Fractional Hedonic Games: Individual and Group Stability

Fractional Hedonic Game: Individual and Group Stability Florian Brandl Intitut für Informatik TU München, Germany brandlfl@in.tum.de Felix Brandt Intitut für Informatik TU München, Germany brandtf@in.tum.de Martin Strobel Intitut für Informatik TU München, Germany martin.r.trobel@mytum.de ABSTRACT Coalition formation provide a veratile framework for analyzing cooperative behavior in multi-agent ytem. In particular, hedonic coalition formation ha gained coniderable attention in the literature. An intereting cla of hedonic game recently introduced by Aziz et al. [] are fractional hedonic game. In thee game, the utility an agent aign to a coalition i hi average valuation for the member of hi coalition. Three common notion of tability in hedonic game are core tability, Nah tability, and individual tability. For each of thee notion we how that table partition may fail to exit in fractional hedonic game. For core table partition thi hold even when all player only have ymmetric zero/one valuation ( mutual friendhip ). We then leverage thee counter-example to how that deciding the exitence of table partition (and therefore alo computing table partition) i N-hard for all conidered tability notion. Moreover, we how that checking whether the valuation function of a fractional hedonic game induce trict preference over coalition i con-complete. Categorie and Subject Decriptor [Theory of computation]: Algorithmic game theory; [Theory of computation]: Solution concept in game theory; [Theory of computation]: Computational complexity and cryptography; [Computing methodologie]: Multi-agent ytem; [Mathematic of computing]: Graph theory General Term Economic, Theory, and Algorithm Keyword Cooperative game; coalition formation; hedonic game; computational complexity. INTRODUCTION Hedonic game a introduced by Drèze and Greenberg [] and further explored by many other [e.g.,, 9,,,,, 7,, ] preent a natural veratile framework to tudy Appear in: roceeding of the th International Conference on Autonomou Agent and Multiagent Sytem (AAMAS 0), Bordini, Elkind, Wei, Yolum (ed.), May 8, 0, Itanbul, Turkey. Copyright c 0, International Foundation for Autonomou Agent and Multiagent Sytem (www.ifaama.org). All right reerved. the formal apect of coalition formation. In hedonic game, coalition formation i approached from a game-theoretic angle. The outcome are coalition tructure partition of the agent over which the agent have preference. Moreover, the agent have di erent individual or joint trategie at their dipoal to a ect the coalition tructure to be formed. Variou olution concept uch a the core, the trict core, and everal kind of individual tability have been propoed to analyze thee game. The characteritic feature of hedonic game i a non-externalitie condition, which incorporate the ueful but arguably implifying aumption that every agent preference over the coalition tructure are fully determined by hi preference over coalition he belong to and do not depend on how the remaining agent are grouped. Neverthele, the number of coalition an agent can be a member of i exponential in the total number of agent and the development and analyi of concie repreentation a well a intereting ubclae of hedonic game are an ongoing concern in computer cience and game theory. articularly prominent in thi repect are repreentation in which the agent are aumed to entertain preference over the other agent, which are then ytematically lifted to preference over coalition [ee e.g., 9, ]. The work preented in thi paper pertain to fractional hedonic game (FHG), a ubcla of hedonic game in which every agent i aumed to have cardinal utilitie or valuation for the other agent. Thee induce preference over coalition by conidering the average valuation for the member of every coalition. The higher thi valuation, the more preferred the repective coalition i. In other work, the min, max, and um operator have been ued for hedonic game baed on wort agent [9], hedonic game baed on bet agent [8], and additively eparable hedonic game, repectively [ee, e.g., ]. FHG can be repreented by a weighted directed graph where the weight of edge (i, j) denote the valuation of agent i for agent j. The formal tudy of FHG wa initiated by Aziz et al. [] who obtained reult for core tability in variou ubclae of FHG. Some natural economic cenario can be adequately modeled a FHG. Conider, for example, the formation of political partie. The valuation of two agent for each other may be interpreted a to which extent their opinion overlap, e.g., the invere of their ditance in the political pectrum. In political environment, agent need to form coalition and join partie to acquire influence. On the other hand, a partie become larger, the diagreement among their member rie, 9

