Optimal Partitions in Additively Separable Hedonic Games∗

Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence

Haris Aziz Felix Brandt Hans Georg Seedig Department of Informatics Technische Universitat¨ Munchen¨ 85748 Garching bei Munchen,¨ Germany {aziz,brandtf,seedigh}@in.tum.de

Abstract form just as much as the formation of coalitions depends on how the payoffs are distributed. We conduct a computational analysis of fair and Coalition formation games, as introduced by Dreze` and optimal partitions in additively separable hedonic Greenberg [1980], provide a simple but versatile formal games. We show that, for strict preferences, a model that allows one to focus on coalition formation as such. Pareto optimal partition can be found in polynomial In many situations it is natural to assume that a player’s appre- time while verifying whether a given partition is ciation of a coalition structure only depends on the coalition Pareto optimal is coNP-complete, even when pref- he is a member of and not on how the remaining players are erences are symmetric and strict. Moreover, com- grouped. Initiated by Banerjee et al. [2001] and Bogomolnaia puting a partition with maximum egalitarian or util- and Jackson [2002], much of the work on coalition formation itarian social welfare or one which is both Pareto now concentrates on these so-called hedonic games. optimal and individually rational is NP-hard. We The main focus in hedonic games has been on notions of also prove that checking whether there exists a par- stability for coalition structures such as Nash stability, indi- tition which is both Pareto optimal and envy-free vidual stability, contractual individual stability, or core stabil- is Σp-complete. Even though an envy-free partition 2 ity and characterizing conditions under which they are guar- and a Nash stable partition are both guaranteed to anteed to be non-empty [see, e.g., Bogomolnaia and Jackson, exist for symmetric preferences, checking whether 2002]. The most prominent examples of hedonic games are there exists a partition which is both envy-free and two-sided matching games in which only coalitions of size Nash stable is NP-complete. two are admissible [Roth and Sotomayor, 1990]. General coalition formation games have also received at- 1 Introduction tention from the artificial intelligence community, where the focus has generally been on computing partitions that give Ever since the publication of von Neumann and Morgen- rise to the greatest social welfare [see, e.g., Sandholm et stern’s Theory of Games and Economic Behavior in 1944, al., 1999]. The computational complexity of hedonic games coalitions have played a central role within game theory. The has been investigated with a focus on the complexity of crucial questions in coalitional game theory are which coali- computing stable partitions for different models of hedonic tions can be expected to form and how the members of coali- games [Ballester, 2004; Dimitrov et al., 2006; Cechlarov´ a,´ tions should divide the proceeds of their cooperation. Tra- 2008]. We refer to Hajdukova´ [2006] for a critical overview. ditionally the focus has been on the latter issue, which led to Among hedonic games, additively separable hedonic the formulation and analysis of concepts such as Gillie’s core, games (ASHGs) are a particularly natural and succinct rep- the Shapley value, or the bargaining set. Which coalitions resentation in which each player has a value for every other are likely to form is commonly assumed to be settled exoge- player and the value of a coalition to a particular player is nously, either by explicitly specifying the coalition structure, computed by simply adding his values of the players in his a partition of the players in disjoint coalitions, or, implicitly, coalition. by assuming that larger coalitions can invariably guarantee Additive separability satisfies a number of desirable ax- better outcomes to its members than smaller ones and that, as iomatic properties [Barbera` et al., 2004]. ASHGs are the a consequence, the grand coalition of all players will eventu- non-transferable utility generalization of graph games stud- ally form. ied by Deng and Papadimitriou [1994]. Sung and Dimitrov The two questions, however, are clearly interdependent: ff [2010] showed that for ASHGs, checking whether a core sta- the individual players’ payo s depend on the coalitions that ble, strict-core stable, Nash stable, or individually stable par- ∗This material is based on work supported by the Deutsche tition exists is NP-hard. Dimitrov et al. [2006] obtained posi- Forschungsgemeinschaft under grants BR-2312/6-1 (within the Eu- tive algorithmic results for subclasses of additively separable ropean Science Foundation’s EUROCORES program LogICCC) hedonic games in which each player divides other players into and BR 2312/7-1. friends and enemies. Branzei and Larson [2009] examined

