Accessibility and stability of the coalition structure core (original) (raw)

On the stability of an Optimal Coalition Structure

The two main questions in coalition games are 1) what coalitions should form and 2) how to distribute the value of each coalition between its members. When a game is not superadditive, other coalition structures (CSs) may be more attractive than the grand coalition. For example, if the agents care about the total payoff generated by the entire society, CSs that maximize utilitarian social welfare are of interest. The search for such optimal CSs has been a very active area of research. Stability concepts have been defined for games with coalition structure, under the assumption that the agents agree first on a CS, and then the members of each coalition decide on how to share the value of their coalition. An agent can refer to the values of coalitions with agents outside of its current coalition to argue for a larger share of the coalition payoff. To use this approach, one can find the CS s★ with optimal value and use one of these stability concepts for the game with s★. However, it m...

On non-transferable utility games with coalition structure

International Journal of Game Theory, 1991

We introduce a solution function for Non-transferable Utility (NTU) games when prior coalition structure isgiven. This solution function generalizes both the Harsanyi solution function for NTU games and the Owen solution for TU games with coalition structure.

The Cost of Stability in Coalitional Games

Lecture Notes in Computer Science, 2009

A key question in cooperative game theory is that of coalitional stability, usually captured by the notion of the core-the set of outcomes such that no subgroup of players has an incentive to deviate. However, some coalitional games have empty cores, and any outcome in such a game is unstable.

The core of a game with a continuum of players and finite coalitions: The model and some results

Mathematical Social Sciences, 1986

In this paper we develop a new model of a cooperative game with a continuum of players. In our model, only finite coalitions-ones containing only finite numbers of players-are permitted to form. Outcomes of cooperative behavior are attainable by partitions of the players into finite coalitions. This is appropriate in view of our restrictions on coalition formation. Once feasible outcomes fare properly defined, the core concept is standard-no permissible coalition can improve upon its outcome. We provide a sufficient condition for the nonemptiness of the core in the case where the players can be divided into a finite number of types, This result is applied to a market game and the nonemptiness of the core of the market game is stated under considerably weaker conditions (but with finite types). In addition, it is illustrated that the framework applies to assignment games with a continuum of players.

On the complexity of the core over coalition structures

IJCAI International Joint Conference on Artificial Intelligence, 2011

The computational complexity of relevant corerelated questions for coalitional games is addressed from the coalition structure viewpoint, i.e., without assuming that the grand-coalition necessarily forms. In the analysis, games are assumed to be in "compact" form, i.e., their worth functions are implicitly given as polynomial-time computable functions over succinct game encodings provided as input. Within this setting, a complete picture of the complexity issues arising with the core, as well as with the related stability concepts of least core and cost of stability, is depicted. In particular, the special cases of superadditive games and of games whose sets of feasible coalitions are restricted over tree-like interaction graphs are also studied.

E ciency in Coalition Games with Externalities

2006

A natural,extension,of superadditivity,is not su¢ cient to imply that the grand,coalition is e¢ cient when,externalities are present. We provide a con- dition –analogous,to convexity–that,is su¢ cient for the grand coalition to be e¢ cient and show,that this also implies that the (appropriately de…ned) core is nonempty. Moreover, we propose a mechanism which implements the most e¢ cient partition for all coalition formation,games,and characterize the payo¤ division of the mechanism. JEL Classi…cation Numbers: C71, C72, D62 Keywords: Coalition formation, externalities, partition function games, Shapley value, implementation.

Solutions for Games with General Coalitional Structure and Choice Sets

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.

Cooperative Games with Overlapping Coalitions Georgios Chalkiadakis Evangelos Markakis

In the usual models of cooperative game theory, the outcome of a coalition formation process is either the grand coalition or a coalition structure that consists of disjoint coalitions. However, in many domains where coalitions are associated with tasks, an agent may be involved in executing more than one task, and thus may distribute his resources among several coalitions. To tackle such scenarios, we introduce a model for cooperative games with overlapping coalitions-or overlapping coalition formation (OCF) games. We then explore the issue of stability in this setting. In particular, we introduce a notion of the core, which generalizes the corresponding notion in the traditional (non-overlapping) scenario. Then, under some quite general conditions, we characterize the elements of the core, and show that any element of the core maximizes the social welfare. We also introduce a concept of balancedness for overlapping coalitional games, and use it to characterize coalition structures that can be extended to elements of the core. Finally, we generalize the notion of convexity to our setting, and show that under some natural assumptions convex games have a non-empty core. Moreover, we introduce two alternative notions of stability in OCF that allow a wider range of deviations, and explore the relationships among the corresponding definitions of the core, as well as the classic (non-overlapping) core and the Aubin core. We illustrate the general properties of the three cores, and also study them from a computational perspective, thus obtaining additional insights into their fundamental structure.

Coalitional Games on Graphs: Core Structure, Substitutes

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.