., (a) HG/E/SC/VERIF, (b) HG/E/C/VERIF, and j ∈ C0 (j 6= i) is with at most 3 friends. Each member (c) HG/F/SC/VERIF. We address the complexity of these in C4 is also with at most 4 friends (since they are not with i). Then C ∪ C is a deviation. problems in the next sections. 4 0 Class NP corresponds to the set of decision problems • When π(i) ∩ C4 6= ∅: each member of C4 is with at where ‘yes’-instances allow a polynomially-sized certificate most 3 friends. Also, each member of C3 is with at most verifiable in polynomial-time. For instance, in a further proof, 4 friends. Then, C ∪ C is a deviation. 3 4 we utilize problem MAXCLIQUE, which is among the com- • When π(i) = {i}: the only case for which C0 ∪ C1 (or putationally most intractable problems in class NP. Indeed, it C0 ∪C4) is not a deviation is when (C0 \{i})∪C1 forms is NP-complete: it is in NP and NP-hard.5 a coalition (such that at least one member in C1 does not Definition 2 (Problem MAXCLIQUE). Given graph G = strictly prefer C ∪ C ). Here, each member in C is 0 1 1 (V, A) and lower threshold k ∈ N, does a subset of k ver- with at most 4 friends. Then, consider the members of tices W ⊆ V exist such that subgraph G[W] is a clique? C2. The only way that a member of C2 is with 4 or more than 4 friends is to form a coalition with 2 or 3 Complementation consists in transposing the ‘yes’ and ‘no’ members of C3. However, in such a case, each member answers. Consequently, class coNP is symmetric to class of C3 is with at most 4 friends and each member in C4 is NP and corresponds to the set of decision problems where with at most 3 friends, and therefore, C3 ∪ C4 becomes ‘no’-instances allow a polynomially-sized certificate verifi- a deviation. Thus, each member in C2 is with at most 3 able in polynomial-time. For instance, the problem of veri- friends, but then C1 ∪ C2 is a deviation. fying that a given coalition structure is core stable belongs to class coNP, since we can certify ‘no’-instances with a block- Using a similar argument, we can prove that all the members ing coalition. One can show that a problem is coNP-complete in the same group must be in the same coalition in π. Also, by proving that it is in coNP and that it is the complement of it is not possible that two adjoining groups, Cj and Cj+1, are an NP-hard problem, by symmetry of NP and coNP. isolated, i.e., Cj ∈ π and Cj+1 ∈ π, since Cj ∪Cj+1 becomes A decision problem may also neither allow a yes or a no a deviation. Therefore, consider coalition structure π where verification in polynomial-time, falling outside of NP and no two consecutive groups are alone, which implies that ex- coNP. Class NPNP corresponds6 to the decision problems for actly one Cj is alone. Without loss of generality, we assume which ‘yes’-instances allow a polynomially-sized certificate that just C0 is alone. Then the only coalition structure that is that is verifiable in polynomial-time using a constant-time not discarded by Remark 1 is π = {C0,C1 ∪ C2,C3 ∪ C4}. NP-oracle.7 Class coNPNP is its complement.8 Typically, Then each member in C0 is with 2 friends and each mem- a problem in NPNP is extremely hard, since it consists in a ber in C is with 4 friends, and thus, C ∪ C is a deviation. 