Research Paper AAMAS 2020, May 9–13, Auckland, New Zealand

Pareto-Optimality in Cardinal Hedonic Games

Martin Bullinger Technische Universität München München, Germany [email protected]

ABSTRACT number of agents, and therefore it is desirable to consider expres- Pareto-optimality and individual rationality are among the most nat- sive, but succinctly representable classes of hedonic games. This ural requirements in coalition formation. We study classes of hedo- can be established by encoding preferences by means of a complete nic games with cardinal that can be succinctly represented directed weighted graph where an edge weight 푣푥 (푦) is a cardinal by means of complete weighted graphs, namely additively separable value or that agent 푥 assigns to agent 푦. Still, this underlying (ASHG), fractional (FHG), and modified fractional (MFHG) hedo- structure leaves significant freedom, how to obtain (cardinal) pref- nic games. Each of these can model different aspects of dividing a erences over coalitions. An especially appealing type of preferences society into groups. For all classes of games, we give algorithms is to have the sum of values of the other individuals as the value that find Pareto-optimal partitions under some natural restrictions. of a coalition. This constitutes the important class of additively While the output is also individually rational for modified fractional separable hedonic games (ASHGs) [6]. hedonic games, combining both notions is NP-hard for symmetric In ASHGs, an agent is willing to accept an additional agent into ASHGs and FHGs. In addition, we prove that welfare-optimal and her coalition as long as her valuation of this agent is non-negative Pareto-optimal partitions coincide for simple, symmetric MFHGs, and in this sense, ASHGs are not sensitive to the intensities of single- solving an open problem from Elkind et al. [9]. For general MFHGs, agent preferences. In particular, if all valuations are non-negative, our algorithm returns a 2-approximation to welfare. Interestingly, forming the grand coalition consisting of all agents is a best choice welfare-optimal partitions in MFHGs only require coalitions of at for every agent. In contrast, fractional hedonic games (FHGs) define most three agents. preferences over coalitions by dividing the sum of values by the size of the coalition [1]. This incentivizes agents to form dense cliques, KEYWORDS and therefore appropriately models like-mindedness in the sense of clustering problems. Coalition Formation; Hedonic Games; Pareto-optimality; Welfare On the other hand, agents in FHGs may improve with a new ACM Reference Format: agent whose valuation is below the average of their current coalition Martin Bullinger. 2020. Pareto-Optimality in Cardinal Hedonic Games. In partners. To avoid this, it is also natural to define the value ofa Proc. of the 19th International Conference on Autonomous Agents and Mul- coalition as the average value of other agents, i.e., the denominator tiagent Systems (AAMAS 2020), Auckland, New Zealand, May 9–13, 2020, in the definition of FHGs is replaced by the size of the coalition IFAAMAS, 9 pages. minus 1. This defines the class of modified fractional hedonic games (MFHGs) [17]. For all measures of stability and optimality that have 1 INTRODUCTION been investigated, these games do not guarantee the formation of Coalition formation is a central problem in multi-agent systems large cliques and are therefore less suitable for clustering problems. and has been extensively studied, ever since the publication of In fact, partitions of agents into coalitions satisfying many desirable von Neumann and Morgenstern’s Theory of Games and Economic properties can be computed in polynomial time, often simply by Behavior in 1944. In coalition formation, every agent of a group computing maximum (weight) matchings. However, these games seeks to be in a desirable coalition. As an important special case ensure a certain degree of homogeneity of the agents, as agents tend therein, in clustering problems, a society is observed, for which to contribute uniformly to each other’s utility in stable coalitions. a structuring into like-minded groups, or communities, is to be Having selected a representation for modeling the coalition for- identified [16]. Coalition scenarios can be modeled by letting the mation process, one needs a measure for evaluating the quality of a agents submit preferences, subject to which the happiness of an partition. Various such measures, also called solution concepts, have individual agent with her coalition, or the like-mindedness of a been proposed in the literature. Most of them aim to guarantee coalition, can be measured. The goal of every individual agent is to a certain degree of stability—preventing single agents or groups maximize the value of her coalition. from agents to break apart from their coalitions—and optimality— In many settings, it is natural to assume that an agent is only guaranteeing a globally measured outcome that is good for the concerned about her own coalition, i.e., externalities are ignored. society as entity. A good overview of solution concepts is by Aziz As a consequence, much of the research on coalition formation and Savani [5]. The most undisputed measure of optimality is Pareto- now concentrates on these so-called hedonic games [7]. Still, the optimality, i.e., there should be no other partition, such that every number of coalitions an agent can be part of is exponential in the agent is weakly, and some agent strictly, better off. Apart from its optimality guarantee, Pareto-optimality can also be seen as a Proc. of the 19th International Conference on Autonomous Agents and Multiagent Systems measure of stability, because a Pareto-optimal partition disallows (AAMAS 2020), B. An, N. Yorke-Smith, A. El Fallah Seghrouchni, G. Sukthankar (eds.), May an agent to propose a partition she prefers without having another 9–13, 2020, Auckland, New Zealand. © 2020 International Foundation for Autonomous Agents and Multiagent Systems (www.ifaamas.org). All rights reserved. agent vetoing this proposal. A stronger notion of optimality is that

