A key question in cooperative game theory is that of coalitional stability, usually captured by the notion of the --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. In this paper, we investigate the possibility of stabilizing a coalitional game by using external payments. We consider a scenario where an external party, which is interested in having the players work together, offers a supplemental payment to the grand coalition (or, more generally, a particular coalition structure). This payment is conditional on players not deviating from their coalition(s). The sum of this payment plus the actual gains of the coalition(s) may then be divided among the agents so as to promote stability. We define the as the minimal external payment that stabilizes the game. We provide general bounds on the cost of stability in several classes of games, and explore its algorithmic properties. To develop a better intuition for the concepts we introduce, we provide a detailed algorithmic study of the cost of stability in weighted voting games, a simple but expressive class of games which can model decision-making in political bodies, and cooperation in multiagent settings. Finally, we extend our model and results to games with coalition structures.
Weighted voting games are a popular model of collaboration in multiagent systems. In such games, each agent has a weight (intuitively corresponding to resources he can contribute), and a coalition of agents wins if its total weight meets or exceeds a given threshold. Even though coalitional stability in such games is important, existing research has nonetheless only considered the stability of the grand coalition. In this paper, we introduce a model for weighted voting games with coalition structures. This is a natural extension in the context of multiagent systems, as several groups of agents may be simultaneously at work, each serving a different task. We then proceed to study stability in this context. First, we define the CS-core, a notion of the core for such settings, discuss its non-emptiness, and relate it to the traditional notion of the core in weighted voting games. We then investigate its computational properties. We show that, in contrast with the traditional setting, it is computationally hard to decide whether a game has a non-empty CS-core, or whether a given outcome is in the CS-core. However, we then provide an efficient algorithm that verifies whether an outcome is in the CS-core if all weights are small (polynomially bounded). Finally, we also suggest heuristic algorithms for checking the non-emptiness of the CS-core. Coalition formation is a key aspect of automated negotiation among self-interested agents. In order for coalitions to be stable, a key question that must be answered is how the gains from cooperation are to be distributed. Various solution concepts (such as the Shapley value, core, least core, and nucleolus) have been proposed. In this paper, we demonstrate how these concepts are vulnerable to various kinds of manipulations in open anonymous environments such as the Internet. These manipulations include submitting false names (one acting as many), collusion (many acting as one), and the hiding of skills. To address these threats, we introduce a new solution concept called the anonymity-proof core, which is robust to these manipulations. We show that the anonymity-proof core is characterized by certain simple axiomatic conditions. Furthermore, we show that by relaxing these conditions, we obtain a concept called the least anonymity-proof core, which is guaranteed to be non-empty. We also show that computational hardness of manipulation may provide an alternative barrier to manipulation. We present a new approach to representing coalitional games based on rules that describe the marginal contributions of the agents. This representation scheme captures characteristics of the interactions among the agents in a natural and concise manner. We also develop efficient algorithms for two of the most important solution concepts, the Shapley value and the core, under this representation. The Shapley value can be computed in time linear in the size of the input. The emptiness of the core can be determined in time exponential only in the treewidth of a graphical interpretation of our representation.  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. Coalition formation is a key problem in automated negotiation among self-interested agents. In order for coalition formation to be successful, a key question that must be answered is how the gains from cooperation are to be distributed. Various solution concepts have been proposed, but the computational questions around these solution concepts have received little attention. We study a concise representation of characteristic functions which allows for the agents to be concerned with a number of independent issues that each coalition of agents can address. For example, there may be a set of tasks that the capacity-unconstrained agents could undertake, where accomplishing a task generates a certain amount of value (possibly depending on how well the task is accomplished). Given this representation, we show how to quickly compute the Shapley value--a seminal value division scheme that distributes the gains from cooperation fairly in a certain sense. We then show that in (distributed) marginal-contribution based value division schemes, which are known to be vulnerable to manipulation of the order in which the agents are added to the coalition, this manipulation is NP-complete. Thus, computational complexity serves as a barrier to manipulating the joining order. Finally, we show that given a value division, determining whether some subcoalition has an incentive to break away (in which case we say the division is not in the core) is NP-complete. So, computational complexity serves to increase the stability of the coalition. We study the complexity of influencing elections through bribery: How computationally complex is it for an external actor to determine whether by a certain amount of bribing voters a specified candidate can be made the election's winner? We study this problem for election systems as varied as scoring protocols and Dodgson voting, and in a variety of settings regarding homogeneous-vs.-nonhomogeneous electorate bribability, bounded-size-vs.-arbitrary-sized candidate sets, weighted-vs.-unweighted voters, and succinct-vs.-nonsuccinct input specification. We obtain both polynomial-time bribery algorithms and proofs of the intractability of bribery, and indeed our results show that the complexity of bribery is extremely sensitive to the setting. For example, we find settings in which bribery is NP-complete but manipulation (by voters) is in P, and we find settings in which bribing weighted voters is NP-complete but bribing voters with individual bribe thresholds is in P. For the broad class of elections (including plurality, Borda, k-approval, and veto) known as scoring protocols, we prove a dichotomy result for bribery of weighted voters: We find a simple-to-evaluate condition that classifies every case as either NP-complete or in P. 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 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 paper we define a concise, natural, general representation for games in characteristic form that relies on superadditivity. In our representation, individual agents' values are given as well as values for those coalitions that introduce synergies. We show that this representation allows for efficient checking of whether a given outcome is in the core. We then show that determining whether the core is nonempty is NP-complete both with and without transferable utility. We demonstrate that what makes the problem hard in both cases is determining the collaborative possibilities (the set of outcomes possible for the grand coalition); we do so by showing that if these are given, the problem becomes solvable in time polynomial in the size of the representation in both cases. However, we then demonstrate that for a hybrid version of the problem, where utility transfer is possible only within the grand coalition, the problem remains NP-complete even when the collaborative possibilities are given. Finally, we show that for convex characteristic functions, a solution in the core can be computed efficiently (in O(nl^2) time, where n is the number of agents and l is the number of synergies), even when the collaborative possibilities are not given in advance. 