Polynomial calculation of the Shapley value based on sampling

Castro, Javier; Gómez, Daniel; Tejada, Juan

doi:10.1016/j.cor.2008.04.004

Cited by 454 publications

(359 citation statements)

References 7 publications

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“…Polynomial methods, based on sampling theory, have been proposed in [19] for approximating the Shapley value; these estimations, though, are efficient only if the worth of any coalition S can be calculated in polynomial time, which is not the case for our problem.…”

Section: Theorem 1 In the Cooperative Network Formation Game The Sementioning

confidence: 90%

“…In fact, even using the approximation methods proposed for example in [19], it is necessary to compute the worth of an extremely large number of coalitions, which is computationally very cumbersome, while as we see next, our proposed Nash bargaining solution needs only computing the worth of the grand coalition.…”

Section: Theorem 1 In the Cooperative Network Formation Game The Sementioning

confidence: 99%

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A Nash Bargaining Solution for Cooperative Network Formation Games

et al. 2011

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Abstract. The Network Formation problem has received increasing attention in recent years. Previous works have addressed this problem considering almost exclusively networks designed by selfish users, which can be consistently suboptimal. This paper addresses the network formation issue using cooperative game theory, which permits to study ways to enforce and sustain cooperation among agents. Both the Nash bargaining solution and the Shapley value are widely applicable concepts for solving these games. However, we show that the Shapley value presents three main drawbacks in this context: (1) it is non-trivial to define meaningful characteristic functions for the cooperative network formation game, (2) it can determine for some players cost allocations that are even higher than those at the Nash Equilibrium (i.e., if players refuse to cooperate), and (3) it is computationally very cumbersome. For this reason, we solve the cooperative network formation game using the Nash bargaining solution (NBS) concept. More specifically, we extend the NBS approach to the case of multiple players and give an explicit expression for users' cost allocations. Furthermore, we compare the NBS to the Shapley value and the Nash equilibrium solution, showing its advantages and appealing properties in terms of cost allocation to users and computation time to get the solution. Numerical results demonstrate that the proposed Nash bargaining solution approach permits to allocate costs fairly to users in a reasonable computation time, thus representing a very effective framework for the design of efficient and stable networks.

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Section: Theorem 1 In the Cooperative Network Formation Game The Sementioning

confidence: 90%

Section: Theorem 1 In the Cooperative Network Formation Game The Sementioning

confidence: 99%

A Nash Bargaining Solution for Cooperative Network Formation Games

et al. 2011

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“…In the following definition, which generalizes the previous one, we consider the case in which only a fixed proportion ( 1], is equally allocated among the incumbents.…”

Section: Definitionmentioning

confidence: 99%

“…2 Otherwise, if the relative interior of the core is non-empty and the Shapley value is stable and lies in the core's relative interior, then there exist α e and α p in [0, 1) such that E P α (v) ∈ C(v), for all α ≥ α e , and P P α (v) ∈ C(v), for all α ≥ α p .…”

Section: Definitionmentioning

confidence: 99%

Pyramidal values

2013

Self Cite

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Abstract:We propose a new type of values for cooperative TU-games, which we call pyramidal values. Assuming that the grand coalition is sequentially formed, and all orderings are equally likely, we define a pyramidal value to be any expected payoff in which the entrant player receives a salary and the right to get part of the benefits derived from subsequent incorporations to the just formed coalition, whereas the remaining benefit is distributed among the incumbent players. To be specific, we consider some parametric families of pyramidal values: the egalitarian pyramidal family, which coincides with the a-consensus value family introduced by Ju et al. in (2007), the proportional pyramidal family, and the weighted pyramidal family, which in turn includes the other two families as special cases. We also analyze the properties of these families, as well as their relationships with other previously defined values. Keywords AbstractWe propose a new type of values for cooperative TU-games, which we call pyramidal values. Assuming that the grand coalition is sequentially formed, and all orderings are equally likely, we define a pyramidal value to be any expected payoff in which the entrant player receives a salary and the right to get part of the benefits derived from subsequent incorporations to the just formed coalition, whereas the remaining benefit is distributed among the incumbent players. To be specific, we consider some parametric families of pyramidal values: the egalitarian pyramidal family, which coincides with the α-consensus value family introduced by Ju et al. (2007), the proportional pyramidal family, and the weighted pyramidal family, which in turn includes the other two families as special cases. We also analyze the properties of these families, as well as their relationships with other previously defined values.

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“…There exist approximation techniques for the Shapley value (Policy 5) where the computational complexity increases linearly with the number of players [Fatima et al 2008;Castro et al 2009]. There are also exact solutions that take advantage of some particular game representation to reduce the complexity (e.g., O(n) and O(n 2 ) for a tree [Megiddo 1978] and weighted graph representation respectively [Deng and Papadimitriou 1994]).…”

Section: 22mentioning

confidence: 99%

Fairness and Incentive Considerations in Energy Apportionment Policies

Vergara

Nadjm‐Tehrani

Asplund

2016

ACM Trans. Model. Perform. Eval. Comput. Syst.

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The energy consumption of a system is determined by the system component usage patterns and interactions between the coexisting entities and resources. Energy accounting plays an essential role to reveal the contribution of each entity to the total consumption and for energy management. Unfortunately, energy accounting inherits the apportionment problem of accounting in general, which does not have a general single best solution. In this paper we leverage cooperative game theory commonly used in cost allocation problems to study the energy apportionment problem, i.e., the problem of prescribing the actual energy consumption of a system to the consuming entities (e.g., applications, processes or users of the system).We identify five relevant fairness properties for energy apportionment and present a detailed categorisation and analysis of eight previously proposed energy apportionment policies from different fields in computer and communication systems. In addition, we propose two novel energy apportionment policies based on cooperative game theory which provide strong fairness notion and a rich incentive structure. Our comparative analysis in terms of the identified five fairness properties as well as information requirement and computational complexity shows that there is a trade-off between fairness and the other evaluation criteria. We provide guidelines to select an energy apportionment policy depending on the purpose of the apportionment and the characteristics of the system.

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Polynomial calculation of the Shapley value based on sampling

Cited by 454 publications

References 7 publications

A Nash Bargaining Solution for Cooperative Network Formation Games

A Nash Bargaining Solution for Cooperative Network Formation Games

Pyramidal values

Fairness and Incentive Considerations in Energy Apportionment Policies

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