213 Research Paper AAMAS 2020, May 9–13, Auckland, New Zealand

of (utilitarian) welfare-optimality, which aims to maximize the sum of hedonic games that interpolates between two profiles of utilities of all agents. and can contain games not implementable as an ASHG or FHG, Pareto-optimal outcomes still might be extremely disadvanta- respectively. In fact, for ASHGs and FHGs, perfect partitions can be geous for single agents that receive large negative utility in or- computed in polynomial time (Theorem 5.4, Theorem 6.6), while der to give small positive utility to another agent. Therefore, it is computing Pareto-optimal and individually rational partitions is also desirable that agents receive at least the utility they would NP-hard (Theorem 5.2, Theorem 6.4). receive in a coalition of their own (in all our models this means For cardinal hedonic games, stability and optimality have been non-negative utility). This condition is called individual rationality. studied to some extent. Gairing and Savani [10, 11] settled the Clearly, a Pareto-optimal and individually rational outcome can complexity of the individual-player stability notions of Drèze and always be found by the simple local search algorithm that starts Greenberg for symmetric ASHGs by treating them as local search with the partition of the agents into singleton coalitions and moves problems. For FHGs, hardness and approximation results for wel- to Pareto-improvements as long as these exist. In general, this basic fare are given by Aziz et al. [4], while Aziz et al. [1] study stability. algorithm need not run in polynomial time and the output need Monaco et al. [15] show the tractability of some stability notions not be welfare-optimal. It can even occur that no welfare-optimal and welfare-optimality for simple symmetric MFHGs, and the com- partition is individually rational. As we will see, it is even often putability of partitions in the for weighted MFHGs. The ap- NP-hard to compute a Pareto-optimal and individually rational proximation results for FHGs and most of the positive results for partition (Theorem 5.2, Theorem 6.4). MFHGs rely on computing specific matchings. We study Pareto-optimality in all three classes of games, and Pareto-optimality for cardinal hedonic games was mainly studied give polynomial-time algorithms for computing Pareto-optimal by Elkind et al. [9] in terms of the price of Pareto-optimality (PPO), partitions in important subclasses, including symmetric ASHGs a worst-case ratio of Pareto-optimal and welfare-optimal partitions. and MFHGs. In addition, we prove that welfare-optimal and Pareto- The focus lies on simple symmetric graphs and their main result is optimal partitions coincide for simple symmetric MFHGs, closing to bound the PPO between 1 and 2 for simple, symmetric MFHGs. the bounds on the price of Pareto-optimality for this class of games Since we show that, for these games, every Pareto-optimal parti- left by Elkind et al. [9]. In the weighted case, our algorithm for tion is welfare-optimal, we will close this gap. Pareto-optimality Pareto-optimality in MFHGs gives a 2-approximation of welfare for ASHGs was considered by Aziz et al. [3]. However, they only and its output is always individually rational. While we can prove dealt with a restricted class of preferences that guarantees unique that even in the weighted case, welfare-optimality is attained by top-ranked coalitions and therefore one can apply a simple serial a partition consisting only of coalitions of size two and three, the dictatorship algorithm. complexity of computing a welfare-optimal partition remains open. On the other hand, we prove that computing a Pareto-optimal and 3 PRELIMINARIES individually rational partition is NP-hard for symmetric ASHGs and FHGs, thus extending a result by Aziz et al. [3] for general ASHGs. The primary ingredient of our model is a set of agents 푁 that Note that symmetry is a significant restriction for hardness reduc- assign hedonic preferences over partitions of 푁 (also called coalition tions, because non-symmetric games allow for the phenomenon of structures), where the only information of a partition an agent non-mutual interest. is interested in, is her own coalition. Preferences of agent 푖 are therefore given over N푖 = {퐶 ⊆ 푁 : 푖 ∈ 퐶}, i.e., the subsets of agents including herself, by valuation functions 푣푖 : N푖 → R. We 2 RELATED WORK investigate a partition 휋 of the agents for notions of optimality and Hedonic games were first introduced by Drèze and Greenberg [7]. stability, most importantly Pareto-optimality. Given a partition 휋, Since then, a great amount of research has been devoted to the study we denote by 휋 (푖) the partition of agent 푖 and the utility she received ′ of algorithmic and mathematical properties of axiomatic concepts from this partition by 푣푖 (휋) = 푣푖 (휋 (푖)). A partition 휋 is a Pareto- ′ regarding stability and optimality, representability of preferences, improvement over 휋 if, for all agents 푖 ∈ 푁 , 푣푖 (휋 ) ≥ 푣푖 (휋) and ′ and the discovery of well-behaved, yet expressive classes of hedonic there exists an agent 푗 ∈ 푁 with 푣 푗 (휋 ) > 푣 푗 (휋). In this case, we games. The survey by Hajduková [12] gives a critical overview also say that 휋 ′ Pareto-dominates 휋. A partition 휋 is Pareto-optimal of preference representations and conditions that allow for the if it is not Pareto-dominated. existence and efficient computability of central stability notions, A stronger requirement is that of (utilitarian) welfare-optimality. Í such as Nash and core stability. For a subset 푀 ⊆ 푁 of agents, we denote 푣푀 (휋) = 푖 ∈푀 푣푖 (휋). Pareto-optimality can be studied in many classes of hedo- The social welfare of a coalition 퐶 is SW(퐶) = 푣퐶 (휋). The social Í Í nic games by exploiting a strong relationship between Pareto- welfare of a partition 휋 is SW(휋) = 퐶 ∈휋 SW(퐶) = 푖 ∈푁 푣푖 (휋). optimality and perfection, i.e., partitions that put every agent in one A partition 휋 is called welfare-optimal if it maximizes the function of her most preferred partitions [2]. This gives rise to the preference SW amongst all partitions of agents. Welfare-optimal partitions refinement algorithm (PRA) which finds Pareto-optimal partitions are Pareto-optimal. under certain conditions by means of a perfection-oracle. The re- A partition 휋 is individually rational for agent 푖 if 푣푖 (휋) ≥ 푣푖 ({푖}), sulting Pareto-optimal partitions are even individually rational. i.e., agent 푖 does not prefer to stay alone. In addition, 휋 is individually The assumptions required for the algorithm include the efficient rational if it is individually rational for every agent. Partitions that computation of preference refinements, which is not possible for are welfare-optimal or individually rational and Pareto-optimal ASHGs and FHGs. Indeed, during the PRA, one has to search a set always exist.