43 ff ∈N the tradeo between stability and social welfare in ASHGs. coalition as i and if i is in coalition S i, then i gets utility , ∈N j∈S \{i} vi( j). For coalitions S T i, S i T if and only if ≥ j∈S \{i} vi( j) j∈T\{i} vi( j). Contribution In this paper, we analyze concepts from fair = division in the context of coalition formation games. We A preference profile is symmetric if vi( j) v j(i) for any two players i, j ∈ N and is strict if vi( j) 0 for all i, j ∈ N present the first systematic examination of the complexity of computing and verifying optimal partitions of hedonic games, such that i j. We consider ASHGs (additively separable specifically ASHGs. We examine various standard criteria hedonic games) in this paper. Unless mentioned otherwise, from the social sciences: Pareto optimality, utilitarian social all our results are for ASHGs. welfare, egalitarian social welfare, and envy-freeness [see, e.g., Moulin, 1988]. 2.2 Fair and optimal partitions In Section 3, we show that computing a partition with max- In this section, we formulate concepts from the social sci- imum egalitarian social welfare is NP-hard. Similarly, com- ences, especially the literature on fair division, for the context puting a partition with maximum utilitarian social welfare is of hedonic games. A partition π satisfies individual rational- NP-hard in the strong sense even when preferences are sym- ity if each player does as well as by being alone, i.e., for all metric and strict. i ∈ N, π(i) i {i}. For a utility-based hedonic game (N, P) and In Section 4, the complexity of Pareto optimality is studied. partition π, we will denote the utility of player i ∈ N by uπ(i). We prove that checking whether a given partition is Pareto op- The different notions of fair or optimal partitions are defined timal is coNP-complete in the strong sense, even when pref- as follows.2 erences are strict and symmetric. By contrast, we present a 1. The utilitarian social welfare of a partition is defined as polynomial-time algorithm for computing a Pareto optimal the sum of individual utilities of the players: u (π) = partition when preferences are strict.1 Interestingly, comput- ut uπ(i). A maximum utilitarian partition maximizes ing an individually rational and Pareto optimal partition is i∈N the utilitarian social welfare. NP-hard in general. In Section 5, we consider complexity questions regarding 2. The elitist social welfare is given by the utility of the envy-free partitions. Checking whether there exists a parti- player that is best off: uel(π) = max{uπ(i) | i ∈ N}.A tion which is both Pareto optimal and envy-free is shown to maximum elitist partition maximizes the utilitarian so- be Σp-complete. We present an example which exemplifies cial welfare. 2 ff the tradeo between satisfying stability (such as Nash sta- 3. The egalitarian social welfare is given by the utility of bility) and envy-freeness and use the example to prove that the agent that is worst off: ueg(π) = min{uπ(i) | i ∈ N}. checking whether there exists a partition which is both envy- A maximum egalitarian partition maximizes the egali- free and Nash stable is NP-complete even when preferences tarian social welfare. are symmetric. 4. A partition π of N is Pareto optimal if there exists no partition π of N which Pareto dominates π, that is for 2 Preliminaries all i ∈ N, π (i) i π(i) and there exists at least one player

In this section, we provide the terminology and notation re- j ∈ N such that j ∈ N, π ( j) j π( j). quired for our results. 5. Envy-freeness is a notion of fairness. In an envy-free 2.1 Hedonic games partition, no player has an incentive to replace another player. A hedonic coalition formation game is a pair (N, P) where N is a set of players and P is a preference profile which specifies For the sake of brevity, we will call all the notions de- for each player i ∈ N the preference relation i, a reflexive, scribed above “optimality criteria” although envy-freeness is complete and transitive binary relation on set Ni = {S ⊆ N | rather concerned with fairness than optimality. We consider i ∈ S }. the following computational problems with respect to the S i T denotes that i strictly prefers S over T and S ∼i T optimality criteria defined above. that i is indifferent between coalitions S and T.Apartition π is a partition of players N into disjoint coalitions. By π(i), we Optimality: Given (N, P) and a partition π of N,isπ optimal? denote the coalition in π which includes player i. Existence: Does an optimal partition for a given (N, P) exist? A game (N, P)isseparable if for any player i ∈ N and any Search: If an optimal partition for a given (N, P) exists, find coalition S ∈Ni and for any player j not in S we have the one. following: S ∪{j}i S if and only if {i, j}i {i}; S ∪{j}≺i S xistence if and only if {i, j}≺i {i}; and S ∪{j}∼i S if and only if E is trivially true for all criteria of optimality con- {i, j}∼i {i}. cepts. By the definitions, it follows that there exist partitions In an additively separable hedonic game (N, P), each which satisfy maximum utilitarian social welfare, elitist so- player i ∈ N has value vi( j) for player j being in the same cial welfare, and egalitarian social welfare respectively.

