A linear approximation method for the Shapley value

Fatima, Shaheen; Wooldridge, Michael; Jennings, Nicholas R.

doi:10.1016/j.artint.2008.05.003

Cited by 178 publications

(99 citation statements)

References 24 publications

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“…Still, the hardness of our problems rests on the hardness to compute the related power indices. Note that there are good approximation schemes and dynamic methods known for computing the Shapley-Shubik index (see, e.g., Bachrach, Markakis, Resnik, Procaccia, Rosenschein, & Saberi, 2010;Fatima, Wooldridge, & Jennings, 2008;Bilbao, Fernández, Jiménez, & López, 2000;Matsui & Matsui, 2000;Shapley, 1953), though there is not much known about the exact case.…”

Section: Discussionmentioning

confidence: 99%

False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time

Rey¹,

Rothe²

2014

jair

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False-name manipulation refers to the question of whether a player in a weighted voting game can increase her power by splitting into several players and distributing her weight among these false identities. Relatedly, the beneficial merging problem asks whether a coalition of players can increase their power in a weighted voting game by merging their weights. For the problems of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley-Shubik and the normalized Banzhaf index, merely NP-hardness lower bounds are known, leaving the question about their exact complexity open. For the Shapley-Shubik and the probabilistic Banzhaf index, we raise these lower bounds to hardness for PP, "probabilistic polynomial time," a class considered to be by far a larger class than NP. For both power indices, we provide matching upper bounds for beneficial merging and, whenever the new players' weights are given, also for beneficial splitting, thus resolving previous conjectures in the affirmative. Relatedly, we consider the beneficial annexation problem, asking whether a single player can increase her power by taking over other players' weights. It is known that annexation is never disadvantageous for the Shapley-Shubik index, and that beneficial annexation is NP-hard for the normalized Banzhaf index. We show that annexation is never disadvantageous for the probabilistic Banzhaf index either, and for both the Shapley-Shubik index and the probabilistic Banzhaf index we show that it is NP-complete to decide whether annexing another player is advantageous. Moreover, we propose a general framework for merging and splitting that can be applied to different classes and representations of games.

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Section: Discussionmentioning

confidence: 99%

False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time

Rey¹,

Rothe²

2014

jair

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“…For the latter (i.e., computing P or S), we discuss a sampling based randomized approximation method. 3 Note that the function P/S gives the 'average' number of W-MEDIUM/B-MEDIUM coalitions in a game, the average taken over all possible coalitions. We consider average instead of the 'total' because a game is PROPER/STRONG if the total or average number of W-MEDIUM/B-MEDIUM coalitions is zero, but taking the average offers of ease of discussion of approximation method.…”

Section: Approximation Methodsmentioning

confidence: 99%

“…They have long been studied in game theory, and have more recently been used by researchers in multi-agent systems [3]. In a WVG, the individual players are assigned weights (a player's weight is the number of votes he or she has).…”

Section: Introductionmentioning

confidence: 99%

Automated analysis of weighted voting games

Fatima

Wooldridge

Jennings

2011

Proceedings of the 13th International Conference on Electronic Commerce

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Weighted voting games (WVGs) are an important mechanism for modeling scenarios where a group of agents must reach agreement on some issue over which they have different preferences. However, for such games to be effective, they must be well designed. Thus, a key concern for a mechanism designer is to structure games so that they have certain desirable properties. In this context, two such properties are PROPER and STRONG. A game is PROPER if for every coalition that is winning, its complement is not. A game is STRONG if for every coalition that is losing, its complement is not. In most cases, a mechanism designer wants games that are both PROPER and STRONG. To this end, we first show that the problem of determining whether a game is PROPER or STRONG is, in general, NP-hard. Then we determine those conditions (that can be evaluated in polynomial time) under which a given WVG is PROPER and those under which it is STRONG. Finally, for the general NP-hard case, we discuss two different approaches for overcoming the complexity: a deterministic approximation scheme and a randomized approximation method.

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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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A linear approximation method for the Shapley value

Cited by 178 publications

References 24 publications

False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time

False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time

Automated analysis of weighted voting games

Fairness and Incentive Considerations in Energy Apportionment Policies

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