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Preferences are succinctly represented by a family of cardinal exploited by Monaco et al. [15] for computing welfare-optimal utility functions (푣푖 )푖 ∈푁 where 푣푖 : 푁 → R with 푣푖 (푖) = 0 that partitions. can be aggregated to preferences over coalitions. A natural repre- Theorem 4.2. MaxCliqueMatching is equivalent to finding a sentation is by means of a complete, directed, and weighted graph Pareto-optimal partition on a symmetric MFHG (under Turing reduc- 퐺 = (푁, 퐸, 푣) where the weights are defined by the utility func- tions). Moreover, if we can solve MaxCliqueMatching in polynomial tions. A game is called symmetric if, for all pairs of agents 푖, 푗 ∈ 푁 , time, we can even compute a Pareto-optimal and individually rational 푣 (푗) = 푣 (푖). In this case, the underlying graph is symmetric and 푖 푗 partition for a symmetric MFHG in polynomial time. we denote 푣 (푒) = 푣 (푖, 푗) = 푣푖 (푗) = 푣 푗 (푖) for a 2-elementary set of agents 푒 = {푖, 푗}. A game is called simple if 푣푖 (푗) ∈ {0, 1} for Proof. Assume first that we are given an algorithm tofind all agents 푖, 푗 ∈ 푁 . A hedonic game with simple and symmetric a Pareto-optimal partition on a symmetric MFHG and let 퐺 = preferences can therefore be represented by an unweighted and (푉, 퐸) be an instance of MaxCliqueMatching. We transform 퐺 into undirected (but incomplete) graph. an MFHG with the underlying weighted symmetric graph 퐺 ′ = We define the aggregated utilities for ASHGs, FHGs, and MFHGs (푉, 퐸′, 푣) where 퐸′ = {푒 ⊆ 푉 : |푒| = 2} and for partition 휋 and agent 푖 by ( 1 if 푒 ∈ 퐸 퐴푆퐻퐺 Õ 푣 (푒) = 푣푖 (휋) = 푣푖 (푗) −Δ − 1 else, ∈ ( ) 푗 휋 푖 where Δ is the maximum degree of a vertex in 퐺. 퐴푆퐻퐺 ( ) 퐹퐻퐺 푣푖 휋 Let 휋 be a Pareto-optimal partition of vertices into coalitions 푣 (휋) = , and ′ 푖 |휋 (푖)| for the symmetric MFHG with underlying graph 퐺 . Define C = ( 푣퐴푆퐻퐺 (휋) {푃 ∈ 휋 : |푃 | ≥ 2}. Then, C consists of cliques in 퐺. Otherwise, by 푖 ( ) ≠ { } 푀퐹퐻퐺 |휋 (푖) |−1 for 휋 푖 푖 construction of the utilities, there is one agent who receives negative 푣푖 (휋) = 0 for 휋 (푖) = {푖}. utility. Assume for contradiction that there exists a coalition 푃 ∈ C such that some agents in 푃 receive negative utility. Let 푆 = {푝 ∈ If it is clear from the context which game is considered, we omit 푃 : 푣 (푝, 푝′) = 1 for all 푝′ ∈ 푃\{푝}} be the set of agents that receive the superscripts of the utility functions. non-negative utility from all other agents in 푃. Since some agents We use the following notation from graph theory. For an arbi- receive negative utility, there exists an agent 푞 ∈ 푃\푆. But then, trary graph 퐺 = (푉, 퐸), a vertex set 푊 ⊆ 푉 and an edge set 퐹 ⊆ 퐸, the coalition 푆 = 푆 ∪ {푞} forms a clique in 퐺 and the partition denote by 퐺 [푊 ] and 퐺 [퐹] the subgraph of 퐺 induced by 푊 and 퐹, 푞 휋 ′ = (휋\{푃}) ∪ {푆 } ∪ {{푝′} : 푝′ ∈ 푃\푆 } is a Pareto improvement respectively, and denote by 퐸(퐺) its edge set. 푞 푞 over 휋. Hence, 휋 consists only of cliques and, by design of the MFHG 4 MODIFIED FRACTIONAL HEDONIC GAMES utilities, assigns utility 1 to agents in a clique of size at least 2, and 0 to agents in singleton coalitions. Consequently, C is inclusion- In this section we focus on symmetric MFHGs. The analysis of maximal w.r.t. vertices, because every clique matching that covers Pareto-optimality on this class of games relies on an extension of strictly more agents gives rise to a Pareto-improvement that assigns maximum matchings to cliques. Given a graph, a set C of vertex- utility 1 to a strict superset of agents that already receive utility 1. disjoint cliques, each of size at least 2, is called a clique match- Hence, we can solve MaxCliqueMatching by computing a Pareto- ing. A vertex that is part of any clique in C is called covered or optimal partition of the MFHG induced by 퐺 ′. matched. We are interested in clique matchings that cover a maxi- Conversely, assume that we can solve MaxCliqueMatching. Con- mum number of vertices. We call the corresponding search problem sider a symmetric MFHG induced by a complete weighted graph MaxCliqueMatching. Interestingly, a clique matching is maximum 퐺 = (푁, 퐸, 푣). Algorithm 1 computes a Pareto-optimal partition if and only if it is inclusion-maximal w.r.t. vertices, i.e., there ex- in polynomial time given an algorithm MaxCliqueMatching that ists no other clique matching covering a strict superset of vertices computes a vertex-maximal clique matching in polynomial time. (Theorem 4.5), and can be computed in polynomial time. We can The idea is to restrict attention to the unweighted subgraph induced further simplify the analysis to the special case that only triangles by edges with the largest positive weight still available. and edges are allowed. The running time of the algorithm is clearly polynomial, includ- Given a graph, a set of vertex disjoint cliques, each of size ing polynomially many calls of MaxCliqueMatching. We prove its 2 or 3, is called a 3-clique matching. By splitting larger cliques, correctness. Let 휋 be the output of the algorithm. First, note that MaxCliqueMatching is equivalent to Max3CliqueMatching, i.e., all non-singleton coalitions are cliques in 퐺 with identical positive finding a maximum 3-clique matching. Given a 3-clique match- utility within each clique. Hence, the output is individually rational. ing C, we denote by 푀 (C) and 푇 (C) its cliques of size 2 (edges) For the proof of Pareto optimality, we assume that the while loop and size 3 (triangles), respectively. = S ∪ Ð푚 C C took 푚 iterations and we subdivide 휋 푘=1 푘 , where 푘 is S Theorem 4.1 (Hell and Kirkpatrick [13]). The problem the clique matching in iteration 푘, and consists of the singleton Max3CliqueMatching can be solved in polynomial time. coalitions that are added to 휋 after the while loop. We will show by induction over 푚 that if the algorithm uses 푚 iterations of the while We prove that MaxCliqueMatching is equivalent to finding a loop, then the output is Pareto-optimal. Let 휋 ′ be any coalition such ′ Pareto-optimal partition on an MFHG. Note that a relationship be- that, for all agents 푖 ∈ 푁 , 푣푖 (휋) ≤ 푣푖 (휋 ). We will prove that this ′ tween clique matchings and simple symmetric MFHGs was already implies, for all agents 푖 ∈ 푁 , 푣푖 (휋) = 푣푖 (휋 ).

215 Research Paper AAMAS 2020, May 9–13, Auckland, New Zealand

Input: Symmetric MFHG induced by graph 퐺 = (푁, 퐸, 푣) Input: Coalition 퐶 ∈ 휋∗ Output: Pareto-optimal and individually rational partition 휋 Output: Order (푐1, . . . , 푐푘 ) of 퐶 and weights (푤1, . . . ,푤푘 )