unretricted ymmetric imple ymmetric IS (N-c.)? + NS (N-c.) (N-c.) + CS (N-h.) (N-h.) Table : Summary of reult. + indicate that the exitence of table partition i guaranteed for the repective cla of game, indicate that there are FHG in the repective cla of game in which no table partition exit, and N-h. and Nc. indicate N-hardne and N-completene of deciding whether a table partition exit, repectively. Aziz et al. [] howed that core table partition in unretricted FHG may not exit and Bilò et al. [] howed that Nah table partition in imple ymmetric FHG alway exit. making them uceptible to plit-o. Thu, one could aume that agent eek to maximize the average agreement with the member of their coalition. In thi paper, we tudy table partition in three hierarchical neted ubclae of FHG: unretricted FHG (arbitrary valuation), ymmetric game (mutually equal valuation), and imple ymmetric game (zero/one valuation). Simple game, a conidered by Aziz et al. [], can be conveniently repreented a directed graph. Our contribution i twofold. Firt we tudy for variou tability notion whether a table partition alway exit. We conider core tability, Nah tability, and individual tability. The latter two are baed on movement of a ingle agent, wherea core tability allow a group of agent to deviate. We provide a clear picture for which tability notion a table partition may fail to exit in the three ubclae of FHG introduced above. In the econd part of the paper we examine the computational complexity of deciding whether a table partition exit in a given FHG. Our reult ugget a trong connection to the exitence reult obtained in the firt part. More preciely, we could how for everal cae that when a table partition may fail to exit for ome tability notion in ome cla of FHG, it i N-hard to decide whether a given game in thi cla admit a table partition. Thi alo implie hardne of the important problem of computing a table partition and tand in harp contrat to everal ubclae of FHG conidered by Aziz et al. [], where exitence of a table partition wa alway aociated with an e cient algorithm for computing it. We alo how that checking whether the valuation function of an FHG induce trict preference over coalition i con-complete. Our main finding are ummarized in Table.. RELATED WORK FHG were tudied for the firt time by Aziz et al. [], who focued on core tability. They how that ome FHG fail to admit a core table partition and that for variou ubclae of FHG, e.g., game given by complete multipartite graph or game given by undirected tree, a core table partition alway exit. Aziz et al. left open whether imple ymmetric FHG alway admit a core table partition. We anwer thi quetion in the negative. Bilò et al. [] tarted to analyze FHG from the viewpoint of non-cooperative game theory. They how that Nah table partition may not exit. Furthermore, they give bound on the price of anarchy and the price of tability. For FHG given by unweighted graph the grand coalition i alway Nah table, hence, they examine whenever finer Nah table partition exit in thee game. Our work i connected to both paper. We advance the reult for core tability and Nah tability and initiate the tudy of individual tability a weakening of Nah tability. In particular, we how that, even for very retricted ubclae of FHG, core table and Nah table partition may not exit. For game in thee clae it turn out to be N-hard to decide whether a table partition exit. FHG are related to additively eparable hedonic game [ee e.g., ]. In both, FHG and additively eparable hedonic game, every agent ha a cardinal valuation for every other agent. In additively eparable hedonic game, the valuation for a coalition i derived by adding the valuation for all agent in the coalition. By contrat, in FHG, the valuation for a coalition i derived by adding the valuation for all agent in the coalition and then dividing the um by the total number of agent in the coalition. Although conceptually additively eparable hedonic game and FHG are imilar, their formal propertie are quite di erent. For example, in unweighted and undirected graph, the grand coalition i trivially core table for additively eparable hedonic game, which i not the cae for FHG. An FHG approach to ocial network with only non-negative weight may therefore help to detect like-minded and denely connected communitie. Aziz et al. [] dicu the relationhip between FHG and network clutering in more detail. Stability in hedonic game give rie to many computationally intereting problem, e.g., deciding the exitence of, verifying the tability of, and finding table partition. Thee quetion were extenively tudied in the context of core tability [ee, e.g., 0, ] and additively eparable hedonic game [ee, e.g.,, ]. Aziz et al. [] howed