Partition Function Games Research Articles

Expectation formation rules and the core of partition function games

This paper proposes axiomatic foundations of expectation formation rules, by which deviating players anticipate the reaction of external players in a partition function game. The projection rule is the only rule satisfying subset consistency and responsiveness to the original partition of non-deviating players. It is also the only rule satisfying subset consistency, independence of the original partition of deviating players, and coherence of expectations. Exogenous rules are the only rules satisfying subset consistency and independence of the original partition, and the pessimistic rule is the only rule generating superadditive coalitional games.

Anytime coalition structure generation in multi-agent systems with positive or negative externalities

Much of the literature on multi-agent coalition formation has focused on Characteristic Function Games, where the effectiveness of a coalition is not affected by how the other agents are arranged in the system. In contrast, very little attention has been given to the more general class of Partition Function Games, where the emphasis is on how the formation of one coalition could influence the performance of other co-existing coalitions in the system. However, these inter-coalitional dependencies, called externalities from coalition formation, play a crucial role in many real-world multi-agent applications where agents have either conflicting or overlapping goals.Against this background, this paper is the first computational study of coalitional games with externalities in the multi-agent system context. We focus on the Coalition Structure Generation (CSG) problem which involves finding an exhaustive and disjoint division of the agents into coalitions such that the performance of the entire system is optimized. While this problem is already very challenging in the absence of externalities, due to the exponential size of the search space, taking externalities into consideration makes it even more challenging as the size of the input, given n agents, grows from O(2n) to O(nn).Our main contribution is the development of the first CSG algorithm for coalitional games with either positive or negative externalities. Specifically, we prove that it is possible to compute upper and lower bounds on the values of any set of disjoint coalitions. Building upon this, we prove that in order to establish a worst-case guarantee on solution quality it is necessary to search a certain set of coalition structures (which we define). We also show how to progressively improve this guarantee with further search.Since there are no previous CSG algorithms for games with externalities, we benchmark our algorithm against other state-of-the-art approaches in games where no externalities are present. Surprisingly, we find that, as far as worst-case guarantees are concerned, our algorithm outperforms the others by orders of magnitude. For instance, to reach a bound of 3 given 24 agents, the number of coalition structures that need to be searched by our algorithm is only 0.0007% of that needed by Sandholm et al. (1999) [1], and 0.5% of that needed by Dang and Jennings (2004) [2]. This is despite the fact that the other algorithms take advantage of the special properties of games with no externalities, while ours does not.

Games with externalities: games in coalition configuration function form

In this paper we introduce a model of cooperative game with externalities which generalizes games in partition function form by allowing players to take part in more than one coalition. We provide an extension of the Shapley value (1953) to these games, which is a generalization of the Myerson value (1977) for games in partition function form. This value is derived by considering an adaptation of an axiomatic characterization of the Myerson value (1977).

THE CLASS OF EFFICIENT LINEAR SYMMETRIC VALUES FOR GAMES IN PARTITION FUNCTION FORM

In this paper we study linear symmetric solutions for the space of games in partition function form with n players. In particular, we provide an expression for all linear, symmetric and efficient solutions. Furthermore, adding an additional axiom, we identify a unique value satisfying these properties.

Strategic Aspects of the 1995 and 2004 EU Enlargements

While the 1995 entrants to the EU are by now fully integrated, those joining in 2004 still “enjoy” a secondary status for a number of years. We attribute this difference to the fact that unlike the former EFTA members joining in 1995, the 2004 entrants formed a group with heterogenous interests, one that lacked the same strong internal economic ties. Not being able to act as a unified block they had a considerably weaker bargaining position. We support our arguments by qualitative results from a simple model, a dynamic partition function game based on Yi (Games Econ Behav 20:201–237, 1997) and Morelli and Penelle (Economic integration as a partition function game 1997).

AN AXIOM SYSTEM FOR A VALUE FOR GAMES IN PARTITION FUNCTION FORM

Lucas and Trall (1963) defined the games in partition function form as a generalization of the cooperative games with transferable utility. In our work we propose by means of an axiomatic characterization a solution for such games in partition function form. This solution will be a generalization of the Shapley value (1953).

Stable Organizations with Externalities

We study the stability properties of organizations in partition function games, describing cooperative situations with externalities. An organization is defined as a group of agents, together with a set of bilateral relations, formally, a connected graph. Because of the presence of externalities, the profitability of coalitional threats to an organization depend on the reaction of non coalitional members. This reaction is likely to depend on the links that non coalitional members maintain in the organization. We show that this directly implies that minimally connected organizations emerge under positive externalities, while the fully connected organization emerges under negative. This result is shown to hold independently of the adopted payoff imputation rule. Sharper predictions are possible for the specific case of the egalitarian rule. Here, if only coalitions that are connected in the organization can effectively object to it, the star organization prevails under positive externalities, and the wheel, a non fully connected organization, prevails under negative.

The Core in the Presence of Externalities

We generalise the coalition structure core to partition function games. Our definition relies only on a single crucial assumption, namely that there is some internal consistency in the game. Residuals of the deviation play a game similar to the initial one, and — whenever this is possible — they come to a residual core outcome. Deviating players form their optimistic or pessimistic expectations with this in mind. This leads to a recursive definition of the core. Compared to existing approaches, our core concept displays a reduced sensitivity to behavioural assumptions. We consider the core of an economy with a common pool resource defined by Funaki and Yamato [4], and show that for a number of numerical examples our core concept resolves the puzzle arising when more naive approaches were used. We outline possibilities for further extensions.

The Core of a Partition Function Game

LASZL´ O´ A. K´ OCZY´Abstract. We consider partition function games and introduce new defini-tions of the core that include the effects of externalities. We assume that allplayers behave rationally and that all stable outcomes arising are consistentwith the appropriate generalised concept of the core. The result is a recursivedefinition of the core where residual subgames are considered as games withfewer players and with a partition function that captures the externalities ofthe deviating coalition. Some properties of the new concepts are discussed.

A set of axioms for a value for partition function games

Myerson (1977) derived an efficient value for games in partition function form. In this paper, we present a set of axioms which characterize a different efficient value for such games. This latter value assigns value 0 to dummies and assigns nonnegative values to players in monotone simple games.

Values of games in partition function form

Shapley's value axioms are adapted to games in partition function form, and the natural extension of theShapley value for partition function games is derived.

Solutions for a class of n‐person games in partition function form

AbstractA unique polyhedral solution is given for most n‐person games in partition function form when only partitions into coalitions with 1, n‐1, and n players have large payoffs.