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Nicholas R Jennings - One of the best experts on this subject based on the ideXlab platform.
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efficient computation of the Shapley Value for game theoretic network centrality
arXiv: Computer Science and Game Theory, 2014Co-Authors: Tomasz Michalak, Karthik V Aadithya, Balaraman Ravindran, Piotr L Szczepanski, Nicholas R JenningsAbstract:The Shapley Value---probably the most important normative payoff division scheme in coalitional games---has recently been advocated as a useful measure of centrality in networks. However, although this approach has a variety of real-world applications (including social and organisational networks, biological networks and communication networks), its computational properties have not been widely studied. To date, the only practicable approach to compute Shapley Value-based centrality has been via Monte Carlo simulations which are computationally expensive and not guaranteed to give an exact answer. Against this background, this paper presents the first study of the computational aspects of the Shapley Value for network centralities. Specifically, we develop exact analytical formulae for Shapley Value-based centrality in both weighted and unweighted networks and develop efficient (polynomial time) and exact algorithms based on them. We empirically evaluate these algorithms on two real-life examples (an infrastructure network representing the topology of the Western States Power Grid and a collaboration network from the field of astrophysics) and demonstrate that they deliver significant speedups over the Monte Carlo approach. For instance, in the case of unweighted networks our algorithms are able to return the exact solution about 1600 times faster than the Monte Carlo approximation, even if we allow for a generous 10% error margin for the latter method.
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efficient computation of the Shapley Value for centrality in networks
Workshop on Internet and Network Economics, 2010Co-Authors: Karthik V Aadithya, Balaraman Ravindran, Tomasz Michalak, Nicholas R JenningsAbstract:The Shapley Value is arguably the most important normative solution concept in coalitional games. One of its applications is in the domain of networks, where the Shapley Value is used to measure the relative importance of individual nodes. This measure, which is called node centrality, is of paramount significance in many real-world application domains including social and organisational networks, biological networks, communication networks and the internet. Whereas computational aspects of the Shapley Value have been analyzed in the context of conventional coalitional games, this paper presents the first such study of the Shapley Value for network centrality. Our results demonstrate that this particular application of the Shapley Value presents unique opportunities for efficiency gains, which we exploit to develop exact analytical formulas for Shapley Value based centrality computation in both weighted and unweighted networks. These formulas not only yield efficient (polynomial time) and error-free algorithms for computing node centralities, but their surprisingly simple closed form expressions also offer intuition into why certain nodes are relatively more important to a network.
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a randomized method for the Shapley Value for the voting game
Adaptive Agents and Multi-Agents Systems, 2007Co-Authors: Shaheen Fatima, Michael Wooldridge, Nicholas R JenningsAbstract:The Shapley Value is one of the key solution concepts for coalition games. Its main advantage is that it provides a unique and fair solution, but its main problem is that, for many coalition games, the Shapley Value cannot be determined in polynomial time. In particular, the problem of finding this Value for the voting game is known to be #P-complete in the general case. However, in this paper, we show that there are some specific voting games for which the problem is computationally tractable. For other general voting games, we overcome the problem of computational complexity by presenting a new randomized method for determining the approximate Shapley Value. The time complexity of this method is linear in the number of players. We also show, through empirical studies, that the percentage error for the proposed method is always less than 20% and, in most cases, less than 5%.
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An analysis of the Shapley Value and its uncertainty for the voting game
2006Co-Authors: Shaheen Fatima, Michael Wooldridge, Nicholas R JenningsAbstract:The Shapley Value provides a unique solution to coalition games and is used to evaluate a player's prospects of playing a game. Although it provides a unique solution, there is an element of uncertainty associated with this Value. This uncertainty in the solution of a game provides an additional dimension for evaluating a player's prospects of playing the game. Thus, players want to know not only their Shapley Value for a game, but also the associated uncertainty. Given this, our objective is to determine the Shapley Value and its uncertainty and study the relationship between them for the voting game. But since the problem of determining the Shapley Value for this game is #P-complete, we first present a new polynomial time randomized method for determining the approximate Shapley Value. Using this method, we compute the Shapley Value and correlate it with its uncertainty so as to allow agents to compare games on the basis of both their Shapley Values and the associated uncertainties. Our study shows that, a player's uncertainty first increases with its Shapley Value and then decreases. This implies that the uncertainty is at its minimum when the Value is at its maximum, and that agents do not always have to compromise Value in order to reduce uncertainty.
