The Thirty-Fourth AAAI Conference on Artificial Intelligence (AAAI-20)
The Impact of Selfishness in Hypergraph Hedonic Games
Alessandro Aloisio,1,2 Michele Flammini,2 Cosimo Vinci2 1University of L’Aquila, L’Aquila, Italy 2Gran Sasso Science Institute, L’Aquila, Italy [email protected], michele.fl[email protected], [email protected]
Abstract which each couple of agents is associated to a weight de- termining their level of appreciation for being together, and We consider a class of coalition formation games that can be the overall utility of each agent is simply the sum of the succinctly represented by means of hypergraphs and properly generalizes symmetric additively separable hedonic games. individual values she assigns to the members of her own More precisely, an instance of hypegraph hedonic game con- coalition. Such games can be suitably described by undi- sists of a weighted hypergraph, in which each agent is asso- rected weighted graphs, in which nodes represent agents, ciated to a distinct node and her utility for being in a given and weighted edges represent couples of agents with their coalition is equal to the sum of the weights of all the hyper- appreciation. Besides their nice property of being succinctly edges included in the coalition. We study the performance representable, SASHGs attracted research attention because of stable outcomes in such games, investigating the degrada- they are able to capture several realistic coalition formation tion of their social welfare under two different metrics, the scenarios. However, they cannot handle other real life situ- k-Nash price of anarchy and k-core price of anarchy, where k ations in which coalitions are worth if they reach some crit- is the maximum size of a deviating coalition. Such prices ical mass, as in the formation of political parties. Indeed, are defined as the worst-case ratio between the optimal so- cial welfare and the social welfare obtained when the agents several politicians may be mutually incompatible, but form- reach an outcome satisfying the respective stability criteria. ing coalitions with more than two people can increase the We provide asymptotically tight upper and lower bounds on success of each member; thus, we need subsets of more than the values of these metrics for several classes of hypergraph two people with positive weight together with edges having hedonic games, parametrized according to the integer k, the negative weights (which model the level of incompatibility hypergraph arity r and the number of agents n. Furthermore, of each couple of politicians). Another example comes from we show that the problem of computing the exact value of team formation, where one has to form groups of workers such prices for a given instance is computationally hard, even that have the set of abilities required to perform some tasks, in case of non-negative hyperedge weights. without which the group is unsuccessful. In this case we need to associate a positive weight to each group of workers Introduction able to perform a task when working together, and a weight In several real-life scenarios arising from economics, pol- to each edge modelling the level of compatibility of each itics, and sociology, we assist to phenomena of coalition couple of workers. formation, in which each person, denoted as agent, forms In light of the above examples (and many others), the coalitions with other agents to get some benefit, experienc- graph structure of SASHGs may be not sufficient to describe ing a utility that depends on the particular set of agents several real-life phenomena of coalition formation. To cope she joins to. A simple and efficient model that formally de- with this issue, we consider a superclass of SASHGs that, scribes such phenomena is that of hedonic games (Dreze` while maintaining a succinct representation, can capture the and Greenberg 1980; Banerjee, Konishi, and Sonmez¨ 2001; above more general scenarios. Namely, we focus on the class Bogomolnaia and Jackson 2002), which constitute an inter- of r-hypergraph hedonic games (r-HHGs or simply HHGs), esting class of general coalition formation games, in which in which each subset of at most r ≥ 1 agents is associated to agents mind only their coalitions, regardless of how the oth- a weight determining the level of liking for being a group in- ers aggregate. In hedonic games, we have a finite set of cluded in the same