Stable Outcomes in Modified Fractional Hedonic Games

04/13/2018 ∙ by Gianpiero Monaco, et al. ∙ 0

In coalition formation games self-organized coalitions are created as a result of the strategic interactions of independent agents. For each couple of agents (i,j), weight w_i,j=w_j,i reflects how much agents i and j benefit from belonging to the same coalition. We consider the modified fractional hedonic game, that is a coalition formation game in which agents' utilities are such that the total benefit of agent i belonging to a coalition (given by the sum of w_i,j over all other agents j belonging to the same coalition) is averaged over all the other members of that coalition, i.e., excluding herself. Modified fractional hedonic games constitute a class of succinctly representable hedonic games. We are interested in the scenario in which agents, individually or jointly, choose to form a new coalition or to join an existing one, until a stable outcome is reached. To this aim, we consider common stability notions, leading to strong Nash stable outcomes, Nash stable outcomes or core stable outcomes: we study their existence, complexity and performance, both in the case of general weights and in the case of 0-1 weights. In particular, we completely characterize the existence of the considered stable outcomes and show many tight or asymptotically tight results on the performance of these natural stable outcomes for modified fractional hedonic games, also highlighting the differences with respect to the model of fractional hedonic games, in which the total benefit of an agent in a coalition is averaged over all members of that coalition, i.e., including herself.

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1 Introduction

Teamwork, clustering and coalition formations have been important and widely investigated issues in computer science research. In fact, in many economic, social and political situations, individuals carry out activities in groups rather than by themselves. In these scenarios, it is of crucial importance to consider the satisfaction of the members of the groups.

Hedonic games, introduced in [16], model the formation of coalitions of agents. They are games in which agents have preferences over the set of all possible agent coalitions, and the utility of an agent depends on the composition of the coalition she belongs to. While the standard model of hedonic games assumes that agents’ preferences over coalitions are ordinal, there are several prominent classes of hedonic games where agents assign cardinal utilities to coalitions. Additively separable hedonic games constitute a natural and succinctly representable class of hedonic games. In such setting each agent has a value for any other agent, and the utility of a coalition to a particular agent is simply the sum of the values she assigns to the members of her coalition. Additive separability satisfies a number of desirable axiomatic properties [3] and is the non-transferable utility generalization of graph games studied in [15]. Fractional hedonic games, introduced in [2], are similar to additively separable ones, with the difference that the utility of each agent is divided by the size of her coalition. Arguably, it is more natural to compute the average value of all other members of the coalition [17]. Various solution concepts, such as the core, the strict core, and various kinds of individual stability like Nash Equilibrium have been proposed to analyze these games (see the Related Work subsection).

In this paper we deal with modified fractional hedonic games (MFHGs), introduced in [25], and afterward studied in [17, 23]. MFHGs model natural behavioral dynamics in social environments. Even when defined on undirected and unweighted graphs, they suitably model a basic economic scenario referred to in [2, 10] as Bakers and Millers. Moreover, MFHGs can model other realistic scenarios: (i) politicians may want to be in a party that maximizes the fraction of like-minded members; (ii) people may want to be with an as large as possible fraction of people of the same ethnic or social group.

In MFHGs, slightly differently than in fractional hedonic games, the utility of an agent is divided by the size of the coalition she belongs to minus 1, that indeed corresponds to the average value of all other members than of the coalition. Despite such small difference, we will show that natural stable outcomes in MFHGs perform differently than in fractional hedonic games. Specifically, we adopt Nash stable, Strong Nash stable and core outcomes. Informally, an outcome is Nash stable (or it is a Nash equilibrium) if no agent can improve her utility by unilaterally changing her own coalition. Moreover, an outcome is strong Nash stable if no subset of agents can cooperatively deviate in a way that benefits all of them. Finally, an outcome is in the core or is core stable, if there is no subset of agents , whose members all prefer with respect to the coalition in the outcome. We point out that, (strong) Nash stable outcomes are resilient to a group of agents that can join any coalition and therefore represent a powerful solution concept. However, there are settings in which it is not allowed for one or more agents to join an existent coalition without asking for permission to its members: in these settings the notion of core, where in a non-stable outcome a subset of agents can only form a new coalition itself and cannot join an already non-empty coalition, appears to be more realistic.