4 4 0 coNP problem nested into an NP problem. Problem MIN- Therefore, no core stable coalition structure exists. MAXCLIQUE is typical for this second level of the polyno- Under friends appreciation and considering only friends mial hierarchy. and enemies, both the core and the strict core always exist Definition 3 (Problem MINMAXCLIQUE). Given graph G = [Dimitrov et al., 2006]. However, if we add neutral agents, (V, A), two sets I,J that partition set V into {Vi,j | i ∈ I, j ∈ then we have the following: J}, and lower threshold k ∈ N, does, for every function t : Theorem 2. In an HG/F, the strict core can be empty. I → J, subgraph G[∪i∈I Vi,t(i)], contain a k-sized clique? To prove this theorem, we utilize the following example. Intuitively, V is partitioned into |I|·|J| subsets Vi,j . Ac- cording to function t, |I| subsets are chosen (for each i ∈ I, Example 3. Consider N = {1, 2, 3} under friends apprecia- set Vi,t(i)). Then for the union of these subsets, we consider tion based on partitions F1 = {1, 2}, E1 = {3}, F2 = {2}, MAXCLIQUE. There are |J||I| variations of function t. Prob- ⊥2= {1, 3}, F3 = {2, 3} and E3 = {1}. lem MINMAXCLIQUE lies in class coNPNP, since by infer- Proof of Theorem 2. The core contains all of the coalition ring the correct function t : I → J (a ‘no’ certificate), the NP- structures except {{2}, {1, 3}}. However, there is a weak de- oracle can solve the MAXCLIQUE problem on G[∪i∈I Vi,t(i)] viation from all coalition structures in the core, either {1, 2} and verify the ‘no’ answer. Completeness is defined in a stan- or {2, 3}, and thus, the strict core is empty. dard manner with polynomial-time reductions, and we know: Lemma 1. [Ko and Lin, 1995] Problem MINMAXCLIQUE is 4 Problems and Complexity coNPNP-complete. (Their proof even holds when J = {0, 1} The previous section showed that in an HG/E, both the and |Vi,0| = |Vi,1| for every i ∈ I.) strict core and the core may be empty, and in an HG/F, 5Any problem from class NP can be reduced to MAXCLIQUE in the strict core may be empty. These observations bring polynomial-time; so that solving it efficiently would solve P vs NP. 6 P to surface the following decision problems to decide the Class Σ2 in the second level of the Polynomial Hierarchy. non-emptiness of the strict core and the core for an HG/E 7A blackbox that solves any (co)NP problem in constant-time. 8 P as well as the non-emptiness of the strict core for an Class Π2 in the second level of the Polynomial Hierarchy.

361 Proceedings of the Twenty-Sixth International Joint Conference on Artiﬁcial Intelligence (IJCAI-17)

5 Enemies Aversion and the Strict Core threshold k ∈ N define a restricted instance of MINMAX- In this section, under enemies aversion, we first address the CLIQUE where ∀i ∈ I, |Vi,0| = |Vi,1| holds. We construct complexity of verifying that a given coalition structure is in an instance of coHG/E/SC/EXIST addressing the MINMAX- the strict core. Second, we address the complexity of deciding CLIQUE instance, as follows. Graph GF ⊥ is partially shown whether the strict core is non-empty. We show the following: in Fig. 2, by keeping Remark 1 on necessary cliques in mind. (1) For every set of vertices Vi,j , we introduce set of ERIF Theorem 3. Problem HG/E/SC/V is coNP-complete. vertex-agents Vi,j ≡ Vi,j . For every edge {x, y} in graph Hence the existence problem does not allow a classical ver- G = (V, A), we introduce neutral arcs (x, y) and (y, x) be- S ification procedure in polynomial-time. Indeed, it is not in NP tween the agents of V = i∈I (Vi,0 ∪Vi,1). (There is no edge or coNP, since we show that: between Vi,0 and Vi,1.) (2) We introduce two friend-cliques K0 and K1, each of which contains k −2 mutual friends. Be- Theorem 4. Problem HG/E/SC/EXIST is NPNP-complete. tween them, agent y is a mutual friend with everyone in K0 Therefore, since this problem is at least as intractable as all and K1. (3) Fulcrum-agent ϕ is a mutual friend with agent the problems that nest a coNP problem into an NP problem, y, with everyone in clique K0 (but no one in clique K1) and it is extremely intractable, despite the utter simplicity of ene- with everyone in set V . (4) Between every pair of sets Vi,0 mies aversion as a preference. and Vi,1, we introduce |Vi,0| = |Vi,1| inhibitors (specified below) by pairing the agents of Vi,0 and Vi,1. Each