휋 ← ∅, 퐴 ← 푁 , 퐺푟 ← 퐺 [{푒 ∈ 퐸 : 푣 (푒) > 0}] 퐻 ← 퐶, 퐹 ← {{푥,푦} ∈ 퐸 : 푥,푦 ∈ 퐶, 푣 (푥,푦) > 0}, 푗 ← 1 while 퐸(퐺푟 ) ≠ ∅ do while 퐹 ≠ ∅ do 푣max ← max{푣 (푒) : 푒 ∈ 퐸(퐺푟 )} 푣max ← max{푣 (푒) : 푒 ∈ 퐹 } 퐸퐻 ← {푒 ∈ 퐸(퐺푟 ) : 푣 (푒) = 푣max} Choose {푥,푦} ∈ argmax{푣 (푥,푦) : {푥,푦} ∈ 퐹 } with 푣푥 (휋) ≥ 퐻 ← 퐺푟 [퐸퐻 ] 푣 (푥,푦) C ← MaxCliqueMatching(퐻) 푐 푗 ← 푥, 푤 푗 ← 푣max 휋 ← 휋 ∪ C 퐻 ← 퐻\{푥}, 퐹 ← {푒 ∈ 퐹 : 푥 ∉ 푒}, 푗 ← 푗 + 1 퐴 ← {푎 ∈ 퐴: 푎 not covered by C} Order 퐻 = (푐 푗, . . . , 푐푘 ) arbitrarily, 푤푖 ← 0 for 푖 = 푗, . . . , 푘 퐺푟 ← 퐺푟 [퐴] return (푐1, . . . , 푐푘 ), (푤1, . . . ,푤푘 ) return 휋 ∪ {{푎} : 푎 ∈ 퐴} Algorithm 2: Special ordering for weight distribution Algorithm 1: Pareto-optimal partition of a sym. MFHG

푣 (푐푖, 푐 푗 ) ≤ 푤 푗 and for all 푗 > 푖, 푣 (푐푖, 푐 푗 ) ≤ 푤푖 . Hence, If 푚 = 0, 퐺 contains no edges of positive weight and there- ( ) fore, for all agents 푖 ∈ 푁 , 푣 (휋) = 0 ≥ 푣 (휋 ′). For the induction ∗ Õ 푣 푐푖, 푐 푗 Õ 푤푖 Õ 푤 푗 푖 푖 푣푐 (휋 ) = ≤ + 푖 푘 − 1 푘 − 1 푘 − 1 step, let 푚 ≥ 1. Let 퐻 be the auxiliary graph of the first while 푗≠푖 푗>푖 푗<푖 loop. Note that within 휋, agents in C1 can only be matched with Õ 푤 Õ 푤 푗 Õ 푤 푗 Õ 푤 푗 ≤ 푖 + = 푤 + ≤ 푣 (휋) + . agents in 퐻 since they receive the highest possible utility of any 푘 − 푖 푘 − 푗 푖 푘 − 푗 푐푖 푘 − 푗 agent in any coalition in 퐺 and every other agent diminishes their 푗>푖 푗<푖 푗<푖 푗<푖 MFHG utility. In particular, they cannot be better off. Since C1 is We infer that a vertex-maximal clique matching on 퐻, no agent in 퐻 not cov- 푘 푘 ′ Õ Õ Õ 푤 푗 ered by C1 can be in a coalition with an agent in C1 in 휋 . Define 푣 (휋∗) = 푣 (휋∗) ≤ ©푣 (휋) + ª 퐶 푐푖 ­ 푐푖 푘 − 푗 ® 푊 = {푖 ∈ 푁 : 푖 not covered by C1} and consider 퐺ˆ = 퐺 [푊 ]. Then 푖=1 푖=1 푗<푖 = S ∪ Ð푚−1 C ˆ « ¬ 휋ˆ 푘=1 푘+1 is a possible outcome of the algorithm for 퐺. 푘 푘 ′ Õ Õ 푤 푗 Õ 푤 푗 Furthermore, 휋 restricted to agents in 푊 is weakly better for any = 푣 (휋) + = 푣 (휋) + (푘 − 푗) ≤ 2푣 (휋). 퐶 푘 − 푗 퐶 푘 − 푗 퐶 agent in 푊 than 휋ˆ. Hence, by induction, also no agent outside C1 푖=1 푗<푖 푗=1 can be better off. □ □ Even though Pareto-optimal outcomes may be worse than Note that the approximation guarantee of the theorem extends welfare-optimal outcomes by an arbitrarily large factor, the output to the case of non-symmetric weights, because the symmetrization of the above algorithm guarantees significant social welfare. ′ 1 푣 (푥,푦) = 2 (푣푥 (푦) + 푣푦 (푥)) preserves the welfare. Theorem 4.3. Let a symmetric MFHG be given. Let 휋 be the par- Moreover, the factor of 2 is the best possible approximation tition computed by Algorithm 1 for this game. Then, for any welfare- guarantee of Algorithm 1. Let 휖 > 0 and the complete graph on ∗ ∗ optimal partition 휋 , it holds that 2SW(휋) ≥ SW(휋 ). vertex set 푉 = {푤, 푥,푦, 푧} be given with weights as Proof. Consider a symmetric MFHG induced by graph 퐺 = 1 + 휖 if 푒 = {푥,푦} (푁, 퐸, 푣). Let 휋 be a partition computed by Algorithm 1 for this  game and let 휋∗ be welfare-optimal. We will show that, for any 푣 (푒) = 1 if 푒 ∈ {{푤, 푥}, {푦, 푧}}  coalition 퐶 ∈ 휋∗, it holds that 2푣 (휋) ≥ 푣 (휋∗), which implies the 0 else. 퐶 퐶  assertion. 푤 푥 푦 푧 Before we prove this, we make the observation that, for each 1 1 + 휖 1 edge {푥,푦} = 푒 ∈ 퐸, it holds that 푣 (휋) ≥ 푣 (푥,푦) or 푣 (휋) ≥ 푥 푦 Then, the output of Algorithm 1 is 휋 = {{푥,푦}, {푤}, {푧}} with 푣 (푥,푦). Indeed, if 푣 (푥,푦) ≤ 0, then this follows from the individual SW(휋) = 2 + 2휖 while 휋∗ = {{푤, 푥}, {푦, 푧}} is welfare-optimal rationality of 휋. If we reach a maximum weight 푣 ≤ 푣 (푥,푦) max with SW(휋∗) = 4. in the while loop and 푥 ∉ 퐺 or 푦 ∉ 퐺 then 푣 (휋) > 푣 (푥,푦) or 푟 푟 푥 In the special case of simple symmetric games, the output is 푣 (휋) > 푣 (푥,푦). Otherwise, we reach an iteration where 푣 = 푦 max welfare-optimal. The proof uses the characterization of Pareto- 푣 (푥,푦) and any maximum clique matching matches at least one of optimal partitions by Elkind et al. [9, Lemma 15] that have observed them. that all coalitions in such partitions are stars or cliques. The proof Now let any coalition 퐶 ∈ 휋∗ be given where |퐶| = 푘. The next shows that some welfare-optimal partition is a clique matching and step is to sort the agents in 퐶 by means of Algorithm 2. The result- therefore, all maximum clique matchings are welfare-optimal. We ing order places the agents essentially in decreasing value 푤 of 푖 omit the proof due to space restrictions. an incident edge of high utility. Let (푐1, . . . , 푐푘 ), (푤1, . . . ,푤푘 ) be an outcome of this algorithm. We claim that for every 푖 ∈ {1, . . . , 푘}, Theorem 4.4. Let a simple symmetric MFHG be given. Let 휋 be ( ∗) ≤ ( ) + Í 푤푗 ≤ 푣푐푖 휋 푣푐푖 휋 1≤푗<푖 푘−푗 . Note that for all 1 푗 < 푖, a clique matching of the underlying unweighted graph. Then, 휋 is

