On the Complexity of Cooperative Solution Concepts

Deng, Xin; Papadimitriou, Christos H.

doi:10.1287/moor.19.2.257

Cited by 505 publications

(455 citation statements)

References 8 publications

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“…The problem of computing the Shapley value or the Banzhaf index for voting games is #P-complete [5,12]. In order to overcome this problem, we present new approximation methods to find these indices.…”

Section: Definitionmentioning

confidence: 99%

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An Approximation Method for Power Indices for Voting Games

Fatima

Wooldridge

Jennings

2010

Innovations in Agent-Based Complex Automated Negotiations

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Abstract. The Shapley value and Banzhaf index are two well known indices for measuring the power a player has in a voting game. However, the problem of computing these indices is computationally hard. To overcome this problem, we analyze approximation methods for computing these indices. Although these methods have polynomial time complexity, finding an approximate Shapley value using them is easier than finding an approximate Banzhaf index. We also find the absolute error for the methods and show that this error for the Shapley value is lower than that for the Banzhaf index.

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Section: Definitionmentioning

confidence: 99%

“…Like Shapley value, the Banzhaf index [2] is another way of measuring a player's power. However, a key drawback of both these power indices is that computing them for voting games 1 is, in general, #P-complete [5,12]. In other words, it is practically infeasible to try to compute the exact Shapley value or Banzhaf index.…”

Section: Introductionmentioning

confidence: 99%

An Approximation Method for Power Indices for Voting Games

Fatima

Wooldridge

Jennings

2010

Innovations in Agent-Based Complex Automated Negotiations

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“…However, uncertainty is defined in terms of the Shapley value (i.e., in order to find uncertainty, the Shapley value needs to be determined first). But, as we pointed out, the problem of determining the Shapley value has been shown to be #P-complete [1]. We therefore present a new randomised method (that has polynomial time complexity) for computing the approximate Shapley value.…”

Section: Introductionmentioning

confidence: 97%

“…First, for the weighted voting game that we consider, the problem of determining the Shapley value is #P-complete [1]. Second, it provides the solution only with a limited degree of certainty [11].…”

Section: Introductionmentioning

confidence: 99%

An Analysis of the Shapley Value and Its Uncertainty for the Voting Game

Fatima

Wooldridge

Jennings

2006

Agent-Mediated Electronic Commerce. Designing Trading Agents and Mechanisms

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Abstract.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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“…The standard procedure for computing the nucleolus proceeds by solving a finite (but large) number of related linear programs. As a solution concept, the nucleolus has been analyzed and computed in many OR-games, for instance Okamoto (2008) Granot et al (1996), Deng and Papadimitriou (1994), or Granot and Huberman (1981). An interesting survey on the nucleolus and its computational complexity is given in Greco et al (2015).…”

Section: Introductionmentioning

confidence: 99%

Insights into the Nucleolus of the Assignment Game

2015

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Abstract:We show that the family of assignment matrices which give rise to the same nucleolus form a compact join-semilattice with one maximal element, which is always a valuation (see p.43, Topkis (1998)). We give an explicit form of this valuation matrix. The above family is in general not a convex set, but path-connected, and we construct minimal elements of this family. We also analyze the conditions to ensure that a given vector is the nucleolus of some assignment game.JEL Codes: C71.

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On the Complexity of Cooperative Solution Concepts

Cited by 505 publications

References 8 publications

An Approximation Method for Power Indices for Voting Games

An Approximation Method for Power Indices for Voting Games

An Analysis of the Shapley Value and Its Uncertainty for the Voting Game

Insights into the Nucleolus of the Assignment Game

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