inhibitor, 5.1 Proving Complexity of Verification which is a game connected to one vertex in Vi,0 and one in Proof of Theorem 3. Problem HG/E/SC/VERIF is in coNP, Vi,1, makes the former vertex xor the latter non-available. (5) since ‘no’-instances can be certified with a weakly block- To avoid inhibitors going to two different sides for one Vi,j , ing coalition. For coNP-hardness, we reduce problem MAX- we connect to every set Vi,j a circuit game Li,j (specified be- CLIQUE to the complement of problem HG/E/SC/VERIF. low) in which the strict core is non-empty if and only if the Let graph G = (V, A) and threshold k ∈ N define “every agent in Vi,j is inhibited, or none is inhibited” condi- an instance of MAXCLIQUE. From it, we construct an tion is satisfied. (6) Other arcs are for enemies. HG/E/SC/VERIF instance with vertex-agents V ≡ V, k − 1 Recall that we want to show that our reduction addresses weight-agents in set K, and one fulcrum-agent ϕ; therefore problem MINMAXCLIQUE. Therefore, we want to show: with set of agents N = V ∪ {ϕ} ∪ K. Bearing Remark 1 in ∀t : I → {0, 1}, ∃k-sized clique ∈ G[∪i∈I Vi,t(i)] mind, we depict graph GF ⊥ in Fig. 1, and all other arcs are enemies. Concerning set V , for every edge {i, j} in graph ⇔ ∀π ∈ CN , ∃ weakly blocking coalition X ⊆ N G = (V, A) we construct neutral arcs (i, j) and (j, i). In set K, all k − 1 agents are mutual friends. Between sets V But first, observe that in our construct, assuming strict core and K, fulcrum-agent ϕ shares a mutual friendship with ev- stability, fulcrum-agent ϕ is either grouped with y and K0 eryone. Finally, in given coalition structure π, every vertex- (k − 1 friends) or with a clique in V of at least k friends. agent i ∈ V is in singleton {i}, and the fulcrum-agent forms If agent ϕ goes to a clique in V , then the game on agents a coalition with the k − 1 weight agents. K1 ∪ {y} ∪ K0 is isolated with an empty strict core. (If y K {y} ∪ K In coalition structure π, there is a (weakly) blocking coali- groups with 0, then coalition 1 weakly blocks; y K {y} ∪ K tion if and only if the fulcrum-agent can improve (from k − 1 and if groups with 1, then coalition 0 weakly ϕ to at least k friends) with a k-sized clique in set V , and hence blocks.) If agent goes to the right, then the strict core of {ϕ} ∪ K ∪ {y} ∪ K ϕ, K , y if and only if a k-sized clique exists in graph G. game 1 0 is non-empty: agents 0 group and agents K1 group. We say that an agent is available 5.2 Proving Complexity of Existence for forming a coalition C if she does not worse off. (‘no’⇒‘no’): Let function t∗ : I → {0, 1} be such Proof of Theorem 4. Problem HG/E/SC/EXIST is in NPNP. that subgraph G[∪i∈I Vi,t∗(i)] contains no clique of size k, For ‘yes’-instances, a coalition structure π in the strict core is and construct a coalition structure with no weakly blocking a certificate that can be verified using an NP-oracle. (Recall coalition. For every i ∈ I, we put all the inhibitors be- that the verification problem is coNP-complete.) NP tween Vi,0 and Vi,1 on Vi,1−t∗(i) (so that no circuit game L We prove that problem HG/E/SC/EXIST is NP -hard generates a weakly blocking coalition). Hence, only agents by showing that the coNPNP-complete problem MINMAX- CLIQUE (Lemma 1) can be reduced to the complement of L1,0 L2,0 L3,0 problem HG/E/SC/EXIST. Let graph G = (V, A), set I, K1 which partitions set V into {Vi,0, Vi,1 | i ∈ I} and lower V V V 1,0 2,0 3,0 F F x x x F π: |V | Singletons One coalition ϕ y V x x x x x x F F K F V V V Model of F ϕ F Clique of 1,1 2,1 3,1 G = (V, A) k−1 friends K0 L1,1 L2,1 L3,1 Figure 1: Reducing MAXCLIQUE to co-HG/E/SC/VERIF: The edges of graph G become neutral arcs for agents V . Figure 2: From MINMAXCLIQUE to coHG/E/SC/EXIST: The corresponding graph GF ⊥ specified above.