hardne of two deciion problem for core tability in FHG. Recently, Olen [8] ha examined a variant of imple ymmetric FHG, i.e., game repreented by an unweighted, undirected graph, and invetigated the computation and exitence of Nah table outcome. In the game he conidered, however, every maximal matching i core table and every perfect matching i a bet poible outcome even if there are large clique preent in the graph. By contrat, in our etting agent have an incentive to form large clique. FHG are di erent from but related to another cla of hedonic game called ocial ditance game introduced by Branzei and Laron [7]. In ocial ditance game, an agent not only derive utility from agent he like directly but alo from agent which are at maller ditance from him. FHG alo exhibit ome imilarity with the egregation and tatu-eeking model conidered by Milchtaich and Winter [7] and Lazarova and Dimitrov []. Group formation model baed on type were firt conidered by Schelling [9].. RELIMINARIES Let N be a et {,...,n} of agent or player. Acoalition i a ubet of the agent. For every agent i N, weletn i denote the et {S N : i S} of coalition i i a member of. Every agent i i equipped with a reflexive, complete, and tranitive preference relation % i over the et N i. We 0

ue i and i to refer to the trict and indi erent part of % i, repectively. If % i i alo anti-ymmetric we ay that i preference are trict. A coalition S N i i acceptable for an agent i if i weakly prefer S to being alone, i.e., S % i {i} and unacceptable otherwie. A hedonic game i a pair (N,%), where % =(%,...,% n) i a profile of preference relation % i, modeling the preference of the agent. The valuation function of an agent i i a function v i : N! R aigning a real value to every agent. A valuation function v i can be extended to a valuation function over coalition where, for all S N i, js v i(s) = vi(j). S A hedonic game (N,%) i aid to be a fractional hedonic game (FHG) if, for every agent i in N, there i a valuation function v i uch that for all coalition S, T N i, S % i T if and only if v i(s) v i(t ). Hence, every FHG can be compactly repreented by a tuple of valuation function v =(v,...,v n). It can be hown that every FHG can be induced by valuation function with v i(i) = 0 for all i N. Thu, we aume v i(i) = 0 throughout the paper. We will frequently aociate FHG with weighted digraph G =(N,N N,v) where the weight of the edge (i, j) iv i(j), i.e., the valuation of agent i for agent j. Beide from unretricted FHG, two clae of FHG will be of particular interet to u. An FHG i ymmetric if v i(j) =v j(i) for all i, j N. An FHG i imple if v i(j) {0, } for all i, j N. We note that FHG are not a ubcla of additively eparable hedonic game nor vice vera, i.e., there are FHG that are not additively eparable and vice vera. The outcome of hedonic game are partition of the agent, or coalition tructure. Given a partition = {,..., m} of the agent, (i) denote the coalition in of which agent i i a member. We alo write v i( ) for v i( (i)), which reflect the hedonic nature of the game we conider. By the ame token we obtain preference over partition from preference over coalition. We refer to {N} a the grand coalition. Hedonic game are analyzed uing olution concept, which formalize deirable or optimal way in which the agent can be partitioned (a baed on the agent preference over the coalition). In thi paper, we conider three notion of tability. We ay that a coalition S N block a partition, if every agent i S trictly prefer S to hi current coalition (i), i.e., if S i (i) for all i S. A partition that i not blocked by any coalition i core table (CS). A partition i Nah table (NS) if no agent can benefit from joining another (poibly empty) coalition, i.e., if (i) % i S [ {i} for all S [ {;} and i N. A partition i individually table (IS) if no agent can benefit from joining another (poibly empty) coalition without making ome member of the coalition he join wore o, i.e.,if (i) % i S [ {i} or S j S [ {i} for ome j S for all S [ {;} and i N. Note that no partition where one agent i placed in an unacceptable coalition i core table, Nah table, or individually table, ince thi agent could benefit from forming hi own coalition, i.e., join the empty coalition. By definition, every Nah table partition i alo individually table. However, there i no logical relationhip between core tability and any of the remaining tability notion defined above. In particular, there exit core table partition which are not individually table and Nah table partition which are not core table. Bogomolnaia and Jackon [] provided an example for the firt tatement and the econd tatement can be deduced from Example. Example. Conider the ymmetric FHG given in Figure. Agent are repreented by vertice and the valuation