Hedonic games provide a natural model of coalition formation among self-interested agents. The associated problem of finding stable outcomes in such games has been extensively studied. In this paper, we identify simple conditions on expressivity of hedonic games that are sufficient for the problem of checking whether a given game admits a stable outcome to be computationally hard. Somewhat surprisingly, these conditions are very mild and intuitive. Our results apply to a wide range of stability concepts (core stability, individual stability, Nash stability, etc.) and to many known formalisms for hedonic games (additively separable games, games with W-preferences, fractional hedonic games, etc.), and unify and extend known results for these formalisms. They also have broader applicability: for several classes of hedonic games whose computational complexity has not been explored in prior work, we show that our framework immediately implies a number of hardness results for them.
We consider a class of coalition formation games called hedonic games, i.e., games in which the utility of a player is completely determined by the coalition that the player belongs to. We first define the class of subset-additive hedonic games and show that they have the same representation power as the class of hedonic games. We then define a restriction of subset-additive hedonic games that we call subset-neutral hedonic games and generalize a result by Bogomolnaia and Jackson (2002) by showing the existence of a Nash stable partition and an individually stable partition in such games. We a…
We consider coalition formation games in which each player has preferences over the other players and his preferences over coalitions are based on the best player ($\mathcal{B}$-/B-hedonic games) or the worst player ($\mathcal{W}$/W-hedonic games) in the coalition. We show that for $\mathcal{B}$-hedonic games, an individually stable partition is guaranteed to exist and can be computed efficiently. Similarly, there exists a polynomial-time algorithm which returns a Nash stable partition (if one exists) for $\mathcal{B}$-hedonic games with strict preferences. Both $\mathcal{W}$- and W-hedonic ga…
In this paper, we examine \emph{hedonic coalition formation games} in which each player's preferences over partitions of players depend only on the members of his coalition. We present three main results in which restrictions on the preferences of the players guarantee the existence of stable partitions for various notions of stability. The preference restrictions pertain to \emph{top responsiveness} and \emph{bottom responsiveness} which model optimistic and pessimistic behavior of players respectively. The existence results apply to natural subclasses of \emph{additive separable hedonic game…
We study hedonic coalition formation games in which cooperation among the players is restricted by a graph structure: a subset of players can form a coalition if and only if they are connected in the given graph. We investigate the complexity of finding stable outcomes in such games, for several notions of stability. In particular, we provide an efficient algorithm that finds an individually stable partition for an arbitrary hedonic game on an acyclic graph. We also introduce a new stability concept -in-neighbor stability- which is tailored for our setting. We show that the problem of finding …
Incentive mechanism design is crucial for enabling federated learning. We deal with clustering problem of agents contributing to federated learning setting. Assuming agents behave selfishly, we model their interaction as a stable coalition partition problem using hedonic games where agents and clusters are the players and coalitions, respectively. We address the following question: is there a family of hedonic games ensuring a Nash-stable coalition partition? We propose the Nash-stable set which determines the family of hedonic games possessing at least one Nash-stable partition, and analyze t…
This paper studies the Nash stability in hedonic coalition formation games. We address the following issue: for a general problem formulation, is there any utility allocation method ensuring a Nash-stable partition? We propose the definition of the Nash-stable core. We study the conditions for having a non-empty Nash-stable core. More precisely, we show how relaxed efficiency in utility sharing method allows to ensure a non-empty Nash-stable core.