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welfare-optimal if and only if it is a maximum clique matching. In Proof sketch. Let an MFHG induced by a unweighted graph particular, the output of Algorithm 1 is welfare-optimal. 퐺 = (푉, 퐸) be given. Clearly, welfare-optimal partitions are Pareto- optimal. As we have seen in the above example, Pareto-optimal partitions For the reverse implication, let 휋 be a Pareto-optimal partition need not be welfare-optimal. However, this was shown by Elkind which consists of cliques and stars only [9, Lemma 15]. By splitting et al. [9] for simple, symmetric MFHGs induced by a bipartite graph, larger cliques, we may assume that all cliques are of size 2 and 3 and we will extend their result to simple symmetric MFHGs. While (splitting the cliques yields a partition with identical utilities). Let their results rely on estimating the welfare of partitions in terms of S ⊆ 휋 be its star-coalitions. For any star 푆 with 푘 ≥ 2 leaves, let 푐푆 minimum vertex covers, we will exploit a combinatorial description 푆 be its center and 푙푆, . . . , 푙푆 be its leaves. We define a new partition of Max3CliqueMatching. We will develop it using terminology 1 푘푆 휋 ′ = {퐶 ∈ 휋 : 퐶 ∉ S} ∪ {{푐푆, 푙푆 }, {푙푆 },..., {푙푆 } : 푆 ∈ S}. Then, closely related to the famous blossom algorithm by Edmonds [8] 1 2 푘푆 to show the close relationship of computing maximum cardinality SW(휋 ′) = SW(휋). The core of the proof is that 휋 ′ is a maximum matchings and maximum size 3-clique matchings. clique matching and therefore by Theorem 4.4 welfare-optimal. {{ 푆 푆 } ∈ S} The blossom algorithm deals with odd cycles by finding sub- Define the set 푀S = 푐 , 푙1 : 푆 , i.e., the set of edges that ′ ′ graphs called flowers. Let a graph 퐺 = (푉, 퐸) together with a match- are added to the partition 휋 . Then, 푀 (휋 ) = 푀(휋) ∪ 푀S is the set ing 푀 be given. A path is called 푀-alternating if it alternately of 2-cliques of the 3-clique matching 휋 ′. uses edges of 푀 and outside 푀. An 푀-augmenting path is an 푀- We will prove that the conditions of the combinatorial character- alternating path starting and ending at vertices not covered by 푀. ization of Theorem 4.5 are satisfied. First, assume that there exists An 푀-stem is an 푀-alternating path of even length. The uncovered an 푀 (휋 ′)-augmenting path 푃. An illustration of this step is given endpoint is its root, the covered endpoint is its tip. An 푀-blossom after the proof with the aid of Figure 2. is an odd cycle 퐶 = (퐵, 퐸퐵) of 퐺 such that all vertices except one For 푒 ∈ 푀S, we denote by 푆푒 the star which the edge 푒 originates 푆 are covered by 푀 ∩ 퐸퐵. Let 퐶 = (퐵, 퐸퐵) be an 푀-blossom such that from. An edge 푒 ∈ 푀S ∩ 푃 is called exterior if 푐 푒 , the center vertex 푏 ∈ 퐵 is uncovered and let 푏1,푏2 ∈ 퐵 be the neighbors of 푏 on 퐶. of 푆푒 , is the second or second-last vertex on the path. Otherwise, 퐵 is called 푀-chordal if {푏1,푏2} ∈ 퐸. An 푀-(chordal) flower is the we call the edge interior. union of a stem and a (chordal) blossom that intersect exactly in For an exterior edge 푒, we denote by 푡 (푒) the endpoint of 푃 that the tip of the stem. An example of a chordal flower together with is a neighbor of 푐푆푒 on 푃. The first step is to modify 푃. In a second an augmentation is given in Figure 1. If the matching is clear from step, the modified path will yield a Pareto-improvement over 휋. An the context, we will not specify it for the previous notation. exterior edge 푒 is called saturated if 푡 (푒) is a leaf of 푆푒 . An interior edge 푒 is called saturated if all leaves of 푆푒 are covered by 푃. 푏2 First, we may assume that every exterior edge is saturated or there exists only one exterior edge 푒∗ which corresponds to a star 푆 푙푆 푃 푡 (푒∗) root with two leaves and 2 is the endpoint of that is not . 푏 To this end, assume first that there exist two exterior edges 푒 and 푓 originating from stars 푆 and 푇 , respectively. Replacing 푡 (푒) and ( ) 푆 푇 ∗ 푡 푓 by 푙2 and 푙2 leaves both edges saturated. Otherwise, if 푒 is the only exterior edge originating from star 푆, and is not saturated, 푏1 then 푆 has only two leaves, or we can replace 푡 (푒∗) by a leaf of 푆 uncovered by 푃. In addition, if 푙푆 is not the other endpoint of 푃, we Figure 1: The bold matching indicates a chordal flower that 2 푡 (푒∗) 푙푆 can be augmented via the gray clique cover. can replace by 2 . This establishes the claim. Second, we show that we can additionally assume that all interior edges are saturated. Indeed, if 푒 is an interior edge and 푙 is a leaf Theorem 4.5 (Hell and Kirkpatrick [13]). Let 퐺 = (푉, 퐸) be a of 푆푒 not covered by 푃, then we can replace 푃 by the augmenting graph. Then C is a maximum size 3-clique matching if and only if path that starts with 푙 and proceeds on 푃 with 푒. Assume that the (1) There exists no 푀 (C)-augmenting path. path ends in an exterior edge 푓 . If 푓 is not saturated, we replace (C) ( ) 푆푓 (2) There exists no 푀 -alternating path starting at an uncovered 푡 푓 by 푙2 . Otherwise, we follow the path to the end. In any case, vertex and ending at a vertex covered by a triangle. this procedure yields a path 푃 ′ such that all exterior edges are as (3) There exists no 푀 (C)-chordal flower. after the first step and all interior edges are saturated. In particular, a clique matching is maximum if and only if is vertex- We will show how to obtain a Pareto-improvement over 휋 from maximal. 푃. Label the vertices of the path 푝0, . . . , 푝푚 for some (odd) integer 푚. If the first matching-edge of the path is exterior and saturated, we Note that compared to the classical characterization of maximum delete 푝0 and 푝1 from the path. If the last matching-edge is exterior cardinality matchings, the second condition allows for improvement and saturated, we delete 푝푚−1 and 푝푚 from the path. This leaves a by deleting a triangle, and the third one by creating one in order to ′ ′ ′ path 푃 on vertices 푝0, . . . , 푝푚′ such that all leaves corresponding improve a 3-clique matching. ′ ′ to stars of edges in 푀S ∩ 푃 are covered by 푃 . Let T be the set of 푇 { ′ ′ } Theorem 4.6. Let a simple, symmetric MFHG be given. Let 휋 be a star coalitions 푇 such that 푐 ∉ 푝0, . . . , 푝푚′ , but some leaf of 푇 is partition of the agents. Then, 휋 is welfare-optimal if and only if it is a endpoint of 푃 ′. Pareto-optimal.