function by weighted edge. The only core table partition i {{, }, {, }}. Thi partition i alo individually table but not Nah table, ince agent can benefit from joining the coalition {, }. On the other hand, the partition {{}, {,, }} i Nah (and hence individually) table, but not core table, ince it i blocked by the coalition {, }. If thi game where to be interpreted a an additively eparable hedonic game, the grand coalition would be core table and Nah table. Figure : Example of a ymmetric FHG.. EXISTENCE OF STABLE ARTITIONS Thi ection i divided into three ubection, each correponding to one of the clae of FHG defined above. In thee ection we dicu the exitence of core table, Nah table, and individually table partition, repectively.. Unretricted FHG Aziz et al. [] and Bilò et al. [] howed that core table partition and Nah table partition may not exit in unretricted FHG. Our firt reult i that individually table partition alo may not exit in unretricted FHG. Theorem. In unretricted FHG, core table, Nah table, or individually table partition may not exit. roof. The FHG given in Figure wa ued by Aziz et al. [] to how that core table partition may not exit in unretricted FHG. Furthermore it doe not admit a Nah table or individually table partition. We how that no individually table partition exit. Thi directly implie that no Nah table partition exit. Note that no partition containing a coalition with three or more agent i individually table, ince it i unacceptable for all it member. -

7 7 7 Figure : An FHG in which no core table, Nah table, or individually table partition exit. All miing edge have weight. Alo, no partition in which two agent i and i + are in a ingleton coalition i individually table, ince i would join i + and i + would permit it (or vice vera). Hence, up to ymmetrie, the partition = {{, }, {, }, {}} i the only remaining candidate for an individually table partition. But i not individually table, ince agent can benefit from joining the ingleton coalition {} and agent would permit it.. Symmetric FHG Symmetry capture the idea that agent mutually benefit from each other to the ame extent. Many economic cenario that can be adequately modeled a FHG naturally exhibit ymmetry. In our introductory example, the valuation of two agent for each other i determined by their ditance in the political pectrum. Since ditance function are ymmetric by definition, the aociated FHG i ymmetric, too. We how that even in the context of ymmetric FHG, both core table partition and Nah table partition may not exit. Theorem. In ymmetric FHG, core table or Nah table partition may not exit. roof. For both tatement, we provide game in which no table partition exit. In the FHG depicted in Figure no core table partition exit. The proof i omitted, ince we prove a tronger tatement in Theorem. In the FHG depicted in Figure no Nah table partition exit. Firt, note that no partition with agent and in the ame coalition i table, ince it i unacceptable for both and. Furthermore, no partition in which agent i in a ingleton coalition i table, ince he prefer every coalition to being alone. Hence, up to ymmetrie, we only have to conider = {{, }, {, }}, = {{, }, {}, {}}, = {{,, }, {}}, and = {{, }, {}, {}}. i not table becaue agent can benefit from joining {, }, i not table becaue agent may join {, }, i not table becaue agent may join {}, and i not table becaue agent may join {}. Thi reult i in contrat to the correponding tatement for additively eparable hedonic game. Bogomolnaia and Figure : A ymmetric FHG in which no core table partition exit. All miing edge have weight. - Figure : A ymmetric FHG in which no Nah table partition exit. Jackon [] proved that every ymmetric additively eparable hedonic game admit a core table partition. It remain open whether every ymmetric FHG admit an individually table partition.. Simple FHG In many application it i reaonable to aume that the agent valuation only take the value zero and one. Thi i, for example, the cae in ocial network or exchange economie, if agent only ditinguih between non-friend and friend. Thee o-called imple game can be repreented a unweighted directed graph. There i an edge from one agent to another if the former ha valuation one for the latter. Aziz et al. [] conidered baker and miller game, which form a ubcla of imple game. Thee game correpond to complete -partite graph. More generally, they how that for game that correpond to complete k-partite graph a core table partition alway exit. An open quetion propoed by Aziz et al. [] i whether every imple ymmetric FHG admit a core table partition. We anwer thi quetion in the negative by providing a counter-example with 0 agent. When only requiring non-negative and ymmetric valuation, there i a counter-example with only agent.