coalition, and the utility of each agent is agents, each in turn having a preference order on all the pos- the sum of the values she assigns to the groups in her coali- sible coalitions containing her. tion. Such games can be completely described by weighted A very interesting class of hedonic games is that of sym- hypergraphs, in which again nodes represent agents, while metric additively separable hedonic games (SASHGs), in weighted hyperedges represent groups of agents with their Copyright c 2020, Association for the Advancement of Artificial valuation. As already observed, r-HHGs are a generaliza- Intelligence (www.aaai.org). All rights reserved. tion of SASHGs, and a proper subclass of synchronization
1766 games on hypergraphs (Simon and Wojtczak 2017). • Some of the previous bounds hold even for the NPoAk Depending on the characteristics of the agents (e.g., their (Theorem 3 and Corollary 2). When dealing with non- willingness to cooperate with each other), the selfish be- negative weights, we get a slightly better upper bound, haviour can lead to stable coalitions, in which no agent can holding also for k = r − 1 (Theorem 6). improve her utility according to some deviation criteria. The • For most of the cases for which we have not provided an stability concepts we consider are the k-Nash stability and upper bound, we show that the NPoAk may be unbounded the k-core stability.Inak-Nash stable outcome, no subset of (Theorem 7 and 8). at most k agents can move into other existing or new coali- tions in such a way that each agent of the considered subset See Table 1 and 2 for further details. improves her utility. Similarly, in a k-core stable outcome, no subset of at most k agents can create a new coalition, Standard Non-Standard in such a way that each deviating agent improves. These k ≥ r Θ(ρr−1) ∞ stability criteria are strictly connected with the known con- k
1767 to their graph succinct representation, they are also con- w : E → R associating a real value w(e) with each hy- nected with the graph games introduced by (Deng and Pa- peredge e ∈ E. For simplicity, when referring to weighted padimitriou 1994). Properties guaranteeing the existence of hypergraphs, we omit the term weighted. core allocations (a core is a coalition structure in which no The arity of a hyperedge e is its size |e|.Anr-hypergraph group of agents has an incentive to form a different coali- is a hypergraph such that the arity of each hyperedge is at tion) are studied by (Banerjee, Konishi, and Sonmez¨ 2001), most r, where 2 ≤ r ≤ n.Acomplete r-hypergraph is a hy- while (Bogomolnaia and Jackson 2002), besides the core, pergraph (V,E,w) such that E := {U ⊆ V : |U|≤r}.A consider other forms of stability, such as Nash and indi- uniform r-hypergraph (resp. weakly uniform r-hypergraph) vidual stability. (Olsen 2009) shows that deciding whether is a hypergraph such that the arity of each hyperedge is r a Nash stable outcome of a ASHG exists is NP-complete, (resp. either r,or1). An undirected graph is a uniform 2- as well as deciding the existence of a non-trivial (different hypergraph. from the grand coalition) Nash stable outcome in SASHGs An r-hypergraph hedonic game G(H) (r-HHG, or HHG with non-negative weights. (Gairing and Savani 2010) prove when r is not specified), is a strategic game based on an r- that computing a Nash stable outcome in SASHGs is PLS- hypergraph H, defined as follows: complete, while (Sung and Dimitrov 2010) show that in ASHGs determining the existence of a core stable, strict Agents: The set of agents is V =[n], i.e., each node corre- core stable, Nash stable, or individually stable outcome is sponds to an agent. We reasonably assume that n ≥ 2. NP-hard. (Aziz, Brandt, and Seedig 2013) show that com- Strategy profiles: An outcome or coalition structure is a par- puting a core or a strict core stable outcome is NP-hard tition C = {C1,C2,...,Ch} of V . Each set Cl is a coali- even for symmetric and strict preferences. Furthermore, they tion of C, and h is the size of C. show that the problem of computing optimal outcomes is NP-hard. As SASHGs are a subclass of HHGs, the results Utility functions: Given a coalition structure C and an agent obtained in (Aziz, Brandt, and Seedig 2013) imply that the i ∈ V , let C(i) be the coalition of C such that i ∈ C(i). problem of computing a core stable outcome or an optimal Given U ⊆ V , let E(U):={e ∈ E : e ⊆ U}.Given