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′ Consider 휋 = {퐶 ∈ 휋 : 퐶 ∩ {푝0, . . . , 푝푚 } = ∅} ∪ allow for 0-weights, and unique top-ranked coalitions are therefore { \{ ′ ′ } ∈ T } ∪ ( ′\ ( ′)) 푇 푝0, 푝푚′ : 푇 푃 푀 휋 , which is a Pareto- guaranteed [3, Theorem 11]. We extend this result to a very general improvement over 휋. Hence, 휋 is not Pareto-optimal, which is class of ASHGs that includes symmetric ASHGs. a contradiction. Therefore, the first condition of the combinatorial An ASHG is called mutually indifferent if 푣푖 (푗) = 0 implies characterization is satisfied. 푣 푗 (푖) = 0 for every pair of agents 푖, 푗. Note that every symmetric An example for this case is given in Figure 2. The path 푃 is formed ASHG is mutually indifferent. by the straight lines and the partition 휋 ′ by the bold edges. Dashed Theorem 5.1. A Pareto-optimal outcome for mutually indiffer- edges indicate leaves of the stars 푆 and 푇 . The edge {푐푆, 푙푆 } is exte- 1 ent ASHGs can be computed in polynomial time. In particular, a {푐푇 , 푙푇 } rior and saturated, and the edge 1 is interior and saturated. Pareto-optimal outcome for symmetric ASHGs can be computed in The gray partition yields a Pareto-improvement over 휋. polynomial time.

푆 푇 푙2 푙2 Proof. Consider Algorithm 3. The algorithm can be seen as a variant of serial dictatorship where in every coalition formed through a dictator 푑푖 , the dictator asks the agents in her coalition to improve using a strict rank order such that none of higher rank 푙푆 푐푆 푙푆 푙푇 푐푇 3 1 1 1 1 becomes worse off.

Figure 2: Example of a Pareto-improvement that can be con- structed from an 푀 (휋 ′)-augmenting path. Input: Mutually indifferent ASHG induced by 퐺 = (푁, 퐸, 푣) Output: Pareto-optimal partition 휋 The proofs that the second and third condition hold are similar, 휋 ← ∅, 퐷 ← 푁 , 푖 ← 1 and use that the first condition is already proved. while 퐷 ≠ ∅ do Hence, 휋 ′ is a maximum clique matching and is therefore welfare- Pick 푑푖 ∈ 퐷 optimal. Since SW(휋 ′) = SW(휋), it follows that 휋 is welfare- 퐶푖 ← {푑푖 } ∪ {푗 ∈ 퐷 : 푣푑 (푗) > 0}, 퐼푖 ← {푗 ∈ 퐷 : 푣푑 (푗) = 0} optimal. 푖 푖 □ 퐻푖 ← 퐶푖 \{푑푖 } As a corollary, we obtain efficient verification of Pareto- while 퐻푖 ≠ ∅ ∧ 퐼푖 ≠ ∅ do optimality for simple symmetric MFHGs. Pick ℎ ∈ 퐻푖 퐶푖 ← 퐶푖 ∪ {푗 ∈ 퐼푖 : 푣ℎ (푗) > 0} Theorem 4.7. The problem of verifying Pareto-optimality can be 퐻푖 ← (퐻푖 ∪ {푗 ∈ 퐼푖 : 푣ℎ (푗) > 0})\{ℎ} done in polynomial time for simple symmetric MFHGs. 퐼푖 ← {푗 ∈ 퐼푖 : 푣ℎ (푗) = 0} Proof. Let an MFHG be given and a partition 휋 that is to be 휋 ← 휋 ∪ {퐶푖 }, 퐷 ← 퐷\퐶푖 , 푖 ← 푖 + 1 checked for Pareto-optimality. Simply compute a partition 휋∗ via return 휋 Algorithm 1 and compare their social welfare. By Theorem 4.2 and Algorithm 3: Pareto-optimality for mutual indifference Theorem 4.1, this runs in polynomial time. By Theorem 4.4, 휋∗ is welfare-optimal. Finally, by Theorem 4.6, comparing social welfare checks for Pareto-optimality. □ The running time is polynomial since every edge in the graph underlying the ASHG is checked at most once. For general, weighted MFHGs, it is still of interest as to whether For correctness, denote by 퐶1, . . . ,퐶푘 the coalitions that form one can also find a welfare-optimal partition in polynomial time. in the order of the algorithm, and 퐼푖 the respective indifference While this question remains open, we can at least narrow down sets (some may be empty) at the end of the inner while-loop. Note the search to coalitions of small size. Then, a weighted version of that for all 푖 ∈ 1, . . . , 푘, 푎 ∈ 퐶푖,푏 ∈ 퐼푖 holds that 푣푎 (푏) = 0. For Max3CliqueMatching might give rise to an efficient algorithm. correctness, assume that 휋 ′ is a partition such that for every agent ′ ′ The proof of the proposition relies on the fact that we can split 푖 ∈ 푁 , 푣푖 (휋 ) ≥ 푣푖 (휋). If |퐶푖 | > 1, then 퐶푖 ⊆ 휋 (푑푖 ) ⊆ 퐶푖 ∪ 퐼푖 . a coalition 퐶 of size at least 4 into an edge 푒 and the remainder 퐶\푒 Hence, no agent in such a coalition will be better off. In addition, such that SW(푒) + SW(퐶\푒) ≥ SW(퐶). The edge 푒 maximizes no agent in a singleton coalition 퐶푖 = {푑푖 } ∈ 휋 will be better off, a cleverly chosen objective function that relies on the weight of 푒, since they can only form coalitions with other singleton coalitions the weight of the cut between 푒 and 퐶\푒, and the welfare of 퐶\푒. or with coalitions such that they are in the set 퐼푗 , both of which Proposition 4.8. Let a partition 휋 of the agents of a general give them 0 utility. Therefore, no agent’s utility has improved. □ ′ | | ≤ MFHG be given. Then, there exists a partition 휋 with 퐶 3 for Slight modifications of the above algorithm give computational ∈ ′ SW( ′) ≥ SW( ) all 퐶 휋 with 휋 휋 . In particular, there exists a tractability of Pareto-optimality even for more general classes of welfare-optimal partition consisting of coalitions of size at most 3. ASHGs. The same algorithm works for the class of ASHGs such that 푣푖 (푗) = 0 implies 푣 푗 (푖) ≤ 0 for every pair of agents 푖, 푗. Hence, 5 ADDITIVELY SEPARABLE HEDONIC the only edges remaining for the full domain of ASHGs are critical GAMES edges of the form {푖, 푗} such that 푣푖 (푗) > 0 while 푣 푗 (푖) = 0. One In this section, we will survey Pareto-optimality on ASHGs. Positive idea towards obtaining an algorithm for a more general class of results exist so far only for a very restrictive class that does not ASHGs is to use a pivoting rule that selects dictators. This allows,