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AAMAS - An analysis of the Shapley Value and its uncertainty for the voting game
Proceedings of the fourth international joint conference on Autonomous agents and multiagent systems - AAMAS '05, 2005Co-Authors: Shaheen Fatima, Michael Wooldridge, Nicholas R JenningsAbstract:The Shapley Value provides a unique solution to coalition games and is used to evaluate a player's prospects of playing a game. Although it provides a unique solution, there is an element of uncertainty associated with this Value. This uncertainty in the solution of a game provides an additional dimension for evaluating a player's prospects of playing the game. Thus, players want to know not only their Shapley Value for a game, but also the associated uncertainty. Given this, our objective is to determine the Shapley Value and its uncertainty and study the relationship between them for the voting game. But since the problem of determining the Shapley Value for this game is #P-complete, we first present a new polynomial time randomized method for determining the approximate Shapley Value. Using this method, we compute the Shapley Value and correlate it with its uncertainty so as to allow agents to compare games on the basis of both their Shapley Values and the associated uncertainties. Our study shows that, a player's uncertainty first increases with its Shapley Value and then decreases. This implies that the uncertainty is at its minimum when the Value is at its maximum, and that agents do not always have to compromise Value in order to reduce uncertainty.
Henk Norde - One of the best experts on this subject based on the ideXlab platform.
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The Shapley Value for Partition Function Form Games
International Game Theory Review, 2007Co-Authors: Kim Hang Pham, Henk NordeAbstract:Different axiomatizations of the Shapley Value for TU games can be found in the literature. The Shapley Value has been generalized in several ways to the class of games in partition function form. In this paper we discuss another generalization of the Shapley Value and provide a characterization.
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The Shapley Value for Partition Function Form Games
SSRN Electronic Journal, 2002Co-Authors: Kim Hang Pham, Henk NordeAbstract:Different axiomatic systems for the Shapley Value can be found in the literature. For games with a coalition structure, the Shapley Value also has been axiomatized in several ways. In this paper, we discuss a generalization of the Shapley Value to the class of partition function form games. The concepts and axioms, related to the Shapley Value, have been extended and a characterization for the Shapley Value has been provided. Finally, an application of the Shapley Value is given.
Claus-jochen Haake - One of the best experts on this subject based on the ideXlab platform.
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The Shapley Value of Phylogenetic Trees
2016Co-Authors: Claus-jochen Haake, Akemi KashiwadaAbstract:Every weighted tree corresponds naturally to a cooperative game that we call a tree game; it assigns to each subset of leaves the sum of the weights of the minimal subtree spanned by those leaves. In the context of phylogenetic trees, the leaves are species and this assignment captures the diversity present in the coalition of species considered. We consider the Shapley Value of tree games and suggest a biological interpretation. We determine the linear transformation M that shows the dependence of the Shapley Value on the edge weights of the tree, and we also compute a null space basis of M. Finally, we characterize the Shapley Value on tree games by five axioms, a counterpart to Shapley's original theorem on the larger class of cooperative games. We also include a brief discussion of the core of tree games.
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Comments on: Transversality of the Shapley Value
TOP, 2008Co-Authors: Claus-jochen HaakeAbstract:The paper by Stefano Moretti and Fioravante Patrone celebrates the possibly most influential solution concepts from cooperative game theory: the Shapley Value. With their excellent collection of applications, in which the Shapley Value provides a meaningful solution, the authors in fact not only highlight the importance of this particular solution but also the relevance of cooperative game theory as a “technique” usable in other fields of research.
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the Shapley Value of phylogenetic trees
Journal of Mathematical Biology, 2008Co-Authors: Claus-jochen Haake, Akemi KashiwadaAbstract:Every weighted tree corresponds naturally to a cooperative game that we call a tree game; it assigns to each subset of leaves the sum of the weights of the minimal subtree spanned by those leaves. In the context of phylogenetic trees, the leaves are species and this assignment captures the diversity present in the coalition of species considered. We consider the Shapley Value of tree games and suggest a biological interpretation. We determine the linear transformation M that shows the dependence of the Shapley Value on the edge weights of the tree, and we also compute a null space basis of M. Both depend on the split counts of the tree. Finally, we characterize the Shapley Value on tree games by four axioms, a counterpart to Shapley's original theorem on the larger class of cooperative games. We also include a brief discussion of the core of tree games.
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The Shapley Value of Phylogenetic Trees
Journal of mathematical biology, 2007Co-Authors: Claus-jochen Haake, Akemi KashiwadaAbstract:Every weighted tree corresponds naturally to a cooperative game that we call a "tree game"; it assigns to each subset of leaves the sum of the weights of the minimal subtree spanned by those leaves. In the context of phylogenetic trees, the leaves are species and this assignment captures the diversity present in the coalition of species considered. We consider the Shapley Value of tree games and suggest a biological interpretation. We determine the linear transformation M that shows the dependence of the Shapley Value on the edge weights of the tree, and we also compute a null space basis of M. Both depend on the "split counts" of the tree. Finally, we characterize the Shapley Value on tree games by four axioms, a counterpart to Shapley's original theorem on the larger class of cooperative games.
Akemi Kashiwada - One of the best experts on this subject based on the ideXlab platform.