outcome in HHGs is NP-hard, too. Furthermore, as strongly an agent i ∈ V , the utility ui(C) of i in C is the sum of Nash stable outcomes are a subset of core stable outcomes, the weights of all the hyperedges e containing i such that we also get that strongly Nash stable outcomes may not ex- e ⊆ C(i), i.e., ui(C):= e∈E(C(i)):i∈e w(e). ist in HHGs. The problem of computing an outcome ver- ifying particular optimality or stability criteria in ASHGs The grand coalition structure is the coalition structure de- C = {V } has been considered in (Aziz, Brandt, and Seedig 2011; fined as , that is obtained when all agents belong Peters and Elkind 2015). to the same coalition. The singleton coalition structure is C = {{1}, {2},...,{n}} Another relevant class of hedonic games is that of frac- the coalition structure , that is ob- tional hedonic games, studied in (Aziz et al. 2019; Pe- tained when each coalition contains exactly one agent. ters and Elkind 2015; Olsen 2012; Monaco, Moscardelli, A complete r-hypergraph (resp. uniform r-hypergraph , and Velaj 2018; Carosi, Monaco, and Moscardelli 2019; resp. weakly uniform r-hypergraph, resp. graph) hedonic Bilo` et al. 2018; Flammini et al. 2018; Monaco, Moscardelli, game is an r-HHG such that the underlying r-hypergraph and Velaj 2019). The inefficiency of these and other related is complete (resp. uniform, resp. weakly uniform, resp. a classes of hedonic games under different stability and opti- uniform 2-hypergraph). A standard r-hypergraph hedonic mality notions has been considered in (Elkind, Fanelli, and game is an r-HHG such that, for any U ⊆ V with |U|≥2, w(e) ≥ 0 Flammini 2016; Balliu et al. 2019; Balliu, Flammini, and we have that either (i) e∈E(U):|e|=s for any Olivetti 2017; Flammini, Monaco, and Zhang 2017). s ∈ [r] \{1}, or (ii) e∈E(U):|e|=s w(e) ≤ 0 for any Finally, we remark that other games related to HHGs s ∈ [r] \{1}. are the group activity selection games, studied in (Dar- Remark 1. The class of standard HHGs is well-motivated mann et al. 2018; Igarashi, Bredereck, and Elkind 2017; by real-life scenarios. Indeed, if we consider that a positive Igarashi, Peters, and Elkind 2017; Bilo` et al. 2019)), and (resp. negative) weight of a hyperedge e models the fact that max-cut games, studied in (Gourves` and Monnot 2009; all the agents belonging to e are happy (resp. unhappy) to Gourves` and Monnot 2010; Feldman and Friedler 2015; stay together in the same coalition, a HHG is standard if, Carosi et al. 2019). given an arbitrary subset U of agents, either they are happy on the average with respect to all the hyperedges of U with Preliminaries arity s>1 (independently from s), or they are unhappy on Given two integers r ≥ 1 and n ≥ 1, let [n]={1, 2,...,n} the average with respect to such hyperedges. and (n)r := n · (n − 1) · ...· (n − r +1)be the falling fac- Observe that weakly uniform r-HHGs and r-HHGs with torial. Moreover, given a set A, let χA denote the indicator non-negative weights are standard. In Figure 1, we ex- function, i.e., χA(x)=1if x ∈ A, χA(x)=0otherwise. hibit a non-uniform game with both positive and nega- tive weights that is standard. The social welfare SW(C) C r-hypergraph hedonic games. A weighted hypergraph is of a coalition structure is defined as the sum of all the C SW(C):= u (C)= a triple H =(V,E,w) consisting of a finite set V =[n] agents’ utilities in , i.e., i∈V i E ⊆ 2V h r t w(e). of nodes, a collection of hyperedges, and a weight l=1 t=1 e⊆Cl:|e|=t
1768 Given k ≤ n, a coalition structure C is k-core stable if and only if, for any Z ⊆ V with |Z|≤k, there exists at least one agent i ∈ Z such that ui(C) ≥ ui(C|Z). According to the definitions often adopted in several existing works on hedonic games, an n-core stable coalition structure is equiv- alently called core stable. ForagameG(H), let CSk(G(H)) be the set of k- core stable coalition structures of G(H). Observe that NSk(G(H)) ⊆ CSk(G(H)). Remark 2. If w(e) ≥ 0 for any e ∈ E such that |e| =1, Figure 1: A hypergraph H with V =[4]and then each agent gets a non-negative utility in any k-core sta- E = {{1, 2}, {2, 3}, {3, 4}, {4, 1}, {1, 2, 3}, {2, 3, 4}}, ble coalition structure (and thus, in any k-Nash stable coali- w({1, 2, 3})=w({2, 3, 4})=3, w({1, 2})= tion structure), otherwise she would deviate in a singleton w({1, 4})=−1, and w({2, 3})=w({3, 4})=2. coalition. Furthermore, the social optimum is guaranteed to