362 Proceedings of the Twenty-Sixth International Joint Conference on Artiﬁcial Intelligence (IJCAI-17)

∪i∈I Vi,t∗(i) are available for grouping with fulcrum-agent ϕ, x1 x2 x x ⊥ ⊥ but never in a clique larger that k − 1 (based on the premise). F F F F Therefore, by grouping agents ϕ, y, and K0 into one coali- ⊥ ⊥ F ⊥ α β1 β2 tion, fulcrum-agent ϕ would worse off by deviating within ⊥ ∪ V ∗ . To conclude, by forming singletons with each F ⊥ i∈I i,t (i) y F ⊥ F ⊥ agent in ∪i∈I Vi,t∗(i) and a coalition with agents K1, this y coalition structure admits no weakly blocking coalition. y1 y2 (‘yes’⇒‘yes’): Assume that for every function t : I → Figure 3: Gates NOT, OR and DUPLIC (inputs x, outputs y) {0, 1}, subgraph G[∪i∈I Vi,t(i)] contains a clique of size k, and for the sake of contradiction let coalition structure π ad- ⊥ u (π) mit no weakly blocking coalition. Then there exists t such Li,j : combination of gates for ⊥ F V s x y z i,j V x W V ¬x that for every i ∈ I, all the inhibitors between Vi,0 and Vi,1 x∈Vi,j x∈Vi,j F ⊥ (π) F v go to side 1−t (i), or otherwise circuit games Li,0 and Li,1 Figure 4: Connections of circuit game L contain a weakly blocking coalition. Consequently, fulcrum- i,j agent ϕ is not grouped with agents y and K0, but at least with a k-sized clique in GF ⊥[∪i∈I Vi,t(π)(i)] that exists based on agent; when x is available, both x and s can join the fulcrum. the premise. Then the game on agents K1 ∪ {y} ∪ K0 is ⊥ ⊥ isolated. However, its strict core is empty, a contradiction. 0 Vi,j x s x Li,j F F Inhibitors pair every vertex x in set Vi,0 with a vertex y in Vi,1. Their construction is depicted in graph GF ⊥ below. Crucially, each inhibitor is an en- Finally, the validity of the formula and the availability of agent y make z non-available for u and v who remain sta- Vi,0 emy of the other agents, preventing x x xor y to participate in the game. ble singletons. Otherwise, the game on agents {u, z, v} is We show that inhibitor-clique isolated and has an empty strict core. To summarize, the circuit game has a non-empty strict core if and only if formula F ⊥ {a, c, b} is either grouped with agent F (∧ x) ∨ (∧ ¬x) holds. ⊥ x xor agent y, and if it is grouped x∈Vi,j x∈Vi,j F with agent x (resp. y), then agent x (resp. y) prefers to stay with the 6 Extension to Enemies Aversion and the Core a c b F F inhibitor upon every other coalition, In this section, under enemies aversion, we extend the results since she has two friends in it. on complexity of existence obtained for the strict core to the F ⊥ If agent c joins x, then a and b fol- core. First, the complexity of verification was addressed in ⊥ F low. Similarly, if c joins y, then b and the proof of (Th.1) [Sung and Dimitrov, 2007]: a follow. Finally, if c is not grouped [ ] y with x or y, then a and b join x and Theorem 5 ( Sung and Dimitrov, 2007 , Proof of Th.1). Problem HG/E/C/VERIF is coNP-complete. Vi,1 y, and c is interested in joining x or y, and a and b follow c. There might be no polynomial-time verification procedure for the existence problem. Indeed, we show the following: Each circuit game Li,j is connected to the agents of set Vi,j and constructed so that its strict core is non-empty if and only Theorem 6. Problem HG/E/C/EXIST is NPNP-complete. if “every agent in Vi,j is inhibited, or none is inhibited.” This NP construction relies on a combination of smaller gadget games Proof (sketch). First, problem HG/E/C/EXIST is in NP , that model any logical gate under the interpretation where an since, for ‘yes’-instances, a coalition structure π in the core is a certificate that can be verified easily using an NP-oracle. agent available amounts to Boolean true. Gates NOT, OR, NP and DUPLIC with inputs x and outputs y (Fig 3) are sufficient As previously, we prove that HG/E/C/EXIST is NP - to