Our aim is to study the existence, performance and computability of natural stable outcomes for MFHGs. In particular, we evaluate the performance of Nash, strong Nash, and core stable outcomes for MFHGs, by means of the widely used notions of price of anarchy (resp. strong price of anarchy and core price of anarchy), and price of stability (resp. strong price of stability and core price of stability), which are defined as the ratio between the social optimal value and the social value of the worst (resp. best) stable outcome.

An instance of MFHG can be effectively modeled by means of a weighted undirected graph , where nodes in represent the agents, and the weight of an edge represents how much agents and benefit from belonging to the same coalition.

1.1 Related Work

To the best of our knowledge, only few papers dealt with stable outcomes for MFHGs. Olsen [25] considers unweighted undirected graphs and investigates computational issues concerning the problem of computing a Nash stable outcome different than the trivial one where all the agents are in the same coalition. The author proves that the problem is NP-hard when we require that a coalition must contain a given subset of the agents, and that it is polynomial solvable for any connected graph containing at least four nodes. Kaklamanis et al. [23] show that the price of stability is for unweighted graphs. Finally, Elkind et al. [17] study the set of Pareto optimal outcomes for MFHGs.

Fractional hedonic games have been introduced by Aziz et al. [2]. They prove that the core can be empty for games played on general graphs and that it is not empty for games played on some classes of undirected and unweighted graphs (that is, graphs with degree at most , multipartite complete graphs, bipartite graphs admitting a perfect matching and regular bipartite graphs). Brandl et al. [12], study the existence of core and individual stability in fractional hedonic games and the computational complexity of deciding whether a core and individual stable partition exists in a given fractional hedonic game. Bilò et al. [9] initiated the study of Nash stable outcomes for fractional hedonic games and study their existence, complexity and performance for general and specific graph topologies. In particular they show that the price of anarchy is , and that for unweighted graphs, the problem of computing a Nash stable outcome of maximum social welfare is NP-hard, as well as the problem of computing an optimal (not necessarily stable) outcome. Furthermore, the same authors in [10] consider unweighted undirected graphs and show that -Strong Nash outcomes, that is, an outcome such that no pair of agents can improve their utility by simultaneously changing their own coalition, are not always guaranteed. They also provide upper and lower bounds on the price of stability for games played on different unweighted graphs topologies. Finally, Aziz et al. [4] consider the computational complexity of computing welfare maximizing partitions (not necessarily Nash stable) for fractional hedonic games. We point out that fractional hedonic games played on unweighted undirected graphs model realistic economic scenarios referred to in [2, 10] as Bakers and Millers.

Hedonic games have been introduced by Dréze and Greenberg [16], who analyzed them under a cooperative perspective. Properties guaranteeing the existence of core allocations for games with additively separable utility have been studied by Banerjee, Konishi and Sönmez [8], while Bogomolnaia and Jackson [11] deal with several forms of stable outcomes like the core, Nash and individual stability. Ballester [5] considers computational complexity issues related to hedonic games, and show that the core and the Nash stable outcomes have corresponding NP-complete decision problems for a variety of situations, while Aziz et al. [3] study the computational complexity of stable coalitions in additively separable hedonic games. Moreover, Olsen [24] proves that the problem of deciding whether a Nash stable coalitions exists in an additively separable hedonic game is NP-complete, as well as the one of deciding whether a non-trivial Nash stable coalitions exists in an additively separable hedonic game with non-negative and symmetric preferences (i.e., unweighted undirected graphs).