The work we present in this paper initiated the formal study of fractional hedonic games, coalition formation games in which the utility of a player is the average value he ascribes to the members of his coalition. Among other settings, this covers situations in which players only distinguish between friends and non-friends and desire to be in a coalition in which the fraction of friends is maximal. Fractional hedonic games thus not only constitute a natural class of succinctly representable coalition formation games, but also provide an interesting framework for network clustering. We propose…
Hedonic games provide a general model of coalition formation, in which a set of agents is partitioned into coalitions, with each agent having preferences over which other players are in her coalition. We prove that with additively separable preferences, it is $Σ_2^p$-complete to decide whether a core- or strict-core-stable partition exists, extending a result of Woeginger (2013). Our result holds even if valuations are symmetric and non-zero only for a constant number of other agents. We also establish $Σ_2^p$-completeness of deciding non-emptiness of the strict core for hedonic games with dic…
We consider a team formation setting where agents have varying levels of expertise in a global set of required skills, and teams are ranked with respect to how well the expertise of teammates complement each other. We model this setting as a hedonic game, and we show that this class of games possesses many desirable properties, some of which are as follows: A partition that is Nash stable, core stable and Pareto optimal is always guaranteed to exist. A contractually individually stable partition (and a Nash stable partition in a restricted setting) can be found in polynomial-time. A core stabl…
Hedonic games formalize coalition formation scenarios where players evaluate an outcome based on the coalition they are contained in. Due to a large number of possible coalitions, compact representations of these games are crucial. We complement known compact representation models by a distance-based approach: Players' preferences are encoded in a bipolar manner by ordinal preferences over a small set of known neighbouring players, coalitions are represented by adequate preference orders from a player's perspective, and preferences over coalitions are extended based on a directed form of Hausd…
We study how stability can be maintained even after any set of at most k players leave their groups, in the context of hedonic games. While stability properties ensure an outcome to be robust against players' deviations, it has not been considered how an unexpected change caused by a sudden deletion of players affects stable outcomes. In this paper, we propose a novel criterion that reshapes stability form robustness aspect. We observe that some stability properties can be no longer preserved even when a single agent is removed. However, we obtain positive results by focusing on symmetric frie…
We study the computational complexity of finding stable outcomes in hedonic games, which are a class of coalition formation games. We restrict our attention to symmetric additively-separable hedonic games, which are a nontrivial subclass of such games that are guaranteed to possess stable outcomes. These games are specified by an undirected edge- weighted graph: nodes are players, an outcome of the game is a partition of the nodes into coalitions, and the utility of a node is the sum of incident edge weights in the same coalition. We consider several stability requirements defined in the liter…
We consider a coalition formation setting where each agent belongs to one of the two types, and agents' preferences over coalitions are determined by the fraction of the agents of their own type in each coalition. This setting differs from the well-studied Schelling's model in that some agents may prefer homogeneous coalitions, while others may prefer to be members of a diverse group, or a group that mostly consists of agents of the other type. We model this setting as a hedonic game and investigate the existence of stable outcomes using hedonic games solution concepts. We show that a core sta…
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 coalit…
Individual Rationality in Constrained Hedonic Games: Additively Separable and Fractional Preferences
Czech Technical University in Prague（布拉格捷克理工大学）；Ben-Gurion University of the Negev（内盖夫本-古里安大学）；Charles University（查理大学）；AGH University of Krakow（克拉科夫AGH科技大学）
Gran Sasso Science Institute（格兰萨索科学研究所）；University of L’Aquila（拉奎拉大学）；University of Warsaw（华沙大学）
Additively separable hedonic games and fractional hedonic games have received considerable attention. They are coalition forming games of selfish agents based on their mutual preferences. Most of the work in the literature characterizes the existence and structure of stable outcomes (i.e., partitions in coalitions), assuming that preferences are given. However, there is little discussion on this assumption. In fact, agents receive different utilities if they belong to different partitions, and thus it is natural for them to declare their preferences strategically in order to maximize their ben…