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The Shapley Value of Phylogenetic Trees
2016Co-Authors: Claus-jochen Haake, Akemi KashiwadaAbstract:Every weighted tree corresponds naturally to a cooperative game that we call a tree game; it assigns to each subset of leaves the sum of the weights of the minimal subtree spanned by those leaves. In the context of phylogenetic trees, the leaves are species and this assignment captures the diversity present in the coalition of species considered. We consider the Shapley Value of tree games and suggest a biological interpretation. We determine the linear transformation M that shows the dependence of the Shapley Value on the edge weights of the tree, and we also compute a null space basis of M. Finally, we characterize the Shapley Value on tree games by five axioms, a counterpart to Shapley's original theorem on the larger class of cooperative games. We also include a brief discussion of the core of tree games.
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the Shapley Value of phylogenetic trees
Journal of Mathematical Biology, 2008Co-Authors: Claus-jochen Haake, Akemi KashiwadaAbstract:Every weighted tree corresponds naturally to a cooperative game that we call a tree game; it assigns to each subset of leaves the sum of the weights of the minimal subtree spanned by those leaves. In the context of phylogenetic trees, the leaves are species and this assignment captures the diversity present in the coalition of species considered. We consider the Shapley Value of tree games and suggest a biological interpretation. We determine the linear transformation M that shows the dependence of the Shapley Value on the edge weights of the tree, and we also compute a null space basis of M. Both depend on the split counts of the tree. Finally, we characterize the Shapley Value on tree games by four axioms, a counterpart to Shapley's original theorem on the larger class of cooperative games. We also include a brief discussion of the core of tree games.
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The Shapley Value of Phylogenetic Trees
Journal of mathematical biology, 2007Co-Authors: Claus-jochen Haake, Akemi KashiwadaAbstract:Every weighted tree corresponds naturally to a cooperative game that we call a "tree game"; it assigns to each subset of leaves the sum of the weights of the minimal subtree spanned by those leaves. In the context of phylogenetic trees, the leaves are species and this assignment captures the diversity present in the coalition of species considered. We consider the Shapley Value of tree games and suggest a biological interpretation. We determine the linear transformation M that shows the dependence of the Shapley Value on the edge weights of the tree, and we also compute a null space basis of M. Both depend on the "split counts" of the tree. Finally, we characterize the Shapley Value on tree games by four axioms, a counterpart to Shapley's original theorem on the larger class of cooperative games.
Udi Weinsberg - One of the best experts on this subject based on the ideXlab platform.
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the Shapley Value in knapsack budgeted games
Workshop on Internet and Network Economics, 2014Co-Authors: Smriti Bhagat, Anthony Kim, Shanmugavelayutham Muthukrishnan, Udi WeinsbergAbstract:We propose the study of computing the Shapley Value for a new class of cooperative games that we call budgeted games, and investigate in particular knapsack budgeted games, a version modeled after the classical knapsack problem. In these games, the “Value” of a set S of agents is determined only by a critical subset T ⊆ S of the agents and not the entirety of S due to a budget constraint that limits how large T can be. We show that the Shapley Value can be computed in time faster than by the naive exponential time algorithm when there are sufficiently many agents, and also provide an algorithm that approximates the Shapley Value within an additive error. For a related budgeted game associated with a greedy heuristic, we show that the Shapley Value can be computed in pseudo-polynomial time. Furthermore, we generalize our proof techniques and propose what we term algorithmic representation framework that captures a broad class of cooperative games with the property of efficient computation of the Shapley Value. The main idea is that the problem of determining the efficient computation can be reduced to that of finding an alternative representation of the games and an associated algorithm for computing the underlying Value function with small time and space complexities in the representation size.
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The Shapley Value in Knapsack Budgeted Games
arXiv: Computer Science and Game Theory, 2014Co-Authors: Smriti Bhagat, Anthony Kim, Shanmugavelayutham Muthukrishnan, Udi WeinsbergAbstract:We propose the study of computing the Shapley Value for a new class of cooperative games that we call budgeted games, and investigate in particular knapsack budgeted games, a version modeled after the classical knapsack problem. In these games, the "Value" of a set $S$ of agents is determined only by a critical subset $T\subseteq S$ of the agents and not the entirety of $S$ due to a budget constraint that limits how large $T$ can be. We show that the Shapley Value can be computed in time faster than by the na\"ive exponential time algorithm when there are sufficiently many agents, and also provide an algorithm that approximates the Shapley Value within an additive error. For a related budgeted game associated with a greedy heuristic, we show that the Shapley Value can be computed in pseudo-polynomial time. Furthermore, we generalize our proof techniques and propose what we term algorithmic representation framework that captures a broad class of cooperative games with the property of efficient computation of the Shapley Value. The main idea is that the problem of determining the efficient computation can be reduced to that of finding an alternative representation of the games and an associated algorithm for computing the underlying Value function with small time and space complexities in the representation size.