Feldman et al. [18] investigate some interesting subclasses of hedonic games from a non-cooperative point of view, by characterizing Nash equilibria and providing upper and lower bounds on both the price of stability and the price of anarchy. It is worth noticing that in their model they do not have an underlying graph, but agents lie in a metric space with a distance function modeling their distance or “similarity”. Peters [26] considers “graphical” hedonic games where agents form the vertices of an undirected graph, and each agent’s utility function only depends on the actions taken by her neighbors (with general value functions). It is proved that, when agent graphs have bounded treewidth and bounded degrees, the problem of finding stable solutions, i.e., Nash equilibria, can be efficiently solved. Finally, hedonic games have also been considered by Charikar et al. [13] and by Demaine et al. [14] from a classical optimization point of view (i.e, without requiring stability for the solutions) and by Flammini et al. in an online setting [19].

Peters et al. [27] consider several classes of hedonic games and identify simple conditions on expressivity that are sufficient for the problem of checking whether a given game admits a stable outcome to be computationally hard.

From a different perspective, strategyproof mechanisms for additively separable hedonic and fractional hedonic games have been proposed in [20, 28].

Finally, hedonic games are being widely investigated also under different utility definitions: For instance, in [6, 7], coalition formation games, in which agent utilities are proportional to their harmonic centralities in the respective coalitions, are considered.

1.2 Our Results

We start by dealing with strong Nash stable outcomes. We first prove that there exists a simple star graph with positive edge weights that admits no strong Nash stable outcomes. Therefore we focus on unweighted graphs, and present a polynomial time algorithm that computes an optimum outcome that can be transformed in a strong Nash stable one with the same social welfare, implying that strong Nash stable outcomes always exist and that the strong price of stability is . We further prove that the strong price of anarchy is exactly . In particular, we are able to show that, even for jointly cooperative deviations of at most agents, the strong price of anarchy is at most (we emphasize that, as we will describe in the next paragraph, the price of anarchy for Nash stable outcomes that are resistant to deviations of one agent grows linearly with the number of agents), while it is at least for jointly cooperative deviations of any subsets of agents.

We subsequently turn our attention on Nash stable outcomes. We notice that Nash stable outcomes are guaranteed to exist only if edge weights are non-negative. In a similar way as in [9], we prove that the price of anarchy is at least , where is the number of agents, even for unweighted paths, and that it is at most for the more general case of non-negative edge-weighted graphs, thus giving an asymptotically tight characterization. We also prove a matching lower bound of to the price of stability.

We finally consider core stable outcomes and show that they always exist, and in particular that an outcome that is core stable can be computed in polynomial time, even in the presence of negative weights, i.e., for general undirected weighted graphs. We then establish that the core price of stability is . We further show that the core price of anarchy is at most . We also provide a tight analysis for unweighted graphs.

In the next subsection we emphasize the differences between MFHGs and fractional hedonic games.

1.3 Main Differences between MFHGs and Fractional Hedonic Games

Roughly speaking, we say that an outcome is a -strong Nash equilibrium if no subset of at most agents can jointly change their strategies in a way that all of the agents strictly improve their utility. It is easy to see that, for any , such that , a -strong Nash equilibrium is also a -strong Nash equilibrium. It is known that -strong Nash stable outcomes are not guaranteed to exist for fractional hedonic games, even for unweighted graphs [10]. In this paper we show that for MFHGs played on unweighted graphs, -strong Nash equilibrium always exists and can be computed in polynomial time, for any , where is the number of agents, and provide a tight analysis on the strong price of anarchy and stability.

For both MFHGs and Fractional Hedonic Games, Nash stable outcomes (or equivalently -strong Nash stable) are guaranteed to exist [9] for positive weights, but not for negative ones; moreover, the price of stability grows linearly with the number of agents. For fractional hedonic games played on unweighted graphs, it is known [10] that the price of stability is greater than even for simple graphs and that computing an optimum is NP-hard. For MFHGs we show that it is possible to compute in polynomial time a (strong) Nash equilibrium that is also optimum.

Finally, it is known that the core can be empty even for fractional hedonic games played on unweighted graphs and that it is NP-hard deciding the existence [12]. In this paper we show that for MFHGs the core is not empty for any graphs (this result was also observed in [1] for unweighted graphs), and that a core stable outcome can be computed in polynomial time. We further provide a tight and an almost tight analysis for the core price of stability and anarchy, respectively.

2 Preliminaries

For an integer , denote with the set .