G(H) is a standard 3-HHG. Indeed, if U ∈ be non-negative, as it achieves at least the social welfare of {{1, 2, 3, 4}, {1, 2, 3}, {2, 3, 4}, {3, 4, 1}, {2, 3}, {3, 4}} the singleton coalition structure. w(e) ≥ 0 s ∈ [r] \{1} we have that e∈E(U):|e|=s for any , In light of the previous remark and as commonly assumed and if U ∈{{1, 2, 4}, {1, 2}, {1, 4}} we have that in hedonic games, it is reasonable to suppose that for any e∈E(U):|e|=s w(e) ≤ 0 for any s ∈ [r] \{1}. agent is better staying alone than into a coalition in which she is unhappy; thus we do the following assumption: Assumption 1. w(e) ≥ 0 for any e ∈ E such that |e| =1. k-Nash stability. Given a coalition structure C = {C1,C2,...,Ch} and a subset of nodes Z ⊆ V , a function ϕ : Z → C ∪ [n] Z C Quality of stable coalitions. ForagameG(H), an opti- is called deviation function of from , ∗ and associates each agent of Z with a coalition of C, or with mal coalition structure C , also called social optimum,isan an integer of [n] that will be used to define a new coalition. outcome maximizing the social welfare. The k-Nash price of −1 G(H) NPoA (G(H)) Given l ∈ [h] and t ∈ [n], let Z := ϕ [Cl] (i.e., the set anarchy of (denoted k ) is the worst-case l C∗ of agents associated with coalition Cl by function ϕ), and ratio between the social welfare of a social optimum , and ˆ −1 the social welfare of a k-Nash stable coalition structure, that let Zt := ϕ [t] (i.e., the set of agents associated with some ∗ NPoA (G(H)) := max SW(C ) ≥ 1 new coalition by function ϕ). Given a deviation function ϕ is, k C∈NSk(G(H)) SW(C) , with of Z from C, let (C|ϕ):={{C1 \ Z}∪Z1, {C2 \ Z}∪ the convention that ∞ := c/0 for any c>0, 1:=0/0, and ˆ ˆ ˆ NPoA (G(H)) := 1 G(H) k Z2,...,{Ch \Z}∪Zh, Z1, Z2,...,Zn}\{∅}. Observe that k if does not admit -Nash stable (C|ϕ) defines a new coalition structure. A generic coalition coalition structures.1 Analogously, the k-core price of an- structure (C|ϕ) is called deviation coalition structure of Z archy of G(H) (denoted as CPoAk(G(H))) is the worst-case ∗ from C, and it is equivalently defined as a coalition struc- ratio between the social welfare of a social optimum C , and ture in which each agent i of Z moves from coalition C(i) the social welfare of a k-core stable coalition structure, that SW(C∗) to coalition ϕ(i) if ϕ(i) ∈ C, and moves into a new coali- is, CPoAk(G(H)) := maxC∈CS (G(H)) , and again ˆ ˆ k SW(C) tion Zϕ(i) if ϕ(i) ∈ [n], where Zϕ(i) is the coalition made we adopt the convention that CPoAk(G(H)) := 1 if G(H) of all the agents associated by ϕ to the same integer ϕ(i). does not admit k-core stable coalition structures. Given an integer k ≤ n, a coalition structure C is k-Nash If NPoAk(G(H)) = ∞ (resp. CPoAk(G(H)) = ∞), we stable if and only if, for any Z ⊆ V with |Z|≤k, and any say that the k-Nash (resp. k-core) price of anarchy of G(H) deviation function ϕ of Z from C, there exists at least one is unbounded. Since NSk(G(H)) ⊆ CSk(G(H)),wehave agent i ∈ Z such that ui(C) ≥ ui(C|ϕ). According to the that NPoAk(G(H)) ≤ CPoAk(G(H)). Furthermore, given definitions often adopted in several existing works on hedo- two integers h and k with 1 ≤ h ≤ k, since each k-Nash nic games, a 1-Nash stable coalition structure is equivalently stable (resp. k-core stable) coalition structure is also h-Nash called Nash stable, while an n-Nash stable coalition struc- stable (resp. h-core stable), we have that NPoAk(G(H)) ≤ ture is called strongly Nash stable. For a game G(H), let NPoAh(G(H)) (resp. CPoAk(G(H)) ≤ CPoAh(G(H))). NSk(G(H)) be the set of k-Nash stable coalition structures of G(H). The k-core price of anarchy In this section, we show upper and lower bounds on the k- k-core stability. A deviation coalition structure (C|ϕ) of core price of anarchy of any standard r-HHG with k ≥ r, Z from C is called core deviation coalition structure of Z which are asymptotically tight in the variable ρ := n/k. from C if ϕ(i)=∅ for any i ∈ Z, and we equivalently (C|Z) denote it as . Informally, a core deviation coalition 1We set the price of anarchy equal to 1 (that is the lowest possi- structure is obtained when all the agents of set Z move into ble value), in such a way that instances not admitting stable coali- the empty coalition, i.e., they form a new coalition made of tion structures are implicitly discarded when computing the price all the agents of Z, only. of anarchy of an arbitrary HHG.
