On the complexity of core, kernel, and bargaining set

Synthesis Lectures on Artificial Intelligence and Machine Learning, 2011

Cooperative game theory is a branch of (micro-)economics that studies the behavior of self-interested agents in strategic settings where binding agreements among agents are possible. Our aim in this book is to present a survey of work on the computational aspects of cooperative game theory. We begin by formally defining transferable utility games in characteristic function form, and introducing key solution concepts such as the core and the Shapley value. We then discuss two major issues that arise when considering such games from a computational perspective: identifying compact representations for games, and the closely related problem of efficiently computing solution concepts for games. We survey several formalisms for cooperative games that have been proposed in the literature, including, for example, cooperative games defined on networks, as well as general compact representation schemes such as MC-nets and skill games. As a detailed case study, we consider weighted voting games: a widely-used and practically important class of cooperative games that inherently have a natural compact representation. We investigate the complexity of solution concepts for such games, and generalizations of them. We briefly discuss games with non-transferable utility and partition function games. We then overview algorithms for identifying welfare-maximizing coalition structures and methods used by rational agents to form coalitions (even under uncertainty), including bargaining algorithms. We conclude by considering some developing topics, applications, and future research directions.

2004

We study mechanisms that can be modelled as coalitional games with transferable utilities, and apply ideas from mechanism design and game theory to problems arising in a network design setting. We establish an equivalence between the game-theoretic notion of agents being substitutes and the notion of frugality of a mechanism. We characterize the core of the network design game and relate it to outcomes in a sealed bid auction with VCG payments. We show that in a game, agents are substitutes if and only if the core of the forms a complete lattice. We look at two representative games-Minimum Spanning Tree and Shortest Path-in this light.

SSRN Electronic Journal, 2013

In this paper we introduce the concept of quasi-building set that may underlie the coalitional structure of a cooperative game with restricted communication between the players. Each feasible coalition, including the set of all players, contains a nonempty subset called the choice set of the coalition. Only players that are in the choice set of a coalition are able to join to feasible subcoalitions to form the coalition and to obtain a marginal contribution. We demonstrate that all restricted communication systems that have been studied in the literature take the form of a quasi-building set for an appropriate set system and choice set. Every quasi-building set determines a nonempty collection of maximal strictly nested sets and each such set induces a rooted tree satisfying that every node of the tree is a player that is in the choice set of the feasible coalition that consists of himself and all his successors in the tree. Each tree corresponds to a marginal vector of the underlying game at which each player gets as payoff his marginal contribution when he joins his successors in the tree. As solution concept of a quasi-building set game we propose the average marginal vector (AMV) value, being the average of the marginal vectors that correspond to the trees induced by all maximal strictly nested sets of the quasi-building set. Properties of this solution are also studied. To establish core stability we introduce appropriate convexity conditions of the game with respect to the underlying quasi-building set. For some specifications of quasi-building sets, the AMV-value coincides with solutions known in the literature, for example, for building set games the solution coincides with the gravity center solution and the Shapley value recently defined for this class. For graph games it therefore differs from the well-known Myerson value. For a full communication system the solution coincides with the classical Shapley value.

Ec, 2002

We study mechanisms that can be modelled as coalitional games with transferable utilities, and apply ideas from mechanism design and game theory to problems arising in a network design setting. We establish an equivalence between the game-theoretic notion of agents being substitutes and the notion of frugality of a mechanism. We characterize the core of the network design game and relate it to outcomes in a sealed bid auction with VCG payments. We show that in a game, agents are substitutes if and only if the core of the forms a complete lattice. We look at two representative games-Minimum Spanning Tree and Shortest Path-in this light.