We model a coalition formation game by means of a undirected graph. For an undirected edge-weighted graph , denote with the number of its nodes. For the sake of convenience, we adopt the notation and to denote the edge and its weight , respectively. Say that is unweighted if for each . We denote by , the sum of the weights of all the edges incident to . Moreover, let be the maximum edge-weight incident to . We will omit to specify when clear from the context. Given a set of edges , denote with the total weight of edges in . Given a subset of nodes , is the subgraph of induced by the set , i.e., .

Given an undirected edge-weighted graph , the modified fractional hedonic game induced by , denoted as , is the game in which each node is associated with an agent. We assume that agents are numbered from to and, for every , each agent chooses to join a certain coalition among candidate ones: the strategy of agent is an integer , meaning that agent is selecting candidate coalition . A coalition structure (also called outcome or partition) is a partition of the set of agents into coalitions such that for each , and for any with . Notice that, since the number of candidate coalitions is equal to the number of agents (nodes), some coalition may be empty. If , we say that agent is a member of the coalition . We denote by the coalition in of which agent is a member. In an outcome , the utility of agent is defined as . We notice that, for any possible outcome , we have that .

Each agent chooses the coalition she belongs to with the aim of maximizing her utility. We denote by , the new coalition structure obtained from by moving agent from to ; formally, . An agent deviates if she changes the coalition she belongs to. Given an outcome , an improving move (or simply a move) for agent is a deviation to any coalition that strictly increases her utility, i.e., . Moreover, agent performs a best-response in coalition by choosing a coalition providing her the highest possible utility (notice that a best-response is also a move when there exists a coalition such that ). An agent is stable if she cannot perform a move. An outcome is (pure) Nash stable (or a Nash equilibrium) if every agent is stable. An improving dynamics, or simply a dynamics, is a sequence of moves, while a best-response dynamics is a sequence of best-responses. A game has the finite improvement path property if it does not admit an improvement dynamics of infinite length. Clearly, a game possessing the finite improvement path property always admits a Nash stable outcome. We denote with the set of Nash stable outcomes of .

An outcome is a -strong Nash equilibrium if, for each obtained from , when a subset of at most agents (with ) jointly change (or deviate from) their strategies (not necessarily selecting the same candidate coalition), for some belonging to , that is, after the joint collective deviation, there always exists an agent in the set of deviating ones who does not improve her utility. We denote with the set of strong Nash stable outcomes of . We simply say that an outcome is a strong Nash equilibrium if is an -strong Nash equilibrium. It is easy to see that, for any graph and any , , while the vice versa does not in general hold. Clearly, . Analogously to the notion of Nash equilibrium, also for strong Nash equilibria it is possible to define a dynamics as a sequence of improving moves, where each move performed by agents in leading from outcome to outcome is such that all of them improve their utility, i.e. for every .

We say that a coalition strongly blocks an outcome , if each agent strictly prefers , i.e., strictly improve her utility with respect to her current coalition . An outcome that does not admit a strongly blocking coalition is called core stable and is said to be in the core. We denote with the core of .

The social welfare of a coalition structure is the summation of the agents’ utilities, i.e., . We overload the social welfare function by applying it also to single coalitions to obtain their contribution to the social welfare, i.e., for any , so that . It is worth noticing that, equivalently, for any , and .

Given a game , an optimum coalition structure is one that maximizes the social welfare of . The price of anarchy (resp. strong price of anarchy and core price of anarchy) of a modified fractional hedonic game is defined as the worst-case ratio between the social welfare of a social optimum outcome and that of a Nash equilibrium (resp. strong Nash equilibrium and core). Formally, for any , (resp. and ). Analogously, the price of stability (resp. strong price of stability and core price of stability) of is defined as the best-case ratio between the social welfare of a social optimum outcome and that of a Nash equilibrium (resp. strong Nash equilibrium and core). Formally, for any , (resp. and ). Clearly, for any game it holds that (resp. and ).

3 Strong Nash stable outcomes

In this section we consider strong Nash stable outcomes. We start by showing that even the existence of -strong nash equilibria is not guaranteed for non-negative edge-weights graphs.