1769 Furthermore, we show that, if either the considered r- counted as many times as the number of leaders contained in hypergraph is not standard, or k 1770 ∗ ∗ Given a coalition Cl and a hyperedge e ∈ E(Cl ),wehave Proof sketch. Consider an r-HHG G(H), where H is a com- that |{Z ⊆ V : |Z| = k, e ⊆ Z}| is equal to the number plete and uniform r-hypergraph with n nodes, and each hy- k −|e| V \ e C of ways to select elements within set , that is peredge has unitary weight. Let be the coalition structure n−|e| . Thus, by continuing from (7), we get in which one coalition C has k = n − k n/k agents, and k−|e| n/k k C∗ ∗ coalitions have each agents, and let be the grand h coalition structure. One can easily observe that C is k-core w(e) ·|{Z ⊆ V : |Z| = k, e ⊆ Z}| SW(C∗) stable. Finally, we show that CPoAk(G(H)) ≥ SW(C) = l=1 ∗ e∈E(Cl ) n(n−1)r−1 ∗ kn/k(k−1) +k(k−1) . h n −|e| r−1 r−1 = w(e) k −|e| The following theorem provides a better lower bound l=1 e∈E(C∗) ⎛ l ⎞ when k is close to n, holding even for the k-Nash price of k ≥ r − 1 ∗ anarchy and if . r ⎜ h ⎟ 1 n − s = ⎜s w(e)⎟ , Theorem 3. For any integers r ≥ 2, k ≥ r − 1, and n ≥ ⎝ ⎠ s k − s max{k, r} r G(H) s=1 ∗ , there exists an -HHG with non-negative l=1 e∈E(Cl ): |e|=s weights such that CPoAk(G(H)) ≥ NPoAk(G(H)) ≥ 1+ (n/2−1) and the claim follows. r−1 . 2(min{k,n/2−1})r−1 By (1), Lemma 1, and Lemma 2, we get Remark 3. If r is constant w.r.t. the input instance, the upper xs λ bound of Theorem 1 and the lower bounds of Theorem 2 ⎛ ⎞ ρ := n/k r h and 3 are asymptotically tight in variable , and n − 1 their value increase as Θ(ρr−1). ⎝s w(e)⎠ (8) k − 1 In the following theorems we show that, without the as- s=1 l=1 e∈E(C ):|e|=s l sumptions done in the Theorem 1, there always exists a k k k ∗ Z HHG with unbounded -core price of anarchy. = uiZ (C) ≥ uiZ (C|Zj ∩ C (ij )) j j Theorem 4. For any integers r ≥ 2, k 1771 If weights are non-negative, we have the following Because of Theorem 5 and the following results, we have slightly better upper bound, holding also for k = r − 1. that, without the assumptions done in Corollary 2 and The- Theorem 6. For any r-HHG G(H) with non-negative orem 6, in most of the cases there exists a HHG with n k n−1 ≥ NPoAk = ∞. 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