1Thus, we identify a natural problem in coalitional game theory 2All welfare notions considered in this paper (utilitarian, elitist, where verifying a possible solution is presumably harder than actu- and egalitarian) are based on the interpersonal comparison of utili- ally ﬁnding one. ties. Whether this assumption can reasonably be made is debatable.

44 3 Complexity of maximizing social welfare w1 w2 w3 w|S | In this section, we examine the complexity of maximizing 3 3 3 3 social welfare in ASHGs. We ﬁrst observe that computing a x1 −1 x2 −1 x3 −1 ··· x|S | −1 maximum utilitarian partition for strict and symmetric preferences is NP-hard because it is equivalent to the NP-hard 3 3 3 3 problem of maximizing agreements in the context of correla- y1 y2 y3 y|S | tion clustering [Bansal et al., 2004]. 1 Theorem 1 Computing a maximum utilitarian partition is 1 2 3 4 5 6 · ·· |R|−2 |R|−1 |R| NP-hard in the strong sense even with symmetric and strict z z z z z z z z z preferences. Figure 1: A graph representation of an ASHG derived from an Computing a maximum elitist partition is much easier. For instance of E3C. The (symmetric) utilities are given as edge any player i, let F(i) = { j | v ( j) > 0} be the set of play- i weights. Some edges and labels are omitted: All edges be- = ers which i strictly likes and f (i) j∈F(i) vi( j). Both s r ∈ r , r ∈ tween any y and z have weight 1 if r s. All z z with F(i) and f (i) can be computed in linear time. Let k N 1 ≥ ∈ r r are connected with weight |R|−1 . All other edges miss- be the player such that f (k) f (i) for all i N. Then − π = {{{k}∪F(k)}, N \{{k}∪F(k)}} is a partition which maxi- ing in the complete undirected graph have weight 7. mizes the elitist social welfare. As a corollary, we can verify π whether a partition has maximum elitist social welfare by time. We know that uπ∗ (s ) = uπ (s ) = P for all 1 ≤ j ≤ n π∗ j j computing a partition with maximum elitist social welfare and uπ∗ (i) ≤ P for all 1 ≤ i ≤ m. But then from the assump- π π∗ ∗ and comparing uel( ) with uel( ). Just like maximizing the tion we have ueg(π ) > ueg(π ) which is a contradiction. utilitarian social welfare, maximizing the egalitarian social welfare is hard. 4 Complexity of Pareto optimality Theorem 2 Computing a maximum egalitarian partition is We now consider the complexity of computing a Pareto op- NP-hard in the strong sense. timal partition. The complexity of Pareto optimality has already been considered in several settings such as house al- Proof: We provide a polynomial-time reduction from the location [Abraham et al., 2005]. Bouveret and Lang [2008] NP-hard problem MaxMinMachineCompletionTime [Woeg- examined the complexity of Pareto optimal allocations in re- inger, 1997] in which an instance consists of a set of m iden- source allocation problems. We show that checking whether tical machines M = {M1,...,Mm}, a set of n independent a partition is Pareto optimal is hard even under severely re- jobs J = {J1,...,Jn} where job Ji has processing time pi. stricted settings. The problem is to allot jobs to the machines such that the minimum processing time (without machine idle times) of all Theorem 3 The problem of checking whether a partition is machines is maximized. Let I be an instance of MaxMin- Pareto optimal is coNP-complete in the strong sense, even achine ompletion ime = n M C T and let P i=1 pi. From I we when preferences are symmetric and strict. construct an instance I of EgalSearch. The ASHG for instance I consists of N = {i | Mi ∈ M}∪{si | Ji ∈ J} and the Proof: The problem is clearly in coNP as another partition preferences of the players are as follows: for all i = 1,...m which Pareto dominates the given partition π is a witness π and all j = 1,...,n let vi(s j) = p j and vs (i) = P. Also, that is not Pareto optimal. The reduction is from the j for 1 ≤ i, i ≤ m, i i let vi(i ) = −(P + 1) and for NP-complete problem E3C (EXACT-3-COVER) to deciding

1 ≤ j, j ≤ n, j j let vs (vs ) = 0. Each player i corre- whether a given partition is Pareto dominated by another par- j j , sponds to machine Mi and each player s j corresponds to job tition or not. Recall that in E3C, an instance is a pair (R S ), = { ,..., } J j. where R 1 r is a set and S is a collection of sub- | | = Let π be the partition which maximizes ueg(π). We show sets of R such that R 3m for some positive integer m and that players 1,...,m are in separate coalitions and each player |s| = 3 for each s ∈ S . The question is whether there is a ⊆ s j is in π(i) for some 1 ≤ i ≤ m. We can do so by proving sub-collection S S which is a partition of R. two claims. The first claim is that for i, j ∈{1,...m} such that It is known that E3C remains NP-complete even if each r ∈ i j, we have that i π( j). The second claim is that each R occurs in at most three members of S [Garey and Johnson, , , player s j is in a coalition with a player i. The proofs of the 1979]. Let (R S ) be an instance of E3C. (R S ) can be reduced claims are omitted due to space limitations. to an instance ((N, P),π), where (N, P) is an ASHG defined A job allocation Alloc(π) corresponds to a partition π in the following way. Let N = {ws, xs, ys | s ∈ S }∪{zr | r ∈ } where s j is in π(i)ifjobJ j is assigned to Mi for all j and R . The players preferences are symmetric and strict and are s = = defined as follows (as also depicted in Figure 1): v s (x ) = i. Note that the utility uπ(i) s ∈π(i) vi(s j) s ∈π(i) p j of w j j s s s a player corresponds to the total completion time of all jobs vxs (y ) = 3 for all s ∈ S ; vys (w ) = vys (w ) = −1 for all ∗ , ∈ r = ∈ r = − assigned to Mi according to Alloc(π). Let π be a maximum s s S ; vys (z ) 1ifr s and vys (z ) 7ifr s; r egalitarian partition. Assume that there is another partition π vzr (z ) = 1/(|R|−1) for any r, r ∈ R; and va(b) = −7 for any and Alloc(π ) induces a strictly greater minimum completion a, b ∈ N and a b for which va(b) is not already defined.