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for example, for a positive result for the class of ASHGs such that Corollary 5.3. Finding a Pareto-improvement is NP-hard for the critical edges form a directed acyclic graph (using a topological symmetric ASHGs. order on the agents for a pivoting rule). Finally, it is interesting to see why the strong relation between The outcome of the algorithm can, however, have an arbitrar- Pareto-optimality and perfection exploited by Aziz et al. [2] does ily large gap to the maximum social welfare that is obtained in a not hold for ASHGs. The preference refinement algorithm computes welfare-optimal outcome. In addition, all but one agent may obtain an individually rational, Pareto-optimal partition given an oracle a worst coalition. On the other hand, a Pareto-optimal and indi- that decides whether there exists a perfect partition and, in the case vidually rational outcome does always exist, but computing such a there exists one, can compute one. While the former problem is partition is intractable. The following is a strengthening of a result NP-hard, the latter can be solved in polynomial time by forming a by Aziz et al. [3] who dealt with the whole class of ASHGs and top-ranked coalition, adding requisite agents, and adding outside established weak NP-hardness. agents that need an inside agent. This theorem even holds for the The reduction is from the NP-complete problem Exact3Cover more general class of separable hedonic games [3, Theorem 9]. [14]. An instance (푅, 푆) of Exact3Cover (X3C) consists of a ground set 푅 together with a set 푆 of 3-element subsets of 푅. A ‘yes’ instance Theorem 5.4. The problem of, given a general ASHG, computing is an instance so that there exists a subset 푆 ′ ⊆ 푆 that partitions 푅. a perfect partition or deciding that no such partition exists, can be done in polynomial time. Theorem 5.2. Finding a Pareto-optimal and individually ratio- nal partition for symmetric ASHGs is (strongly) NP-hard, even if all 6 FRACTIONAL HEDONIC GAMES weights are integers bounded from above by 3. The serial dictatorship version used for ASHGs in Algorithm 3 Proof sketch. We provide a Turing reduction, illustrated in implicitly exploits the fact that ASHGs have the property that top- Figure 3, from X3C. Given an instance (푅, 푆) of X3C, we con- ranked coalitions in subgames are the restrictions of top-ranked struct the symmetric ASHG with agent set 푁 = 푅 ∪ 푉 where coalitions in the original game. This is not the case anymore for 푉 = {푠푖 : 푖 = 1,..., 5, 푠 ∈ 푆} consists of 5 copies of agents for the FHGs. However, the set of top-ranked coalitions can be described sets in 푆. Preferences are given by weights 푣 as using the following observation. • 푣 (푖, 푗) = 0, 푖, 푗 ∈ 푅, 푖 ≠ 푗 Proposition 6.1. Let an FHG be given based on a graph 퐺 = • 푣 (푖, 푠 ) = 2, 푠 ∈ 푆, 푖 ∈ 푠 1 (푁, 퐸, 푣) and let 푑 ∈ 푁 . Let 퐶 be a top-ranked coalition of 푑 and • 푣 (푠 , 푠 ) = 푣 (푠 , 푠 ) = 푣 (푠 , 푠 ) = 푣 (푠 , 푠 ) = 푣 (푠 , 푠 ) = 3, 1 2 1 3 2 4 3 5 4 5 set 휇 = 푣 (퐶). Then, 퐶 = {푎 ∈ 푁 \{푑} : 푣 (푎) > 휇} ∪ 푊 for some 푣 (푠 , 푠 ) = 0, 푠 ∈ 푆, and 푑 푑 2 3 푊 ⊆ {푎 ∈ 푁 \{푑} : 푣 (푎) = 휇}. • all other weights are set to −13. 푑 Hence, to obtain the top-ranked coalitions of an agent, one can 푖 푗 푘 order the other agents in decreasing value and add them until 푅 0 0 0 0 0 another agent is not beneficial. If there are agents that give exactly 2 2 2 the utility of a top-ranked coalition, they may or may not be added. We will now show efficient computability of Pareto-optimal 푠 1 partitions for certain classes of FHGs. An FHG satisfies the equal 3 3 affection condition if 푣 (푦), 푣 (푧) > 0 implies 푣 (푦) = 푣 (푧) for 푠2 0 푠3 푥 푥 푥 푥 푉 푠 = {푖, 푗, 푘 } every triple of agents 푥,푦, and 푧. An FHG is called generic if 푣푥 (푦) ≠ 3 3 푣푥 (푧) for every triple of agents 푥,푦, and 푧, i.e., the utilities over the

푠4 3 푠5 remaining set of agents are pairwise distinct for every agent. Since the equal affection condition guarantees unique top-ranked Figure 3: ASHG for the reduction. Indicated edges between coalitions for every agent, we obtain the following theorem, which 푅-agents have weight 0, other omitted edges have weight −13. applies in particular to simple FHGs. Theorem 6.2. Finding a Pareto-optimal partition for FHGs satis- We will argue that if we can compute a Pareto-optimal and fying the equal affection condition can be done in polynomial time. individually rational partition, we can decide X3C. Amongst all Another variant of serial dictatorship finds Pareto-optimal parti- individually rational partitions, the highest utility that the agents tions on generic FHGs. can obtain is 2, 6, and 3 for agents in 푅, {푠1 : 푠 ∈ 푆}, and {푠푖 : 푖 = 2,..., 4, 푠 ∈ 푆}, respectively. It can be shown that there exists an Theorem 6.3. Finding a Pareto-optimal partition for generic FHGs individually rational partition that attains these bounds for every can be done in polynomial time. agent if and only if there exists a 3-partition of 푅 through sets in 푆. Proof sketch. Let an FHG be based on the graph 퐺 = (푁, 퐸, 푣). Hence, we can solve X3C in polynomial time by computing We give an algorithm based on serial dictatorship that exploits a dy- a Pareto-optimal and individually rational partition for the corre- namically created hierarchy for the dictatorship. The next dictator sponding ASHG, and check whether every agent receives the utility is chosen based on the top choices of the previous dictator. of a best partition amongst individually rational partitions. □ By Proposition 6.1, we know the structure of the top-ranked By applying a local search algorithm that starts with the single- coalitions of an agent. Let an agent set 푀 ⊆ 푁 be given, that ton partition, we obtain the following corollary. induces the FHG on the subgraph 퐺 [푀], and let 푑 ∈ 푀. There