Theorem. In imple ymmetric FHG, core table partition may not exit. roof. The FHG depicted in Figure doe not admit a core table partition. For two agent i, j N we ay that i i connected to j if i valuation for j i (and vice vera). Let be a core table partition. The firt tep i to how that A l S and C l T for all l {,...,}. We how both tatement for l =. The ret follow from the ymmetry of the game. A S : Aume for contradiction that thi i not the cae. Since A [C i a -clique, at leat one agent i A [C ha a valuation of at leat / for hi coalition (otherwie A [ C i blocking). Aume i A. If (i) contain an agent i i not connected to, then u i( ) apple 9 / < / ince i i connected to at mot 9 agent in any coalition. Hence (i) only contain agent i i connected to. But then A [ (i) i blocking, ince every agent in A i connected to the ame agent a i, a contradiction. Hence i C. A \ (i) = ; implie u i( ) apple /. If (i) containanagenti i not connected to, then u i( ) apple 7 /9 < /, ince i i connected to at mot 7 agent in any coalition. Hence, (i) \ A = S = ; and (i) only contain agent i i connected to. Thu, C (i) (otherwie C [ (i) i blocking). At leat one agent k in A [ B and at leat one agent k in A [ B ha a valuation of at leat / for hi coalition, ince both et are -clique. k,k S, ince u j( ) apple / for all j S. If k A \ S, then (k ) only contain agent k i connected to, otherwie u k ( ) apple /7 < /. Then (k ) [ S i blocking. Hence k B. Analogouly it follow that k B. We how that (k ) = (k ). Aume for contradiction that (k )= (k )=T. If S contain at leat two agent k i not connected to, we have u k (T ) apple 0/ < / (ince k i connected to at mot 0 agent in any coalition). Hence, T contain one agent k i not connected to, namely k. The analogou aertion hold for k. Since u k (T ) /, we have T 0. But then T contain at leat agent k i not connected to, ince there are only agent that both k and k are connected to. Thi implie that u k apple 0 / < /, a contradiction. If B (i), it follow that u j( ) apple /7 < / for all j S. If j A \ S i in a coalition with an agent j i not connected to, u j( ) apple / < /, ince (j) cannot contain an agent from (i) and from both (k ) and (k ) (ince (k ) = (k )). Hence S [ (j) i blocking. If (i) \ B = it follow that S = and u j( ) = / for all j S. At leat one agent k in A [B ha a valuation of at leat / for hi coalition. If k (i) it follow that u k ( ) = / < /, a contradiction. If k B \ (i), then (k) only contain agent k i connected to. If A (k) or A (k), then A [ C A [ C are blocking, repectively. (k) \ A =or (k) \ A = i not poible ince our previou analyi for A and C alo applie to A and C, and A and C, repectively. But then, u k ( ) apple / < /. Hence k A. Thi implie that (k) only contain agent k i connected to. Hence A (k), otherwie A [ (k) i blocking. Alo (k) = A [ B, becaue otherwie A [ C i blocking. Hence A \ (k )=;. Thu, (k ) can only contain agent k i connected to. Thi implie B (k ). A u k ( ) /, (k ) \ C. Thu, if (k ) contain ome agent in A, then A [ C i blocking. If (k )=B [ C, then A [ C i blocking and otherwie C i blocking. Thi contradict that i table. C T : At leat one agent i in B [ C ha a valuation of at leat / for hi coalition (otherwie B [ C i blocking). Aume i B and u j( ) < / for all j C. Then A l (i) for ome l {, }. But then A l [ C l i blocking. Hence the aumption i wrong and i C.Note that (i) cannot contain an agent i i not connected to, otherwie u i( ) apple 7 /9 < /, ince i i connected to at mot 7 agent in any coalition. But then C (i), otherwie C [ (i) i blocking. It cannot be that (i) A l [ B l or (i) A l [ B l for i A l, ince A l [ C l i blocking for all l {,...,}. If A [C [S with ;= S B, then u i( ) apple /7 < / for all i A. Hence B [ C,B [ C, otherwie either A [B or A [B are blocking (u i(a [B )=u i(a [B )= / for all i A ). But then u i( ) apple / for all i A. Hence A [ C i blocking, a contradiction. In any other partition in which ome i A i in a coalition with an agent he i not connected to, we have u i( ) apple 9 / < / for all i A and u j( ) apple / < / for all j C. Hence A [ C i blocking. Hence (i) only contain agent i i connected to for all i A l and l {,...,}. We have hown previouly that at leat one agent i l C l ha a valuation of at leat / for hi coalition for all l {,...,}. Hence, (i l ) cannot contain an agent i l i not connected to. Therefore, either (i l )=A l [ C l or (i l )= B l [ C l for all l {,...,}. If A l [ C l for all j {,...,}, then A [ B [ B i blocking. Hence we can aume without lo of generality that A [ S with S B [ B. If S <, then A [ C i blocking. Hence S. Without lo of generality, B S. If follow that B [ C, ince one agent in C ha a valuation of at leat / for hi coalition. Thi implie that A [ C. Furthermore A [ C,A [ C, otherwie B [ C or B [ C are blocking. Then we get A [ C. But then A [ B [ B i blocking. Hence, i not core table, a contradiction.. COMUTATIONAL COMLEXITY We now focu on the computational complexity of variou deciion problem aociated with FHG. Since the number of coalition an agent can be a member of i exponential in the number of agent, we aume in thi ection that the agent preference are not given explicitly a ranking over coalition but implicitly by valuation function. Firt, we will how that, given the valuation function, it i concomplete to decide whether every agent ha trict preference over coalition. Looely put, our econd main reult in thi ection how that whenever in ome cla of game a table partition may fail to exit for ome tability notion, it i N-hard to decide whether a table partition exit for a game in thi cla. The proof i baed on a generic contruction which ue the counter-example from Section a gadget.. Strictne of reference A problem of independent interet i to decide for given valuation function whether thee induce trict preference over coalition. Clearly, it i eay to verify that an agent i indi erent between two coalition. But ince the number of coalition i exponential in the number of agent, it i not clear how to e ciently verify that no agent i indi erent between ome pair of coalition. We will how that it i unlikely that an e cient algorithm for thi problem exit.