Theorem 1.

There exists a star graph containing only non-negative edge-weights such that admits no -strong Nash stable outcome.

Figure 1: The star graph .
Proof.

Let be a star of order centred in as depicted in Figure 1. The weights of the edges are such that there exists a node leaf such that , while for all the other leafs we have that, , for any , and for small enough positive , (for instance ). First notice that, the grand coalition where all the agents belong to the same coalition, is not a -strong Nash stable outcome since, for instance, the two agents would both get strictly higher utility if they belong to a different coalition containing only them. On the other hand, any outcome where any leaf does not belong to the same coalition containing the center is even not Nash stable (i.e., -strong Nash stable), since would get utility zero but she can improve her utility by selecting the coalition containing the agent . Hence, the claim follows. ∎

Given the above negative result, in the remainder of this section, we focus on unweighted graphs.

Let , and be the unweighted cliques with , and nodes, respectively, i.e., is an isolated node, has nodes and a unique edge and is a triangle with edges. We say that a coalition being isomorphic to , or is a basic coalition.

3.1 Strong Price of Stability

In this subsection we show that, for unweighted graphs, it is possible to compute in polynomial time an optimum outcome and also a strong Nash outcome with the same social value. As consequence we get that the strong price of stability is .

In order to show how to compute in polynomial time an optimal solution, we first need some additional lemmata.

Lemma 2.

Given a coalition with , there exists an edge belonging to such that

Proof.

Let and be the number of edges and nodes in coalition , respectively. Moreover, let be the edge minimizing . Let us assume by contradiction that

By simple calculations, we obtain that

(1)

We denote by and the maximum and the minimum degrees of nodes in , respectively. We have

(2)
(3)

Substituting (2),  (3), in (1), the following holds:

a contradiction, because the maximum degree of a node is at most . ∎

We are now ready to prove the following theorem, showing that it is possible to consider, without decreasing the social welfare of the outcome, only coalition structures formed by basic coalitions.

Theorem 3.

For any coalition structure , there exists a coalition structure containing only basic coalitions and such that .

Figure 2: Possible coalitions with three nodes.
Proof.

Consider any coalition belonging to . In the following we show that either coalition is basic, or the nodes in can be partitioned in basic coalitions such that . This statement proves the claim because we can consider and sum up over all coalitions belonging to .

We prove the statement by induction on the number of nodes in .

The base of the induction is for : For and , is already a basic coalition. For , there are four possible configurations shown in Figure 2. For configurations (a), (b) and (c), again already is a basic coalition (or can be trivially divided in basic coalitions). For configuration (d), let the nodes in ; clearly, . Consider coalitions and . It is easy to check that .

As to the induction step, given any , assume now that the statement holds for ; we want to show that it also holds for .

By Lemma 2, we know that there exists an edge belonging to such that . Since , by the induction hypothesis, coalition can be decomposed in basic coalitions such that . Therefore, given that also is a basic coalition, we have proven the induction step. ∎

By Theorem 3, in order to compute an optimal solution for the coalition structure generation problem (i.e., an outcome maximizing the social welfare), it is possible to exploit a result from [22]:

Theorem 4 ([22]).

Given an unweighted graph , it is possible to compute in polynomial time a partition of the nodes of in sets inducing subgraphs isomorphic to , or (i.e., a coalition structure composed by basic coalitions) maximazing the number of nodes belonging to sets inducing subgraphs isomorphic to or .

In fact, by combining Theorems 3 and 4, it is possible to prove the following result.

Theorem 5.

Given an unweighted graph , there exists a polynomial time algorithm for computing a coalition structure maximizing the social welfare.

Proof.

By Theorem 3, there must exist an optimal outcome in which, for all , is a basic coalition. Notice that any node in a basic coalition isomorphic to does not contribute to the social welfare, while all nodes in other coalitions contribute to . It follows that, in order to maximize the social welfare, the number of nodes belonging to coalitions isomorphic to or has to be maximized, and therefore the solution computed in Theorem 4 is optimal also for our problem. ∎

In [23] the authors show that the price of stability of modified unweighted fractional hedonic games is , without considering complexity issues. The different characterization of the optimum done in Theorem 3 allows us to first compute in polynomial time an outcome that maximizes the social welfare (done in Theorem 5) and then to transform this optimal outcome into a strong Nash without worsening its social welfare, again by a polynomial time transformation. The following theorem completes this picture by providing a polynomial time algorithm for transforming an optical outcome into a strong Nash with the same social welfare, thus also proving that the strong price of stability is .