45 The partition π in the instance ((N, P),π)is{{xs, ys}, {ws}| Sonmez,¨ 1998] is a well-known mechanism in resource allo- s ∈ S }} ∪ {{zr | r ∈ R}}. We see that the utilities of the players cation in which an arbitrary player is chosen as the ‘dictator’ s s s are as follows: uπ(w ) = 0 for all s ∈ S ; uπ(x ) = uπ(y ) = 3 who is then given his most favored allocation and the pro- r for all s ∈ S ; and uπ(z ) = 1 for all r ∈ R. cess is repeated until all players or resources have been dealt Assume that there exists S ⊆ S such that S is a partition with. In the context of coalition formation, serial dictator- of R. Then we prove that π is not Pareto optimal and there ship is well-defined if preferences of players over coalitions exists another partition π of N which Pareto dominates π.We are strict. Serial dictatorship is also well-defined for ASHGs s s s form another partition π = {{x , w }|s ∈ S }∪{{y , zi, z j, zk}| with strict preferences as the dictator forms a coalition with s ∈ S ∧ i, j, k ∈ s}∪{{xs, ys}, {ws}|s ∈ (S \ S )}}. In that all the players he strictly likes who have been not considered s s case, uπ (w ) = 3 for all s ∈ S ; uπ (w ) = 0 for all s ∈ as dictators or are not already in some dictator’s coalition. s s r S \ S ; uπ(x ) = uπ(y ) = 3 for all s ∈ S ; and uπ(z ) = The resulting partition π is such that for any other partition 1 + 2/(|R|−1) for all r ∈ R. Whereas the utilities of no player π , at least one dictator will strictly prefer π to π . Therefore in π decreases, the utility of some players in π is more than π is Pareto optimal. in π. Since π Pareto dominates π, π is not Pareto optimal. A standard criticism of Pareto optimality is that it can lead We now show that if there exists no S ⊆ S such that S is to inherently unfair allocations. To address this criticism, the a partition of R, then π is Pareto optimal. We note that −7is algorithm can be modified to obtain less lopsided partitions. asufficiently large negative valuation to ensure that if v (b) = a Whenever an arbitrary player is selected to become the dic- v (a) = −7, then a, b ∈ N cannot be in the same coalition b tator among the remaining players, choose a player that does in a Pareto optimal partition. For the sake of contradiction, not get extremely high elitist social welfare among the re- assume that π is not Pareto optimal and there exists a partition maining players. Nevertheless, even this modified algorithm π which Pareto dominates π. We will see that if there exists may output an partition that fails to be individually rational. a player i ∈ N such that uπ > uπ, then there exists at least We know that the set of partitions which are both Pareto op- one j ∈ N such that uπ < uπ. The only players whose utility timal and individually rational is non-empty. Repeated Pareto can increase (without causing some other player to be less improvements on individually rational partition consisting of happy) are {xs | s ∈ S }, {ws | s ∈ S } or {zr | r ∈ R}.We singletons leads to a Pareto optimal and individually rational consider these player classes separately. If the utility of player partition. We show that computing a Pareto optimal and indi- xs increases, it can only increase from 3 to 6 so that xs is in vidually rational partition for ASHGs is weakly NP-hard. the same coalition as ys and ws. However, this means that ys gets a decreased utility. The utility of ys can increase or stay the same only if it forms a coalition with some zrs. However Theorem 5 Computing a Pareto optimal and individually ra- in that case, to satisfy all zrs, there needs to exist an S ⊆ S tional partition is weakly NP-hard. such that S is a partition of R. Proof: Consider the decision problem SubsetSumZero in Assume the utility of a player ws for s ∈ S increases. This s s which an instance consists of a set of k integer weights is only possible if w is in the same coalition as x . Clearly, = { ,..., } { s, s} { s, s, s} A a1 ak and the question is whether there exists a the coalition formed