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18
A Tractable and Expressive Class of Marginal Contribution Nets and Its Applications
, 2008
"... Coalitional games raise a number of important questions from the point of view of computer science, key among them being how to represent such games compactly, and how to efficiently compute solution concepts assuming such representations. Marginal contribution nets (MCnets), introduced by Ieong an ..."
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Cited by 24 (3 self)
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Coalitional games raise a number of important questions from the point of view of computer science, key among them being how to represent such games compactly, and how to efficiently compute solution concepts assuming such representations. Marginal contribution nets (MCnets), introduced by Ieong and Shoham, are one of the simplest and most influential representation schemes for coalitional games. MCnets are a rulebased formalism, in which rules take the form pattern − → value, where “pattern ” is a Boolean condition over agents, and “value ” is a numeric value. Ieong and Shoham showed that, for a class of what we will call “basic” MCnets, where patterns are constrained to be a conjunction of literals, marginal contribution nets permit the easy computation of solution concepts such as the Shapley value. However, there are very natural classes of coalitional game that
Approximating power indices: theoretical and empirical analysis
, 2010
"... Many multiagent domains where cooperation among agents is crucial to achieving a common goal can be modeled as coalitional games. However, in many of these domains, agents are unequal in their power to affect the outcome of the game. Prior research on weighted voting games has explored power indice ..."
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Cited by 24 (9 self)
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Many multiagent domains where cooperation among agents is crucial to achieving a common goal can be modeled as coalitional games. However, in many of these domains, agents are unequal in their power to affect the outcome of the game. Prior research on weighted voting games has explored power indices, which reflect how much “real power” a voter has. Although primarily used for voting games, these indices can be applied to any simple coalitional game. Computing these indices is known to be computationally hard in various domains, so one must sometimes resort to approximate methods for calculating them. We suggest and analyze randomized methods to approximate power indices such as the Banzhaf power index and the Shapley–Shubik power index. Our approximation algorithms do not depend on a specific representation of the game, so they can be used in any simple coalitional game. Our methods are based on testing the game’s value for several sample coalitions. We show that no approximation algorithm can do much better for general coalitional games, by providing lower bounds for both deterministic and randomized algorithms for calculating power indices. We also provide empirical results regarding our method, and show that it typically achieves much better accuracy and confidence than those required.
Cooperative Games with Overlapping Coalitions
"... 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 th ..."
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Cited by 23 (9 self)
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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 (nonoverlapping) 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 nonempty 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 (nonoverlapping) 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. 1.
Approximating Power Indices
"... Many multiagent domains where cooperation among agents is crucial to achieving a common goal can be modeled as coalitional games. However, in many of these domains, agents are unequal in their power to affect the outcome of the game. Prior research on weighted voting games has explored power indices ..."
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Cited by 22 (8 self)
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Many multiagent domains where cooperation among agents is crucial to achieving a common goal can be modeled as coalitional games. However, in many of these domains, agents are unequal in their power to affect the outcome of the game. Prior research on weighted voting games has explored power indices, which reflect how much “real power ” a voter has. Although primarily used for voting games, these indices can be applied to any simple coalitional game. Computing these indices is known to be computationally hard in various domains, so one must sometimes resort to approximate methods for calculating them. We suggest and analyze randomized methods to approximate power indices such as the Banzhaf power index and the ShapleyShubik power index. Our approximation algorithms do not depend on a specific representation of the game, so they can be used in any simple coalitional game. Our methods are based on testing the game’s value for several sample coalitions. We also show that no approximation algorithm can do much better for general coalitional games, by providing lower bounds for both deterministic and randomized algorithms for calculating power indices.
Power and stability in connectivity games
 In AAMAS (2
, 2008
"... We consider computational aspects of a game theoretic approach to network reliability. Consider a network where failure of one node may disrupt communication between two other nodes. We model this network as a simple coalitional game, called the vertex Connectivity Game (CG). In this game, each agen ..."
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Cited by 16 (6 self)
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We consider computational aspects of a game theoretic approach to network reliability. Consider a network where failure of one node may disrupt communication between two other nodes. We model this network as a simple coalitional game, called the vertex Connectivity Game (CG). In this game, each agent owns a vertex, and controls all the edges going to and from that vertex. A coalition of agents wins if it fully connects a certain subset of vertices in the graph, called the primary vertices. We show that power indices, which express an agent’s ability to affect the outcome of the vertex connectivity game, can be used to identify significant possible points of failure in the communication network, and can thus be used to increase network reliability. We show that in general graphs, calculating the Banzhaf power index is #Pcomplete, but suggest a polynomial algorithm for calculating this index in trees. We also show a polynomial algorithm for computing the core of a CG, which allows a stable division of payments to coalition agents.