obtain a Boolean algebra. In gate NOT, the availability of hard by showing that MINMAXCLIQUE can be reduced to coHG/E/C/EXIST. The proof follows the same ideas de- x makes y non-available. In gate OR, the availability of x1 or veloped for the strict core, relying on inhibitors and circuit x2 makes α non-available and y available. In gate DUPLIC, games. However, due to the structure of circuit games here, the availability of x is duplicated into y1 and y2. Note that we need to introduce vertex-cliques (resp. a fulcrum-clique) gate AND(x1, x2) equals NOT(OR(NOT(x1), NOT(x2))), and gates OR and AND generalize from binary operators to instead of vertex-agents (resp. a fulcrum-agent), as follows: multinary ones. (1) First, for each vertex x in V we introduce a vertex- clique, K , that contains k0 mutual friends (k0 is specified By combining these gadget games, circuit game Li,j is x below). For all edges {x, y} in graph G = (V, A), we intro- constructed to obtain formula V x W V ¬x 0 0 x∈Vi,j x∈Vi,j duce mutual neutral arcs between each x ∈ Kx and y ∈ Ky. as the following availability of output agent y: “everyone or (2) Second, we introduce a generalization of Example 2 with 00 no-one” (Fig. 4). To ensure that the entire game (of the re- five cliques {C0,...,C4}, each of which contains (k − 1) duction) is not altered by the agents of the logic game, every mutual friends (k00 is specified below). (3) Third, we intro- 00 agent x in set Vi,j is separated from Li,j by a (double nega- duce a fulcrum-clique of k mutual friends, Kϕ. Each agent tion) gate where agent s is mutually neutral with every other in Kϕ is a mutual friend with each agent in C0 and with

363 Proceedings of the Twenty-Sixth International Joint Conference on Artiﬁcial Intelligence (IJCAI-17)

NP Kx1 Kx2 Kx Theorem 8. Problem HG/F/SC/EXIST is NP -complete. Kx F F F F Proof (sketch). The proof follows a similar sketch as for the Kα Kβ1 Kβ2 strict core and enemies aversion (i.e., a fulcrum-agent be- F F tween MINMAXCLIQUE, inhibitors, circuit games and Ex- F F F Ky ample 3). However, Remark 1 does not hold with friends ap- Ky Ky1 Ky2 preciation. However, we are still able to model gadget games and inhibitors. Here is the main idea for a construction: each

Figure 5: Gates NOT, OR and DUPLIC (in: Kx, out: Ky) agent can be forced to choose between two coalitions where her number of friends is the same but the number of enemies everyone in set V . (4) Finally, inhibitors and circuit games differ. Thus, she will prefer to be in the coalition with the (specified below) play the same role as for the strict core. lowest number of enemies if and only if it is available. The main argument for this reduction’s validity is the same Although these results show the extreme intractability of as for the strict core setting. Fulcrum-clique Kϕ is either 00 00 the strict core under friends appreciation, the complexity- grouped with C0 ((k − 1) friends) or with at least a k -sized clique in V , in a core stable coalition. Therefore, we only landscape changes radically to easiness for the core: present the inhibitors and the circuit games. Theorem 9. Given an HG/F,(1) the existence of a core-stable An inhibitor is a clique that contains (k00 + 1) mutual coalition structure is guaranteed, and (2) it can be computed friends and pairs a vertex-clique Kx in Vi,0 with a vertex- in polynomial-time as the strongly connected components of clique Ky in Vi,1. Each agent of the inhibitor-clique is a graph GF = (N,AF ). mutual friend with every agent in K and K . Thus, the x y Proof. The idea of using strongly connected components is inhibitor-clique either groups with Kx or Ky in a core stable coalition structure. If the inhibitor-clique is grouped with similar to a previous result with only friends and enemies [ ] 2 K (resp. K ), then