Theorem 6.

Given an unweighted graph , it is possible to compute in polynomial time an outcome and such that .

Proof.

Let be the optimal outcome computed in polynomial time by Theorem 5. Let the set of agents belonging in to coalitions isomorphic to or . Notice that . No agent in can have an incentive in changing her strategy (and thus can belong to any deviating subset of agents), because and a node can gain at most in any outcome. Therefore, if , then is also a strong Nash equilibrium and the claim directly follows.

In order to complete the proof, it is sufficient to (i) show the existence of a dynamics involving only the set of agents , where , and leading to a strong Nash outcome ; (ii) providing a polynomial time algorithm for computing .

For any , let be the set containing all coalitions of isomorphic to . We first provide some useful properties of nodes in :

  • For any couple of distinct nodes , edge , because otherwise the social welfare of could be improved by putting and in the same coalition: a contradiction to the optimality of .

    Figure 3: Proof of (P2).
  • For any and any vertex belonging to a coalition in , edge , because otherwise the social welfare of could be improved by removing from her current coalition and putting her in the same coalition of : a contradiction to the optimality of ( see Figure 3).

    Figure 4: Proof of (P3).
  • For any couple of distinct nodes and any coalition , if there exists an edge connecting node to a node in (assume without loss of generality to node , i.e. assume that ), then edge , because otherwise the social welfare of could be improved by removing and from their current coalition and putting them in the same coalition of and , respectively: a contradiction to the optimality of (see Figure 4).

    Figure 5: Proof of (P4).
  • For any couple of distinct nodes and any couple of coalitions , if there exist an edge connecting node to a node in (assume without loss of generality to node , i.e. assume that ), and another edge connecting node to a node in (assume without loss of generality to node , i.e. assume that ), then edge , because otherwise the social welfare of could be improved by removing , and from their current coalition and putting them in the same coalition of , and , respectively: a contradiction to the optimality of (see Figure 5).

Consider an initial dynamics, ending in outcome , in which every agent in unilaterally moves in order to increase her utility (that in is ). By properties (P1) and (P2) it follows that, for any , selects a coalition in and by property (P3) it follows that after this initial dynamics, all coalitions in (i.e., all coalitions modified by this initial dynamics) are isomorphic to star graphs, i.e. only one node has degree greater than .

Consider now a sequence of improving moves performed by any subset of agents and such that for any , agent improves her utility after this move. For any , let be the outcome reached after the -th move of this dynamics and be the set of moving agents. We want to show that this dynamics converges, i.e., that a strong Nash equilibrium is eventually reached.

By properties (P3) and (P4) it follows that:

  • For any coalition in , there exists an agent that will always have utility during any dynamics; let the set containing these nodes. Clearly, every agent in , as well as all nodes belonging to coalitions in , will never belong to a subset of nodes performing an improving move and therefore will always remain in the same coalition she belongs in .

  • For any , and any agent (potentially could be an agent of a coalition in or also an agent of a coalition in not belonging to ), is such that there exists a unique and will have a unique edge in connecting her to .

By properties (P5) and (P6), the only nodes participating in the dynamics are nodes either belonging to coalitions in or belonging to coalitions in but not belonging to ; let be the set of these nodes, i.e., for any , .

In order to obtain a strong Nash equilibrium, we notice that the “residual” game played by agents in is equivalent to a singleton congestion game with identical latency functions (CGI), in which we also have a set of resources (i.e. a strong Nash equilibrium in this new game is also a strong Nash equilibrium in our game and vice versa). In a CGI, agent’s strategy consists of a resource. The delay of a resource is given by the number of agents choosing it, and the cost that each agent aims at minimizing is the delay of her selected resource. In particular, the set of agents is and the set of resources is . In fact, in our “residual” game every agent aims at minimizing the cardinality of the star coalition she belongs to. In [21] it has been shown how to compute in polynomial time a strong Nash equilibrium for a class of congestion games including the one of CGI.