Power indices in spanning connectivity games
 in Proc. of the AAIM
, 2009
"... Abstract. The Banzhaf index, ShapleyShubik index and other voting power indices measure the importance of a player in a coalitional game. We consider a simple coalitional game called the spanning connectivity game (SCG) based on an undirected, unweighted multigraph, where edges are players. We ex ..."
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Cited by 14 (2 self)
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Abstract. The Banzhaf index, ShapleyShubik index and other voting power indices measure the importance of a player in a coalitional game. We consider a simple coalitional game called the spanning connectivity game (SCG) based on an undirected, unweighted multigraph, where edges are players. We examine the computational complexity of computing the voting power indices of edges in the SCG. It is shown that computing Banzhaf values is #Pcomplete and computing ShapleyShubik indices or values is NPhard for SCGs. Interestingly, Holler indices and DeeganPackel indices can be computed in polynomial time. Among other results, it is proved that Banzhaf indices can be computed in polynomial time for graphs with bounded treewidth. It is also shown that for any reasonable representation of a simple game, a polynomial time algorithm to compute the ShapleyShubik indices implies a polynomial time algorithm to compute the Banzhaf indices. This answers (positively) an open question of whether computing ShapleyShubik indices for a simple game represented by the set of minimal winning coalitions is NPhard.
Coalitional Structure Generation in Skill Games
"... We consider optimizing the coalition structure in Coalitional Skill Games (CSGs), a succinct representation of coalitional games (Bachrach and Rosenschein 2008). In CSGs, the value of a coalition depends on the tasks its members can achieve. The tasks require various skills to complete them, and age ..."
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Cited by 13 (4 self)
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We consider optimizing the coalition structure in Coalitional Skill Games (CSGs), a succinct representation of coalitional games (Bachrach and Rosenschein 2008). In CSGs, the value of a coalition depends on the tasks its members can achieve. The tasks require various skills to complete them, and agents may have different skill sets. The optimal coalition structure is a partition of the agents to coalitions, that maximizes the sum of utilities obtained by the coalitions. We show that CSGs can represent any characteristic function, and consider optimal coalition structure generation in this representation. We provide hardness results, showing that in general CSGs, as well as in very restricted versions of them, computing the optimal coalition structure is hard. On the positive side, we show that the problem can be reformulated as constraint satisfaction on a hyper graph, and present an algorithm that finds the optimal coalition structure in polynomial time for instances with bounded treewidth and number of tasks.
Mechanisms for MultiLevel Marketing
"... Multilevel marketing is a marketing approach that motivates its participants to promote a certain product among their friends. The popularity of this approach increases due to the accessibility of modern social networks, however, it existed in one form or the other long before the Internet age bega ..."
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Cited by 11 (1 self)
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Multilevel marketing is a marketing approach that motivates its participants to promote a certain product among their friends. The popularity of this approach increases due to the accessibility of modern social networks, however, it existed in one form or the other long before the Internet age began (the infamous Pyramid scheme that dates back at least a century is in fact a special case of multilevel marketing). This paper lays foundations for the study of reward mechanisms in multilevel marketing within social networks. We provide a set of desired properties for such mechanisms and show that they are uniquely satisfied by geometric reward mechanisms. The resilience of mechanisms to falsename manipulations is also considered; while geometric reward mechanisms fail against such manipulations, we exhibit other mechanisms which are falsenameproof.
Power in Normative Systems
, 2009
"... Power indices such as the Banzhaf index were originally developed within voting theory in an attempt to rigorously characterise the influence that a voter is able to wield in a particular voting game. In this paper, we show how such power indices can be applied to understanding the relative importan ..."
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Cited by 8 (4 self)
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Power indices such as the Banzhaf index were originally developed within voting theory in an attempt to rigorously characterise the influence that a voter is able to wield in a particular voting game. In this paper, we show how such power indices can be applied to understanding the relative importance of agents when we attempt to devise a coordination mechanism using the paradigm of social laws, or normative systems. Understanding how pivotal an agent is with respect to the success of a particular social law is of benefit when designing such social laws: we might typically aim to ensure that power is distributed evenly amongst the agents in a system, to avoid bottlenecks or single points of failure. After formally defining the framework and illustrating the role of power indices in it, we investigate the complexity of computing these indices, showing that the characteristic complexity result is #Pcompleteness. We then investigate cases where computing indices is computationally easy.