K (resp. K ) prefers to stay with the Dimitrov et al., 2006 , which can be computed in time O(n ) x y x y [ ] inhibitor upon every other coalition. Tarjan, 1972 . We extend it to neutrals, with a much shorter proof. Let π = {C ,...,C } be the coalition structure Each circuit game L follows the same principle as in 1 m i,j corresponding to a decomposition of graph G = (N,A ) Theorem 4, relying on logic gates NOT, OR, and DUPLIC F F into strongly connected components, and let us show that it (Fig. 5) to obtain a Boolean algebra. However, in gate OR, is strict core stable. Since the condensation graph, where we have to assume that K or K are two friends-cliques x1 x2 each C is contracted into one vertex, is a directed acyclic with identical size t, and we set the size of K and K to j α y graph, it has at least one sink, which we call C . No subset (t − 1). Then the availability of K or K makes K non- s x1 x2 α Y ⊆ C can be part of any blocking coalition X ⊃ Y , since available and K available. In gate DUPLIC, assuming the s y (if Y ⊂ C ) at least one would loose friends or (if Y = C ) size of K is t ≥ 3, we set the sizes of K and K to s s x β1 β2 one may gain enemies. By repeating this argument on the (t − 2), and the size of K and K to (t − 1). Then the y1 y2 condensation graph from which C was removed, we obtain availability of K is duplicated into K and K . s x y1 y2 that no agent can be in a blocking coalition. As in Theorem 4, using these gates, circuit game Li,j can be constructed, and we connect its output to a specific instance of Example 2 with five cliques of size 3. Also, between 8 Conclusion each vertex-clique Kx and Li,j , we introduce a double nega- We studied the computational complexity of coalitional sta- tion, composed of a separating-clique of size k0 and a second bility in hedonic games under enemies aversion and friends clique of size (k0 − 1). To conclude, since the circuit game appreciation, by introducing neutral agents. It was known leads us to set k0 = 9, we set k00 = 18k because the agents that without neutral agents, coalitional stability is just NP- of the separating-clique have to be mutual friends with each hard, while in the very general ASHGs it is NPNP-complete, agent of the fulcrum-clique to guarantee core stability. hence a computationally extremely hard requirement. Be- tween these two models, to the best of our knowledge, our 7 On Friends Appreciation models are among the simplest cases of extreme intractability for coalitional stability in hedonic games. An interest- In this section, we consider hedonic games under friends ap- ing prospect is to explore assumptions that make verification preciation. Even though with only friends and enemies the tractable, in order to bring the existence problem to class NP. existence of a strict core stable coalition structure is guaran- Also, we did not extend our results to if friend/enemy rela- teed, Theorem 2 shows that the existence is not guaranteed tions are symmetric. Furthermore, since very few algorithms with neutral agents. Furthermore, we show the following: and methods have been proposed to achieve coalitional stability, numerical experiments could be a good challenge. Theorem 7. Problem HG/F/SC/VERIF is coNP-complete.

Proof (sketch). Surprisingly, the same reduction as for The- Acknowledgements orem 3 works with a slight modiﬁcation: vertex-agents are This work was partially supported by JSPS KAKENHI Grant neutral toward the fulcrum. Then the preferences of vertex- Number 17H00761 and 15H02751, and JST Strategic Inter- agents and the fulcrum lie in PE ∩ PF . Moreover, the clique national Collaborative Research Program, SICORP. We are of friends cannot be grouped with vertex-agents. grateful to Takamasa Suzuki for his presence.

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