TURKISH JOURNAL OF ELECTRICAL ENGINEERING & COMPUTER SCIENCES

In game-theoretic settings the key notion of analysis is an equilibrium, which is a profile of agent strategies such that no viable coalition of agents can improve upon their coalitional welfare by jointly changing their strategies. A Nash equilibrium, where viable coalitions are only singletons, and a super strong equilibrium, where every coalition is deemed viable, are two extreme scenarios in regard to coalition formation. A recent trend in the literature is to consider equilibrium notions that allow for coalition formation in between these two extremes and which are suitable to model social coalition structures that arise in various real-life settings. The recent literature considered the question on the existence of equilibria under social coalition structures mainly in Resource Selection Games (RSGs), due to the simplicity of this game form and its wide range of application domains. We take the question on the existence of equilibria under social coalition structures from the perspective of computational complexity theory. We study the problem of deciding the existence of an equilibrium in RSGs with respect to a given social coalition structure. For an arbitrary coalition structure, we show that it is computationally intractable to decide whether an equilibrium exists even in very restricted settings of RSGs. In certain settings where an equilibrium is guaranteed to exist we give polynomial-time algorithms to find an equilibrium.

Eumas, 2005

It is self-evident that in numerous Multiagent settings, selfish agents stand to benefit from cooperating by forming coalitions. Nevertheless, negotiating a stable distribution of the payoff among agents may prove challenging. The issue of coalition formation has been investigated extensively in the field of cooperative n-person game theory, but until recently little attention has been given to the complications that arise when the players are software agents. The bounded rationality of such agents has motivated researchers to study the computational complexity of the aforementioned problems.

In many multiagent scenarios, agents distribute resources, such as time or energy, among several tasks. Having completed their tasks and generated profits, task payoffs must be divided among the agents in some reasonable manner. Cooperative games with overlapping coalitions (OCF games) are a recent framework proposed by , generalizing classic cooperative games to the case where agents may belong to more than one coalition. Having formed overlapping coalitions and divided profits, some agents may feel dissatisfied with their share of the profits, and would like to deviate from the given outcome. However, deviation in OCF games is a complicated matter: agents may decide to withdraw only some of their weight from some of the coalitions they belong to; that is, even after deviation, it is possible that agents will still be involved in tasks with non-deviators. This means that the desirability of a deviation, and the stability of formed coalitions, is to a great extent determined by the reaction of non-deviators. In this work, we explore algorithmic aspects of OCF games, focusing on the core in OCF games. We study the problem of deciding if the core of an OCF game is not empty, and whether a core payoff division can be found in polynomial time; moreover, we identify conditions that ensure that the problem admits polynomial time algorithms. Finally, we introduce and study a natural class of OCF games, the Linear Bottleneck Games. Interestingly, we show that such games always have a non-empty core, even assuming a highly lenient reaction to deviations.

Coalition formation is a key topic in multi-agent systems (mas). Coalitions enable agents to achieve goals that they may not have been able to achieve independently, and en- courages resource sharing among agents with different goals. A range of previous studies have found that problems in coalitional games tend to be computationally complex. However, such hardness results consider the entire input as one, ignoring any structural information on the instances. In the case of coalition formation problems, this bundles to- gether several distinct elements of the input, e.g. the agent set, the goal set, the resources, etc. In this paper we re- examine the complexity of coalition formation problems in the coalition resources game model, as a function of their distinct input elements, using the theory of parameterized complexity. The analysis shows that not all parts of the in- put are created equal, and that many instances of the prob- lem are actually tractable. We show that the problem...

Artificial Intelligence, 2010

Coalitional Resource Games (crgs) are a form of Non-Transferable Utility (ntu) game, which provide a natural formal framework for modelling scenarios in which agents must pool scarce resources in order to achieve mutually satisfying sets of goals. Although a number of computational questions surrounding crgs have been studied, there has to date been no attempt to develop solution concepts for crgs, or techniques for constructing solutions. In this paper, we rectify this omission. Following a review of the crg framework and a discussion of related work, we formalise notions of coalition structures and the core for crgs, and investigate the complexity of questions such as determining nonemptiness of the core. We show that, while such questions are in general computationally hard, it is possible to check the stability of a coalition structure in time exponential in the number of goals in the system, but polynomial in the number of agents and resources. As a consequence, checking stability is feasible for systems with small or bounded numbers of goals. We then consider constructive approaches to generating coalition structures. We present a negotiation protocol for crgs, give an associated negotiation strategy, and prove that this strategy forms a subgame perfect equilibrium. We then show that coalition structures produced by the protocol satisfy several desirable properties: Pareto optimality, dummy player, and pseudo-symmetry.

2012