Let us call the obtained strong Nash equilibrium. It remains to show that . Observe that the difference between and is that some coalitions belonging to isomorphic to becomes a coalition isomorphic to a star graph in , and that some coalitions belonging to isomorphic to disappears in . The claim follows by noticing that the contribution to the social welfare of a coalition isomorphic to is zero, and that the contribution to the social welfare of a coalition isomorphic to (whose value is ) is the same as the one of a coalition isomorphic to a star graph. ∎

As a direct consequence of Theorem 6, the following corollary holds.

Corollary 7.

For any unweighted graph and any , .

3.2 Strong Price of Anarchy

In this subsection we study the strong price of anarchy for unweighted graphs.

Theorem 8.

Given any , there exists an unweighted graph such that .

Proof.

Figure 6: The graph .

Let us consider the graph depicted in Figure 6. The number of nodes in is . Specifically, we have agents in the first (upper) layer, and other agents in the second layer. Moreover, the nodes in the upper layer form a clique. It is easy to see that the optimum solution has social welfare at least . In fact, the coalitions structure composed by non-empty coalitions corresponding to the matchings between agents of the first and second layer, i.e., for any , has social welfare exactly . A strong Nash stable outcome is given by the coalition structure composed by two coalitions , where contains all the agents of the clique, while contains all the other agents. Indeed, on the one hand, all the agents belonging to the coalition get utility that is the maximum one they can get, which means that they do not have any interest on deviating from . Therefore, suppose by contradiction that is not strong Nash stable, then the set of deviating agents must be a subset of the agents belonging to the coalition . However, by using the fact that the two non-empty coalitions and of contains the same number of agents, it is easy to see any subset of agents of cannot jointly deviate and all get higher utility with respect to . It follows that is a strong Nash stable outcome. Since , it follows that . ∎

Theorem 9.

For any unweighted graph , .

Proof.

Let the optimal solution computed by Theorem 5, in which all coalitions are basic ones.

Consider any -strong Nash equilibrium .

For any coalition of isomorphic to , on the one hand we have that . On the other hand, since is a -strong Nash stable outcome, or , because otherwise and could jointly perform an improving move. Thus, , whereas .

For any coalition of isomorphic to , on the one hand we have that . On the other hand, since is a -strong Nash stable outcome, at least agents among must have utility in , because otherwise there would exist two agents aiming at jointly perform an improving move: a contradiction to the -strong Nash stability of . Thus, , whereas .

For any , let be such that for any , is isomorphic to . Since agents being in coalitions of isomorphic to do not contribute to , we obtain

From Theorems 8 and 9, we immediately get the following result.

Corollary 10.

The strong price of anarchy for unweighted graphs is .

4 Nash stable outcomes

In this section we consider Nash stable outcomes. We start by showing that there exists a graph containing negative edge-weights such that the game induced by admits no Nash stable outcome. This result is very similar to Lemma of [9].

Theorem 11.

There exists a graph G containing edges with negative weights such that admits no Nash stable outcome.

Figure 7: The graph .
Proof.

Let be the graph in Figure 7 and fix a Nash stable outcome . It is easy to see that, for small enough, agents and cannot belong to the same coalition. By contrast, agents and must belong to the same coalition since otherwise the utility of would be zero. Let be the coalition containing agents and . If , then agent wants to join the coalition and improve her utility from zero to thus contradicting the fact that is Nash stable. If , then, since agents and cannot belong to the same coalition, it must be . Moreover, there exists a coalition containing exactly one between the two agents and . Hence, we get the utility of agent in is , while is the utility in joining coalition , which rises again a contradiction. Since all possibilities for have been considered, it follows that a Nash stable outcome cannot exist. ∎

We further show that there exists a dynamic of infinite length for games played on unweighted graphs.

Theorem 12.