is w x because coalition w x y ⊆ = ubset um s s non-empty S A such that s∈S s 0? Since S S brings a utility of 2 to y . In that case y needs to form a coali- ubset { s, , , } = { , , } s for positive integers is NP-complete, it follows that S - tion y zi z j zk where s i j k .Ify forms a coalition um ero 3 aximal ubset s s S Z is also NP-complete. Therefore, M S - {y , zi, z j, zk}, then all players y for s ∈ (S \{s}) need to form SumZero, the problem of finding a maximal cardinality sub- s coalitions of the form {y , zi , z j , zk } such that s = {i , j , k }. ⊆ = set S A such that s∈S s 0 is NP-hard. Otherwise, their utility of 3 decreases. This is only possible if We prove the theorem by a reduction from MaximalSub- ⊆ there exists a set S S of R such that S is a partition of R. setSumZero. Reduce an instance of I of MaximalSubset- π Assume that there exists a partition that Pareto domi- SumZero to an instance I = (N, P) where (N, P) is an ASHG r r nates π and the utility of a player uπ (z ) > uπ(z ) for some defined in the following way: N = {x, y1, y2}∪Z where r ∈ R. This is only possible if each zr forms the coalition of Z = {z | i ∈{1,...,k}; v (y ) = v (y ) = k + 1; v (z ) = 1 for { r, r , r , s} = { , , } i x 1 x 2 x i the form z z z y where s r r r . This can only all i ∈{1,...,k}; v (z ) = −v (y ) = −v (z ) = v (y ) = a ⊆ y1 i zi 1 y2 i zi 2 i happen if there exists a set S S of R such that S is a par- for all i ∈{1,...,k}; and v (b) = 0 for any a, b ∈ N for which π a tition of R. Thus we have proved that is not Pareto optimal v (b) is not already defined. , a if and only if (R S ) is a ‘yes’ instance. First, we show that in an individually rational partition π, The fact that checking whether a partition is Pareto optimal no player except x gets positive utility, i.e., uπ(b) = 0 for is coNP-complete has no obvious implications on the com- all b ∈ N \{x}. Assume that w.l.o.g y1 gets positive util- plexity of computing a Pareto optimal partition. In fact there ity in π. This implies there exist a subset Z = Z ∩ π(y1) > ∈ is a simple polynomial-time algorithm to compute a partition such that z∈Z vy1 (z) 0. Then there exists z Z such which is Pareto optimal for strict preferences. that vy (z) > 0 which means that vz(y1) < 0. Due to indi- 1 ∈ π = π ∈ π vidual rationality, y2 (z) (y1). But if y1 (y2), then = − < π Theorem 4 For strict preferences, a Pareto optimal partition uπ(y2) z∈Z vy1 (z) 0 and is not individually rational. can be computed in polynomial time. 3We note that in any instance of SubsetSum all zeros in the set A Proof: The statement follows from an application of se- can be omitted to obtain an equivalent problem. Reduce SubsetSum rial dictatorship. Serial dictatorship [Abdulkadiroglu˘ and to SubsetSumZero by adding ak+1 = −W to A.

46 Assume that there exists a zi ∈ Z such that uπ(zi) > 0. Then Proof: The problem is clearly in NP since envy-freeness and > without loss of generality vzi (y1) 0 and due to individual Nash stability can be verified in polynomial time. We re- rationality y1 ∈ π(zi). Again due to individual rationality, duce the problem from E3C. Let (R, S ) be an instance of E3C > y1 needs to be with another z j such that vy1 (z j) 0. And where R is a set and S is a collection of subsets of R such again due to individual rationality, z j needs to be with y2. This that |R| = 3m for some positive integer m and |s| = 3 for each means, that for each zl ∈ π(zi) ∩ Z, uπ(zl) = al − al = 0. s ∈ S . We will use the fact that E3C remains NP-complete We show that in every Pareto optimal and individually ra- even if each r ∈ R occurs in at most three members of S . tional partition π,wehavey1, y2 ∈ π(x). For any other parti- (R, S ) can be reduced to an instance (N, P) where (N, P)is s tion π , in which this does not hold, uπ (x) ≤ 2k+1 < 2k+2 = an ASHG defined in the following way. Let N = {y | s ∈ }∪{ r , r , r | ∈ } uπ(x). S z1 z2 z3 r R . We set all preferences as symmet- ⊆ πS { | ∈ Consider an S A and let z be any partition of