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exists a unique smallest top-ranked coalition, which we denote by Theorem 6.6. The problem of, given an FHG, computing a perfect 푇푑 (푀). Furthermore, for a generic FHG, there exist at most two partition or deciding that no such partition exists, can be done in top-ranked coalitions. Denote in this case by 푡푑 (푀) the number of polynomial time. such coalitions for agent 푑 in the subgame and if 푡푑 (푀) = 2, let 훼푑 (푀) ∈ 푀 be the unique agent such that 푇푑 (푀) ∪ {훼푑 (푀)} is the 7 CONCLUSION other top-ranked coalition. We have investigated Pareto-optimality in three types of cardinal We are ready to formulate the recursive algorithm A that com- hedonic games. The main findings and important related results putes a Pareto-optimal partition by adding iteratively coalitions to a are collected in Table 1. We can efficiently find Pareto-optimal par- partial partition. The actual Pareto-optimal partition is obtained by titions in symmetric MFHGs and AHSGs, and reasonable classes choosing an arbitrary first dictator 푑 ∈ 푁 and calling A(푁, ∅,푑). of FHGs including simple FHGs. The key insight for MFHGs is the equivalence with an extension of matchings to cliques. The combinatorial view of the problem allowed us to completely un- Input: Non-empty agent set 푀, partial partition 휋, pivot agent 푑 derstand Pareto-optimal outcomes on simple, symmetric MFHGs, Output: Partition 휋 where they coincide with welfare-optimal outcomes. This motivates the study of the weighted case, where Pareto-optimal outcomes if 푡푑 (푀) = 1 then have no guarantee on the welfare, and yet our algorithm returns if 푇푑 (푀) = 푀 then a 2-approximation to social welfare. The complexity of welfare- return 휋 ∪ {푇푑 (푀)} else optimization in the weighted case is an interesting open problem. We are at least able to prove that coalitions of size 2 and 3 suffice, Pick 푑new ∈ 푀\푇푑 (푀) which is in line with the research on any other for Execute A(푀\푇푑 (푀), 휋 ∪ {푇푑 (푀)},푑new) else MFHGs. ( ( ) ∪ { ( )}) ( \ ( )) if 푣훼푑 (푀) 푇푑 푀 훼푑 푀 > 푣훼푑 (푀) 푀 푇푑 푀 then if 푇푑 (푀) ∪ {훼푑 (푀)} = 푀 then Pareto optimality Welfare optimality PO PO∧IR Deterministic Approximation return 휋 ∪ {푇푑 (푀) ∪ {훼푑 (푀)}} else MFHG P (sym, Thm. 4.2) P (sym, Thm. 4.2) P (0/1 sym, [15]) 2 (sym, Thm. 4.3) ASHG P (sym, Thm. 5.1) NP-h (sym, Thm. 5.2) NP-h (sym, [3]) open Pick 푑new ∈ 푀\(푇푑 (푀) ∪ {훼푑 (푀)}) FHG P (0/1 sym, Thm. 6.2) NP-h (sym, Thm. 6.4) NP-h (0/1 sym, [4]) 4 (sym, [4]) Execute A(푀\(푇푑 (푀) ∪ {훼푑 (푀)}), 휋 ∪ {푇푑 (푀) ∪ Table 1: Complexity of Pareto- and welfare-optimality for {훼푑 (푀)}},푑new) cardinal hedonic games. Preference restrictions are given ( ( ) ∪ { ( )}) ( \ ( )) else if 푣훼푑 (푀) 푇푑 푀 훼푑 푀 < 푣훼푑 (푀) 푀 푇푑 푀 or in parenthesis, where (0/1) sym denotes (simple) symmet- ( ( ) ∪ { ( )}) = ( \ ( )) 푣훼푑 (푀) 푇푑 푀 훼푑 푀 푣훼푑 (푀) 푀 푇푑 푀 = 0 then ric preferences. For welfare-optimality, the best known ef- Execute A(푀\푇푑 (푀), 휋 ∪ {푇푑 (푀)}, 훼푑 (푀)) ficiently attainable approximation ratio is given. else ∈ { ( ) ∈ \ ( )} Pick 푑new argmax 푣훼푑 (푀) 푥 : 푥 푀 푇푑 푀 ( ) ∈ ( \ ( )) if 훼푑 푀 푇푑new 푀 푇푑 푀 then Execute A(푀\푇푑 (푀), 휋 ∪ {푇푑 (푀)},푑new) The key technique for positive results on ASHGs and FHGs else are refinements of serial dictatorship algorithms. Further enhance- Execute A(푀\(푇푑 (푀) ∪ {훼푑 (푀)}), 휋 ∪ {푇푑 (푀) ∪ ments, e.g., with respect to the order of selection of the dictators, {훼푑 (푀)}},푑new) might yield even better results. On the other hand, computing Pareto-optimal outcomes that satisfy further properties will of- Algorithm 4: Pareto-optimal partition for generic FHG by ten be intractable. Computational hardness is obtained if we re- the recursive algorithm A quire individual rationality in addition. Since it is even hard to compute Pareto-improvements, local search heuristics based on By the top-ranked coalition structure of agents in generic FHGs, Pareto-optimality cannot be exploited. every step of Algorithm 4 can be executed and as argued after Partitions on simple MFHGs can simultaneously satisfy high Proposition 6.1, top-ranked coalitions can be efficiently computed. demands in terms of stability and optimality. Interesting further Hence, the algorithm runs in polynomial time. directions for research therefore concern weighted MFHGs as well It can be checked that 휋 = A(푁, ∅,푑) returns a Pareto-optimal as Pareto-optimality on the general domains of cardinal hedonic partition, provided that the input evolves from a generic FHG. □ games.

Finally, similar statements as for ASHGs hold for FHGs. The ACKNOWLEDGMENTS proofs are similar. This material is based on work supported by the Deutsche Theorem 6.4. Finding a Pareto-optimal and individually rational Forschungsgemeinschaft under grant BR 2312/12-1. The author partition for symmetric FHGs is NP-hard. would like to thank Felix Brandt, Katharina Moench, and Anaëlle Wilczynski for feedback on early versions of the paper and is in- Theorem 6.5. Finding a Pareto-improvement is NP-hard for sym- debted to three anonymous reviewers who have provided very metric FHGs. valuable feedback.

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