A For the other direction, we firt tate (without proof) that, for all l {,...,n}, B B l l (C + m ) > l (C + m+). () l A B C C C C C A B Now uppoe there exit two ditinct coalition S, T N 0 uch that S 0 T. Aume that S > T. Then, is (C + mi) ( S )(C + m ) v 0(S) = S S ( S )(C + m+) ( T )(C + m +) > S T it (C + mi) = v 0(T ). T The trict inequality follow from (). Thi implie S 0 T, contradiction our aumption, and hence S = T. Thu, S 0 T if and only if S = T and is mi = it mi. A B Figure : A imple ymmetric FHG in which no core table partition exit. For all l {,...,}, A l and C l denote clique of agent and B l denote a clique of agent. An edge from one clique to another denote that every agent in the firt clique i connected to every agent in the econd clique. All depicted edge have weight. All miing edge have weight 0. Theorem. It i con-complete to decide whether a given profile of valuation function induce trict preference over coalition. roof. We how that it i N-complete to decide whether, for an agent i N, there are two coalition S, T N i uch that S i T. Thi implie the tatement. Firt, note that thi problem i clearly in N, ince, given two et in N i, it can be checked in linear time whether i ha the ame valuation for both. To prove hardne, we provide a reduction from an intance M = {m 0,...,m k } N k+ of equal um of ubet of equal cardinality (ESS). The anwer to ESS i Ye if there are two ditinct et S, T {,...,k} uch that S = T and is mi = it mi and No otherwie. Cieliebak et al. [0] howed that ESS i Ncomplete. Without lo of generality we aume m 0 =0. Let m + = max {,...,k} m,m = min {,...,k} m, and C =(k +k)(m + m ) +. We define G =(N,N N,v) where N = {0,...,k} and v({i, j}) =C + m i+j (mod k+) for all ditinct i, j {0,...,k}. Without lo of generality, we conider agent 0, who ha valuation C + m i for all i {,...,k}. Suppoe there are two nonempty, ditinct et S, T M uch that S = T, 0 S \ T, and is mi = it mi. Then, we have is (C + mi) v 0(S) = = S C + is mi S + S + = T C + it mi T + Hence, we have S 0 T. = A it (C + mi) = v 0(T ). T +. Exitence of Stable artition In the initial work on FHG, Aziz et al. [] howed hardne of deciding whether a given partition i core table. For Nah tability and individual tability, thi problem can eaily be olved in polynomial time. In thi ection, we dicu problem of a imilar pirit. We conider the problem of deciding whether a given FHG admit a table partition for core tability, Nah tability, and individual tability. It turn out that thi problem i hard whenever it i not trivial. Thi alo implie that finding table partition i intractable. Firt, we define the correponding deciion problem. Definition. For a tability notion E {CS, NS, IS}, the deciion problem (SYMM)FHG-E i given by a (ymmetric) FHG (N,%). The anwer to (SYMM)FHG-E i Ye if there i an E-table partition in (N,%) and No otherwie. Sung and Dimitrov [0] proved that it i N-hard to decide whether a given additively eparable hedonic game admit a core table, Nah table, or individually table partition. To thi end, they provided a polynomial time reduction from the N-complete problem exact cover by -et (EC) [cf. ]. The contruction can be adapted to obtain hardne reult for FHG. We explain the adaption to SYMMFHG- NS and FHG-IS. An intance of EC i a pair (R, S) where R i a et uch that R =m for ome poitive integer m and S i a collection of ubet of R uch that = for every S. The anwer to EC i Ye if there i a ubet of S which partition R, i.e., there i S 0 S uch that S S = R and 0 \ 0 = ; for all ditinct, 0 S 0. EC remain N-complete even if every r R occur in at mot three element of S [cf. ]. Furthermore we can aume without lo of generality that every r R occur in at leat one element of S, otherwie the anwer to the quetion i trivially No. Now, we contruct for a given intance (R, S) ofeca weighted graph repreenting an FHG that admit a table partition if and only the anwer to EC i Ye. We tart by contructing a ubgraph G for every S. For every = {u, v, w}, G = (N,N N,v ) where N = {, u, v, w} and v (i, j) = for all i, j N. Figure illutrate uch a ubgraph. Every ubet S 0 S can be identified with the et of graph {G } S 0. Next we compute, for every r R, the number