In hedonic games, players form coalitions based on individual preferences over the group of players they could belong to. Several concepts to describe the stability of coalition structures in a game have been proposed and analysed in the literature. However, prior research focuses on algorithms with time complexity that is at least linear in the input size. In the light of very large games that arise from, e.g., social networks and advertising, we initiate the study of sublinear time property testing algorithms for existence and verification problems under several notions of coalition stabilit…
Nguyen et al. [1] introduced altruistic hedonic games in which agents' utilities depend not only on their own preferences but also on those of their friends in the same coalition. We propose to extend their model to coalition formation games in general, considering also the friends in other coalitions. Comparing our model to altruistic hedonic games, we argue that excluding some friends from the altruistic behavior of an agent is a major disadvantage that comes with the restriction to hedonic games. After introducing our model and showing some desirable properties, we additionally study some c…
We conduct a computational analysis of fair and optimal partitions in additively separable hedonic games. We show that, for strict preferences, a Pareto optimal partition can be found in polynomial time while verifying whether a given partition is Pareto optimal is coNP-complete, even when preferences are symmetric and strict. Moreover, computing a partition with maximum egalitarian or utilitarian social welfare or one which is both Pareto optimal and individually rational is NP-hard. We also prove that checking whether there exists a partition which is both Pareto optimal and envy-free is $Σ_…
This paper considers the problem of dividing agents among coalitions. We concentrate on Additively Separable Hedonic Games (ASHG's), in which each agent has a non-negative value for every other agent and her utility is the sum of the values she assigns to the members of her coalition. Unlike previous work, we analyze a model where exactly $k$ coalitions must be formed, and the goal is to maximize the utility of the agent which is worst off, i.e., the egalitarian social welfare. We show that this problem is hard, even when the number of agents should be equally divided among the coalitions. We …
NICTA（澳大利亚国家信息与通信技术研究院）；University of New South Wales（新南威尔士大学）；University of Oxford（牛津大学）；Université Paris-Dauphine（巴黎第九大学）
We study hedonic games with dichotomous preferences. Hedonic games are cooperative games in which players desire to form coalitions, but only care about the makeup of the coalitions of which they are members; they are indifferent about the makeup of other coalitions. The assumption of dichotomous preferences means that, additionally, each player's preference relation partitions the set of coalitions of which that player is a member into just two equivalence classes: satisfactory and unsatisfactory. A player is indifferent between satisfactory coalitions, and is indifferent between unsatisfacto…
We consider the coalition formation games with an additional component, `noisy preferences'. Moreover, such noisy preferences are available only for a sample of coalitions. We propose a multiplicative noise model and obtain the prediction probability, defined as the probability that the estimated PAC core-stable partition of the noisy game is also PAC core-stable for the unknown noise-free game. This prediction probability depends on the probability of a combinatorial construct called an `agreement event'. We explicitly obtain the agreement probability for $n$ agent noisy game with l\geq 2 sup…
In hedonic diversity games (HDGs), recently introduced by Bredereck, Elkind and Igarashi (2019), each agent belongs to one of two classes (men and women, vegetarians and meat-eaters, junior and senior researchers), and agents' preferences over coalitions are determined by the fraction of agents from their class in each coalition. Bredereck et al. show that while an HDG may fail to have a Nash stable (NS) or a core stable (CS) outcome, every HDG in which all agents have single-peaked preferences admits an individually stable (IS) outcome, which can be computed in polynomial time. In this work, …
Individual Rationality in Constrained Hedonic Games: Friends, Enemies, and Neutrals
AGH University of Krakow（克拉科夫 AGH 科技大学）；Czech Technical University in Prague（布拉格捷克理工大学）；ELTE Centre for Economic and Regional Studies（布达佩斯罗兰大学经济区域研究中心）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）
The formal study of coalition formation in multi-agent systems is typically realized in the framework of hedonic games, which originate from economic theory. The main focus of this branch of research has been on the existence and the computational complexity of deciding the existence of coalition structures that satisfy various stability criteria. The actual process of forming coalitions based on individual behavior has received little attention. In this paper, we study the convergence of simple dynamics leading to stable partitions in a variety of established classes of hedonic games includin…
An important aspect in systems of multiple autonomous agents is the exploitation of synergies via coalition formation. In this paper, we solve various open problems concerning the computational complexity of stable partitions in additively separable hedonic games. First, we propose a polynomial-time algorithm to compute a contractually individually stable partition. This contrasts with previous results such as the NP-hardness of computing individually stable or Nash stable partitions. Secondly, we prove that checking whether the core or the strict core exists is NP-hard in the strong sense eve…