zi ric. The players preferences are as follows: for all r R, r r r r r a ∈ A \ S }. The claim is that π is a Pareto optimal and v r (z ) = v r (z ) = 3, v r (z ) = 3 and v r (z ) = v r (z ) = −7; i z1 2 z2 1 z1 3 z2 3 z3 2 individually rational partition if and only if π is of the form = { , , }∈ j = k = k = / for all s i j k S , vzi (z ) vzi (z ) v j (z ) 1 10 and {{ , , }∪{ | ∈ }} ∪ πS ⊆ 1 1 1 1 z1 1 x y1 y2 zi ai S z where S A is the maximal i = j = k = / , ∈ subset such that ∈ s = 0. Assume that S ⊆ A is not a vys (z ) vys (z ) vys (z ) 28 10; and for all a b N for s S 1 1 1 = = − maximal subset such that ∈ s = 0. If ∈ s 0, there which valuations have not been defined, va(b) vb(a) 7 s S s S − ffi exists a y ∈{y1, y2} such that uπ(y) < 0. If S is not maximal We note that 7isasu ciently large negative valuation then there is a larger set S and a corresponding partition π = to ensure that if va(b) = vb(a) = −7, then a and b will get S {{x, y1, y2}∪{zi | ai ∈ S }}∪π with uπ(x) = |S | < |S | = uπ (x) negative utility if they are in the same coalition. We show z and uπ(b) = uπ (b) for all b ∈ N \{x}. For any other S ⊆ A that there exists an envy-free and Nash stable partition for , P , such that |S | > |S |, we know that ∈ ≤ 0 which implies (N ) if and only if (R S ) is a ‘yes’ instance of E3C. s S ⊆ that there is a y ∈{y1, y2} which gets negative utility. Assume that there exists S S such that S is a partition π = {{ s, i , j , k}| = of R. Then there exists a partition y z1 z1 z1 s { , , }∈ }∪{{ r }, { r }| ∈ }∪{{ }| ∈ \ } 5 Complexity of envy-freeness i j k S z2 z3 r R s s S S . It is easy π r to see that partition is Nash stable and envy-free. Players z1 r and z3 both had an incentive to be with each other when they Envy-freeness is a desirable property in resource alloca- r tion, especially in cake cutting settings. Lipton et al. [2004] are singletons. However, each z1 now gets utility 3 by being in r r s = { , , }∈ proposed envy-minimization in different ways and examined a coalition with z1 , z1 and y where s r r r S . There- r r r the complexity of minimizing envy in resource allocation set- fore z1 has no incentive to be with z3 and z3 has no incentive to r r r s r r s tings. Bogomolnaia and Jackson [2002] mentioned envy- join {z , z , z , y } because v r (z ) = v r (z ) = v r (y ) = −7. 1 1 1 z3 1 z3 1 z3 freeness in hedonic games but focused on stability. We al- Similarly, no player is envious of another player. ready know that envy-freeness can be easily achieved by the Assume that there exists no partition S ⊆ S of R such that partition of singletons.4 Therefore, in conjunction with envy- S is a partition of R. Then, there exists at least one r ∈ R i { r , r , r , s} freeness, we seek to satisfy other properties such as stabil- such that z1 is not in the coalition of the form z1 z1 z1 y ity or Pareto optimality. A partition is Nash stable if there where s = {r, r , r }∈S . Then the only individually rational r is no incentive for a player to be deviate to another (pos- coalitions which z1 can form and get utility at least 3 are the { r , r } { r , r } r sibly empty) coalition. For symmetric ASHGs, it is known following z1 z3 , z1 z2 . In the first case, z1 wants to deviate { r } r that Nash stable partitions always exist and they correspond to z3 . In the second case, z2 is envious and wants to replace r to partitions for which the utilitarian social welfare is a lo- z3. Therefore, there exists no partition which is both Nash cal optimum [see, e.g., Bogomolnaia and Jackson, 2002]. We stable and envy-free. now show that for symmetric ASHGs, there may not exist any partition which is both envy-free and Nash stable. While the existence of a Pareto optimal partition and an envy-free partition is guaranteed, we show that checking Example 6 Consider an ASHG (N, P) where N = {1, 2, 3} whether there exists a partition which is both envy-free and Pareto optimal is hard. and P is defined as follows: v1(2) = v2(1) = 3,v1(3) = v3(1) = 3 and