There exists an unweighted graph such that does not possess the finite improvement path property, even under best-response dynamics.

Figure 8: The graph .
Proof.

Let us consider the game induced by be the graph depicted in Figure 8. Let us analyze the dynamics that starts from the coalitions structure , where agents are together in a coalition, and agent is alone in another one. It is not difficult to check that, if the agents perform their unique (best) improving moves in the following exact ordering , we get back to the starting coalitions structure . ∎

Despite the above negative results, it is easy to see that, if a graph does not contain negative edge-weights, then the game induced by admits a Nash equilibrium, that is the outcome where all the agents are in the same coalition. Therefore, in the next subsections we characterize the efficiency of Nash stable outcomes in modified fractional hedonic games played on general graphs with non-negative edge-weights.

By definition, we have that .

4.1 Price of Anarchy

We first show that the price of anarchy grows linearly with the number of agents, even for the special case of unweighted paths.

Theorem 13.

There exists an unweighted path such that .

Proof.

Let be an unweighted simple path with an even number of nodes. Notice that, since in this setting the utility of an agent in any outcome is at most , the optimum solution is given by a perfect matching, that is, . However, when all the nodes are in the same coalition, we obtain a Nash stable outcome such that . Hence, the claim follows. ∎

We are able to show an asymptotically matching upper bound, holding for weighted (positive) graphs.

Theorem 14.

For any weighted graph with non-negative edge-weights , .

Proof.

We notice that in any Nash equilibrium , any agent has utility , since agent can always join the coalition containing the agent , where . On the other hand, in the optimal outcome , we have that any agent has utility such that . Hence, by summing over all agents, the theorem follows. ∎

4.2 Price of Stability

On the one hand, since we have proved in Corollary 7 that, for the setting of unweighted graphs, the strong price of stability is , it directly follows that also the price of stability is in this setting, because any strong Nash equilibrium is also a Nash equilibrium.

On the other hand, in the weighted case, given the upper bound to the price of anarchy provided in Theorem 14, the following theorem shows an asymptotically matching lower bound to the price of stability.

Theorem 15.

There exists a weighted star with non-negative edge weights such that .

Figure 9: The star graph .
Proof.

Let be a star with nodes centred in as depicted in Figure 9. The weights of the edges are such that there exists a leaf node such that , while for all the other leaf nodes we have , for an arbitrarily small , (for instance ).

Notice that the grand coalition (i.e., the outcome in which all agents belong to the same coalition) is the unique Nash stable outcome and has social welfare equal to . In fact, in any Nash equilibrium, all the leafs must be in the coalition together with the center . On the other hand, the coalition containing only agents and yields a social value of , and thus the theorem follows. ∎

5 Core stable outcomes

We first show that the strict core of could be empty, even if is unweighted.

Theorem 16.

There exists an unweighted graph such that .

Proof.

Let be a path with nodes .

If , is a blocking coalition. In fact, moving from their coalition in to coalition , both and increase their utility form to .

If , is a weakly blocking coalition. In fact, moving from their coalition in to coalition , increases her utility form to and does not change her utility. The case is symmetric.

Finally, if , is a weakly blocking coalition. In fact, moving from their coalition in to coalition , increases her utility form to and does not change her utility.

Since all possibilities for have been considered, it follows that a strict core stable coalition does not exist. ∎

Given the negative result of Theorem 16 concerning the strict core of modified fractional hedonic games, in the following we focus on the core of this games.

In this section we consider the core of MFHGs. We first show that for any graph , the core of the game in not empty, and that a core stable outcome approximating the optimal social welfare by a factor of can be computed in polynomial time.

Theorem 17.

Given any graph , there exists a polynomial time algorithm for computing a core stable coalition structure such that and all coalitions in are of cardinality at most .

Proof.

Consider the following algorithm, working in phases . Let be the subgraph of such that , that is, has the same vertices as and only contains the edges of of non-negative weight. For any , let be the graph obtained after phase . In any phase , a new coalition isomorphic to is added to as follows: Let be an edge in of maximum weight . We add to the coalition formed by and , i.e., . Moreover, let