Hedonic games are a prominent model of coalition formation, in which each agent's utility only depends on the coalition she resides. The subclass of hedonic games that models the formation of general partnerships, where output is shared equally among affiliates, is referred to as hedonic games with common ranking property (HGCRP). Aside from their economic motivation, HGCRP came into prominence since they are guaranteed to have core stable solutions that can be found efficiently. We improve upon existing results by proving that every instance of HGCRP has a solution that is Pareto optimal, cor…
We investigate verification and existence problems for prominent stability concepts in hedonic games with friends, enemies, and optionally with neutrals [8, 16]. We resolve several (long-standing) open questions [4, 16, 20, 23] and show that for friend-oriented preferences, under the friends and enemies model, it is coNP-complete to verify whether a given agent partition is (strictly) core stable, while under the friends, enemies, and neutrals model, it is NP-complete to determine whether an individual stable partition exists. We further look into natural restricted cases from the literature, …
Gran Sasso Science Institute (GSSI)（格兰萨索科学研究所）；Ca’ Foscari University of Venice（威尼斯大学）
We study PAC learnability and PAC stabilizability of Hedonic Games (HGs), i.e., efficiently inferring preferences or core-stable partitions from samples. We first expand the known learnability/stabilizability landscape for some of the most prominent HGs classes, providing results for Friends and Enemies Games, Bottom Responsive, and Anonymous HGs. Then, having a broader view in mind, we attempt to shed light on the structural properties leading to learnability/stabilizability, or lack thereof, for specific HGs classes. Along this path, we focus on the fully expressive Hedonic Coalition Nets re…
The paper is devoted to game-theoretic methods for community detection in networks. The traditional methods for detecting community structure are based on selecting denser subgraphs inside the network. Here we propose to use the methods of cooperative game theory that highlight not only the link density but also the mechanisms of cluster formation. Specifically, we suggest two approaches from cooperative game theory: the first approach is based on the Myerson value, whereas the second approach is based on hedonic games. Both approaches allow to detect clusters with various resolution. However,…
This paper proposes a novel game-theoretical autonomous decision-making framework to address a task allocation problem for a swarm of multiple agents. We consider cooperation of self-interested agents, and show that our proposed decentralized algorithm guarantees convergence of agents with social inhibition to a Nash stable partition (i.e., social agreement) within polynomial time. The algorithm is simple and executable based on local interactions with neighbor agents under a strongly-connected communication network and even in asynchronous environments. We analytically present a mathematical …
The Bakers and Millers Game with Restricted Locations
Hasso Plattner Institute, University of Potsdam（霍普夫纳研究所、波茨坦大学）；Institute of Computer Science, University of Augsburg（计算机科学研究所、奥格斯堡大学）；Mathematics of Operations Research, University of Twente（运筹学数学研究所、特文特大学）
We revisit the complexity of the well-studied notion of Additively Separable Hedonic Games (ASHGs). Such games model a basic clustering or coalition formation scenario in which selfish agents are represented by the vertices of an edge-weighted digraph $G=(V,E)$, and the weight of an arc $uv$ denotes the utility $u$ gains by being in the same coalition as $v$. We focus on (arguably) the most basic stability question about such a game: given a graph, does a Nash stable solution exist and can we find it efficiently? We study the (parameterized) complexity of ASHG stability when the underlying gra…
In many economic, social and political situations individuals carry out activities in groups (coalitions) rather than alone and on their own. Examples range from households and sport clubs to research networks, political parties and trade unions. The underlying game theoretic framework is known as coalition formation. This survey discusses the notion of core stability in hedonic coalition formation (where each player's happiness only depends on the other members of his coalition but not on how the remaining players outside his coalition are grouped). We present the central concepts and algorit…
This paper addresses the mobile assignment problem in a multi-cell broadcast transmission seeking minimal total power consumption by considering both transmission and operational powers. While the large scale nature of the problem entails to find distributed solutions, game theory appears to be a natural tool. We propose a novel distributed algorithm based on group formation games, called \textit{the hedonic decision algorithm}. This formalism is constructive: a new class of group formation games is introduced where the utility of players within a group is separable and symmetric being a gener…
