Coalition formation is a key problem in automated negotiation among self-interested agents, and other multiagent applications. A coalition of agents can sometimes accomplish things that the individual agents cannot, or can do things more efficiently. However,

This paper deals with the weighted majority game, which is a familiar example of voting systems. In 1960s, U.S. Supreme Court handed down a series of "one person one vote" decisions. After that, calculations of power indices using real data were carried out and presented as evidence in the courtroom. For example, the courts in New York State have accepted the Banzhaf index (also called the Coleman value or Chow parameters) as an appropriate measure for weighted voting systems. The calculation normally requires the aid of a computer and so many counties in U.S. hire specialized consultants, mathematicians or computer scientists (see [13]). 1 In this paper, we discuss some algorithms for calculating power indices. In Section 2, we define weighted majority games and related concepts. Section 3 defines three power indices, the Shapley-Shubik power index, the Banzhaf index and the Deegan-Packel index. Section 4 shows complexity classes of the problems for calculating power indices

Coalition formation is a key problem in automated negotiation among selfinterested agents, and other multiagent applications. A coalition of agents can sometimes accomplish things that the individual agents cannot, or can accomplish them more efficiently. Motivating the agents to abide by a solution requires careful analysis: only some of the solutions are stable in the sense that no group of agents is motivated to break off and form a new coalition. This constraint has been studied extensively in cooperative game theory: the set of solutions that satisfy it is known as the core. The computational questions around the core have received less attention. When it comes to coalition formation among software agents (that represent real-world parties), these questions become increasingly explicit. In this

Weighted threshold games are coalitional games in which each player has a weight (intuitively corresponding to its voting power), and a coalition is successful if the sum of its weights exceeds a given threshold. Key questions in coalitional games include finding coalitions that are stable (in the sense that no member of the coalition has any rational incentive to leave it), and finding a division of payoffs to coalition members (an imputation) that is fair. We investigate the computational complexity of such questions for weighted threshold games. We study the core, the least core, and the nucleolus, distinguishing those problems that are polynomial-time computable from those that are NP-hard, and providing pseudopolynomial and approximation algorithms for the NP-hard problems.

Let N = f1, ..., ng be a finite set of players and KN the complete graph on the node set N [ f0g. Assume that the edges of KN have nonnegative weights and associate with each coalition S ` N of players as cost c(S) the weight of a minimal spanning tree on the node set S [ f0g. Using reduction to EXACT COVER BY 3-SETS, we exhibit the following problem to be NP-complete. Given the vector x 2 ! N with x(N ) = c(N ), decide whether there exists a coalition S such that x(S) ? c(S).

www.elsevier.com/locate/artint We study Coalitional Resource Games (CRGs), a variation of Qualitative Coalitional Games (QCGs) in which each agent is endowed with a set of resources, and the ability of a coalition to bring about a set of goals depends on whether they are collectively endowed with the necessary resources. We investigate and classify the computational complexity of a number of natural decision problems for CRGs, over and above those previously investigated for QCGs in general. For example, we show that the complexity of determining whether conflict is inevitable between two coalitions with respect to some stated resource bound (i.e., a limit value for every resource) is co-NP-complete. We then investigate the relationship between CRGs and QCGs, and in particular the extent to which it is possible to translate between the two models. We first characterise the complexity of determining equivalence between CRGs and QCGs. We then show that it is always possible to translate any given CRG into a succinct equivalent QCG, and that it is not always possible to translate a QCG into an equivalent CRG; we establish some necessary and some sufficient conditions for a translation from QCGs to CRGs to be possible, and show that even where an equivalent CRG exists, it may have size exponential in the number of goals and agents of its source QCG.

A general ordertheoretic linear programming model for the study of matroid-type greedy algorithms is introduced. The primal restrictions are given by so-called weakly increasing submodular functions on antichains. The LP-dual is solved by a Monge-type greedy algorithm. The model offers a direct combinatorial explanation for many integrality results in discrete optimization. In particular, the submodular intersection theorem of Edmonds and Giles is seen to extend to the case with a rooted forest as underlying structure. The core of associated polyhedra is introduced and applications to the existence of the core in cooperative game theory are discussed.

Preference aggregation is used in a variety of multiagent applications, and as a result, voting theory has become an important topic in multiagent system research. However, power indices (which reflect how much “real power ” a voter has in a weighted voting system) have received relatively little attention, although they have long been studied in political science and economics. The Banzhaf power index is one of the most popular; it is also well-defined for any simple coalitional game. In this paper, we examine the computational complexity of calculating the Banzhaf power index within a particular multiagent domain, a network flow game. Agents control the edges of a graph; a coalition wins if it can send a flow of a given size from a source vertex to a target vertex. The relative power of each edge/agent reflects its significance in enabling such a flow, and in real-world networks could be used, for example, to allocate resources for maintaining parts of the network. We show that calculating the Banzhaf power index of each agent in this network flow domain is #P-complete. We also show that for some restricted network flow domains there exists a polynomial algorithm to calculate agents ’ Banzhaf power indices.

We present and formally investigate Cooperative Boolean Games, a new, natural family of coalitional games that are both compact and expressive. In such a game, an agent’s primary aim is to achieve its individual goal, which is represented as a propositional logic formula over some set of Boolean variables. Each agent is assumed to exercise unique control over some subset of the overall set of Boolean variables, and the set of valuations for these variables corresponds to the set of actions the agent can take. However, the actions available to an agent are assumed to have some cost, and an agent’s secondary aim is to minimise its costs. Typically, an agent must cooperate with others because it does not have sufficient control to ensure its goal is satisfied. However, the desire to minimise costs leads to preferences over possible coalitions, and hence to strategic behaviour. Following an introduction to the formal framework of Cooperative Boolean Games, we investigate solution concepts of the core and stable sets for them. In each case, we characterise the complexity of the associated solution concept, and discuss the surrounding issues. Finally, we present a bargaining protocol for cooperation in Boolean games, and characterise the strategies in equilibrium for this protocol.

We introduce cooperative games (N, v) with a description polynomial in n, where n is the number of players. In order to study the complexity of cooperative game problems, we assume that a cooperative game v :2 N ! Q is given by an oracle returning v (S) for each query S N: Finally, we consider several cooperative game problems and we give a list of complexity results.