v2(3) = v3(2) = −7. Then there exists no partition which is both envy-free and Nash stable. Theorem 8 Checking whether there exists a partition which is both Pareto optimal and envy-free is Σp-complete. We use the game in Example 6 as a gadget to prove the 2 following.5 Proof: The problem has a ‘yes’ instance if there exists an envy-free partition that Pareto dominates every other par- Theorem 7 For symmetric preferences, checking whether tition. Therefore the problem is in the complexity class there exists a partition which is both envy-free and Nash sta- coNP =Σp NP 2 . ble is NP-complete in the strong sense. We prove hardness by a reduction from a problem concern- 4 ing resource allocation (with additive utilities) [de Keijzer et The partition of singletons also satisfies individual rationality. , , 5Example 6 and the proof of Theorem 7 also apply to the com- al., 2009]. A resource allocation problem is a tuple (I X w) bination of envy-freeness and individual stability where individual where I is a set of agents, X is a set of indivisible objects stability is a variant of Nash stability [Bogomolnaia and Jackson, and w : I × X → R is a weight function. An a : I → 2X 2002]. is an allocation if for all i, j ∈ I such that i j,wehave

47 a(i) ∩a( j) = ∅. The resultant utility of each agent i ∈ I is editors, Handbook of Utility Theory, volume II, chapter 17, , then x∈a(i) w(i x). It was shown by de Keijzer et al. [2009] pages 893–977. Kluwer Academic Publishers, 2004. ∃ that the problem -EEF-ADD of checking the existence of an A. Bogomolnaia and M. O. Jackson. The stability of hedo- Σp envy-free and Pareto optimal allocation is 2 -complete. nic coalition structures. Games and Economic Behavior, , , ∃ Now, consider an instance (I X w)of -EEF-ADD and re- 38(2):201–230, 2002. duce it to an instance (N, P) of an ASHG where N = I ∪ X S. Bouveret and J. Lang. Efficiency and envy-freeness in and P is specified by the following values: vi(x j) = w(i, x j) fair division of indivisible goods: logical representation and vx (i) = 0 for all i ∈ I, x j ∈ X; vx (x j) = vx (xk) = 0 for , j = = − ·| ∪ k | j, ∈ and complexity. Journal of Artificial Intelligence Research, all x jxk; and vi( j) v j(i) W I X for all i j I where = | , | 32(1):525–564, 2008. W i∈I,x j∈X w(i x j) . It can then be shown that there exists a Pareto optimal and envy-free partition in (N, P) if and only S. Branzei and K. Larson. Coalitional affinity games and the if (I, X, w) is a ‘yes’ instance of ∃-EEF-ADD. The proof is stability gap. In Proc. of 21st IJCAI, pages 1319–1320, omitted due to space limitations. 2009. The results of this section show that, even though envy- K. Cechlarov´ a.´ Stable partition problem. In Encyclopedia of freeness can be trivially satisfied on its own, it becomes much Algorithms. 2008. more delicate when considered in conjunction with other de- B. de Keijzer, S. Bouveret, T. Klos, and Y. Zhang. On the sirable properties. complexity of efficiency and envy-freeness in fair division of indivisible goods with additive preferences. In Proc. of 6 Conclusions 1st International Conference on Algorithmic Decision The- ory, pages 98–110, 2009. We studied the complexity of partitions that satisfy standard criteria of fairness and optimality in additively separable he- X. Deng and C. H. Papadimitriou. On the complexity of co- donic games. We showed that computing a partition with operative solution concepts. Mathematics of Operations maximum egalitarian or utilitarian social welfare is NP-hard Research, 12(2):257–266, 1994. in the strong sense and computing an individually rational D. Dimitrov, P. Borm, R. Hendrickx, and S. C. Sung. Sim- and Pareto optimal partition is weakly NP-hard. A Pareto ple priorities and core stability in hedonic games. Social optimal partition can be computed in polynomial time when Choice and Welfare, 26(2):421–433, 2006. preferences are strict. Interestingly, checking whether a given J. H. Dreze` and J. Greenberg. 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