u v G w Figure : The ubgraph correponding to = {u, v, w} S. All edge have weight. l r = { S: r S}, i.e., the number of et containing r after removing an R-partitioning ubet S 0 form S. Since we aumed that every r R i contained in at leat one S, we have l r 0 for every r R. For every r R we add l r graph G r,k, k {,...,l r}. The exact tructure of the G r,k depend on the actual proof, but in general they have to fulfill the following condition. The FHG induced by G r,k doe not admit a table partition The et of vertice of G r,k contain a vertex r k uch that the FHG induced by G r,k admit a table partition when r k i removed from the game. The lat tep i to connect every r to every r k by an edge of weight (and vice vera) for every r R. All vertice which are not connected by an edge of weight one are connected by an edge of weight M, where M i larger than the um of the weight of adjacent edge with poitive weight for every vertex. Notice that M doe not depend on the given intance of EC. For our purpoe, M = 0u ce. The whole graph now induce an FHG. Notice that thi game i ymmetric if all G r,k are ymmetric. Figure 7 illutrate the contruction for a mall intance of EC. The G r,k are obtained from the graph depicted in Figure, which i an example of an FHG that doe not admit an individually table partition. The main idea behind the whole contruction i the following. For every r R, l r of the r are needed to tabilize the G r,k. On the other hand, the G admit a table partition only if the whole ubgraph form a coalition or if the r are in a coalition with a correponding r. k In a table partition, for every r R, exactly one r i not in a coalition with ome r, k but intead in a coalition coniting of every vertex in G. All thee G together can be identified with a ubet S 0 S which i a partition of R. A table partition exit if and only if uch an S 0 exit. Our finding on the computational complexity of checking whether a table partition exit are ummarized in the following theorem. Theorem. The following hardne reult hold: (i) SYMMFHG-CS i N-hard, (ii) SYMMFHG-NS i N-complete, and (iii) FHG-IS i N-complete. G G Figure 7: The graph belonging to the reduction of an intance (R, S) of EC, where R = {,,,,, } and S = {{,, }, {,, }, {,, }}. All unlabeled edge have weight one. All edge that are not depicted have weight 0. roof. Due to pace contraint we only give a proof for (iii) here. Firt note that it i eay to verify if a partition i individually table. For every agent, one can check in polynomial time if he can deviate without making an agent in hi new coalition wore o. So FHG-IS i in N. Now we will go through the reduction decribed above. Let (R, S) be an intance of EC. An intance of FHG-IS i contructed a follow. Let N = { r : S,r }[{ : S}[{ r, k k k k r, r, r, k r : r R, apple k apple l r}. The r and form the G and the r k to k r the G r,k, repectively. The agent valuation function are defined a follow. (i) For all S,r : v r ( )=v ( r)= (ii) For all S,r,r 0, r = r 0 : v r ( r 0)=v r 0 ( r)= (iii) For all r R, S,r, k {,...,l r}: v r ( k r)= v k r ( r)=

(iv) For all r R, k {,...,l r}: v a k r ( k r )=v k r ( k r )= v k r ( k r ) = v k r ( k r) = v k r ( k r) = and v a k r ( k r) = v k r ( k r )=v k r ( k r )=v k r ( k r )=v k r ( k r)=. (v) For all remaining pair (i, j) we define v i(j) = 0. The number of agent i S + S +( S R ) and all valuation function are bounded by a contant, o thi contruction can be computed in polynomial time. Firt uppoe there exit a ubet S 0 S which i a partition of R. For every r R, let{ r, r,..., lr r } = { S \ S 0 : r } be an enumeration of the et outide of S 0 containing r. Now conider the following partition. = {{ } [ { r : r }: S 0 } [ {{ }: S \ S 0 } [ {{ k r r, k r}, { k r, We claim that i individually table. k r }{ k r, k r}: r R, k {,...,l r}} Conider an agent i { r : r R, S}. If S (i) ={ } [ { r : r } then v i( (i)) = / and for all S k \{ (i)} we have v i(s k [ {i}) < /, and if S (i) ={ k r r, r} k then v i( (i)) = / and for all S k \{ (i)} we have v i(s k [ {i}) apple /. So i ha no incentive to deviate. Conider an agent i { : S}. Then v i( (i)) 0 and for all S k \{ (i)} we have v i(s k [ {i}) < 0. So i ha no incentive to deviate. Conider an agent i { r, k k k k r, r, r, k r : r R, k {,...,l r}}. We have v i( (i)) > 0 and for all S k \{ (i)} we have v i(s k [{i}) < 0. So i ha no incentive to deviate. Hence, i Nah table and thu individually table. For the other direction, uppoe there exit an individually table partition. For every r R and k {,...,l r}, G r,k i iomorphic to the graph depicted in Figure. So in the FHG we contructed from the EC intance, a partition can only be table, if, for every r R and every k {,...,l r}, there exit an agent i { r, k k k k r, r, r, k r} k r, k k r, uch that (i) { r, k r, k r}. By (ii) and (iii) in the definition of the valuation function, the only candidate for thi i r. k From thi we directly get and {{ k r, k r }{ k r, k r}: r R, k {,...,l r}} ( k r) { k r} [ { r : S} for all r R, k {,...,l r}. Suppoe ( r) k >, then there exit ditinct, 0 S uch that { r, 0 r } ( r). k But then cannot be individually table ince r would rather be alone than a member of ( r). k Hence, for every r R and k {,...l r}, there i an agent r uch that ( r)={ k r, k r}. By (v) in the definition of the valuation function, we get ( ) { } [ { r : r }. We can conclude ( )={ } or ( ) = { } [ { r : r }, becaue otherwie there exit r R uch that v r ( ( r)) apple / < / apple v r ( ( ) [ { r}) and v i( ( ) [ { r}) >v i( ( )) for every i ( ). Furthermore, we have, for every i { r : S,r }, that (i) = {i}, otherwie v i( (i)) = 0 < / apple v i( ( ) [ {i}) and v j({,i}) >v j({ }) for all j ( ). Exactly l r agent in { r : S} form a coalition with ome r k for every r R. It follow that, for every r R, there i exactly one S uch that ( r)={ }[{ r 0 : r0 }. Hence, S 0 = { S: ( )={ } [ { r : r }} i a partition of R. The contruction above doe not work for core tability ince the partition given by a olution to the correponding intance of EC would be blocked by the ubgraph G. However, Sung and Dimitrov [0] provided a lightly different contruction for core tability in additively eparable hedonic game which can be adapted to ymmetric FHG. The FHG depicted in Figure erve a gadget for thi contruction. If there exit a ymmetric FHG that doe not admit an individually table partition, the contruction from Theorem can be ued to prove N-completene of SYMMFHG-IS. Every mallet ymmetric FHG that doe not admit an individually table partition could erve a a gadget. If, on the other hand, no uch game exit, the anwer to SYMMFHG- IND i trivially Ye for every game.. CONCLUSIONS We tudied core tability and tability notion baed on deviation of a ingle agent, i.e., Nah tability and individual tability. For thee tability notion we examined whether a table partition may fail to exit for three clae of FHG. In particular, we howed that core table partition may not exit in imple ymmetric FHG. Thi anwer a quetion propoed by Aziz et al. []. In the econd part of the paper, we leveraged the nonexitence example to how that deciding the exitence (and thu alo finding) table partition for the correponding notion of tability and cla of FHG i N-hard. By contrat, Aziz et al. [] propoed a number of clae of FHG in which table partition are guaranteed to exit by providing polynomial-time algorithm for computing uch partition. Thee reult ugget a trong connection between the exitence of table partition and the hardne of finding table partition. It i an intereting problem whether thi connection can be made more precie and extended to more general clae of hedonic game. Since our reult how that for large clae of FHG the exitence of a table partition cannot be guaranteed, it would be deirable to find more natural clae for which the exitence of table partition i guaranteed. In particular, the exitence of individually table partition in ymmetric FHG remain an open problem. Acknowledgment We are grateful to Hari Aziz, aul Harrentein, and the anonymou reviewer for helpful comment. Thi material i baed upon work upported by Deutche Forchunggemeinchaft under grant BR /7- and BR /0-. REFERENCES [] H. Aziz, F. Brandt, and. Harrentein. areto optimality in coalition formation. Game and Economic Behavior, 8: 8,0.

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