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A linear approximation method for the Shapley value

A linear approximation method for the Shapley value
A linear approximation method for the Shapley value
The Shapley value is a key solution concept for coalitional games in general and voting games in particular. Its main advantage is that it provides a unique and fair solution, but its main drawback is the complexity of computing it (e.g for voting games this complexity is #P-complete). However, given the importance of the Shapley value and voting games, a number of approximation methods have been developed to overcome this complexity. Among these, Owen’s multi-linear extension method is the most time efficient, being linear in the number of players. Now, in addition to speed, the other key criterion for an approximation algorithm is its approximation error. On this dimension, the multi-linear extension method is less impressive. Against this background, this paper presents a new approximation algorithm, based on randomization, for computing the Shapley value of voting games. This method has time complexity linear in the number of players, but has an approximation error that is, on average, lower than Owen’s. In addition to this comparative study, we empirically evaluate the error for our method and show how the different parameters of the voting game affect it. Specifically, we show the following effects. First, as the number of players in a voting game increases, the average percentage error decreases. Second, as the quota increases, the average percentage error decreases. Third, the error is different for players with different weights; players with weight closer to the mean weight have a lower error than those with weight further away. We then extend our approximation to the more general k-majority voting games and show that, for n players, the method has time complexity O(k2n) and the upper bound on its approximation error is O(k2/?n).
coalitional game theory, shapley value, approximation method
1673-1699
Fatima, Shaheen S.
993401f1-2dd0-4e9b-86bf-b7fa72639b67
Wooldridge, Michael
94674704-0392-4b93-83db-18198c2cfa3b
Jennings, Nicholas R.
ab3d94cc-247c-4545-9d1e-65873d6cdb30
Fatima, Shaheen S.
993401f1-2dd0-4e9b-86bf-b7fa72639b67
Wooldridge, Michael
94674704-0392-4b93-83db-18198c2cfa3b
Jennings, Nicholas R.
ab3d94cc-247c-4545-9d1e-65873d6cdb30

Fatima, Shaheen S., Wooldridge, Michael and Jennings, Nicholas R. (2008) A linear approximation method for the Shapley value. Artificial Intelligence, 172 (14), 1673-1699. (doi:10.1016/j.artint.2008.05.003).

Record type: Article

Abstract

The Shapley value is a key solution concept for coalitional games in general and voting games in particular. Its main advantage is that it provides a unique and fair solution, but its main drawback is the complexity of computing it (e.g for voting games this complexity is #P-complete). However, given the importance of the Shapley value and voting games, a number of approximation methods have been developed to overcome this complexity. Among these, Owen’s multi-linear extension method is the most time efficient, being linear in the number of players. Now, in addition to speed, the other key criterion for an approximation algorithm is its approximation error. On this dimension, the multi-linear extension method is less impressive. Against this background, this paper presents a new approximation algorithm, based on randomization, for computing the Shapley value of voting games. This method has time complexity linear in the number of players, but has an approximation error that is, on average, lower than Owen’s. In addition to this comparative study, we empirically evaluate the error for our method and show how the different parameters of the voting game affect it. Specifically, we show the following effects. First, as the number of players in a voting game increases, the average percentage error decreases. Second, as the quota increases, the average percentage error decreases. Third, the error is different for players with different weights; players with weight closer to the mean weight have a lower error than those with weight further away. We then extend our approximation to the more general k-majority voting games and show that, for n players, the method has time complexity O(k2n) and the upper bound on its approximation error is O(k2/?n).

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e-pub ahead of print date: 7 July 2008
Published date: September 2008
Keywords: coalitional game theory, shapley value, approximation method
Organisations: Agents, Interactions & Complexity

Identifiers

Local EPrints ID: 265802
URI: http://eprints.soton.ac.uk/id/eprint/265802
PURE UUID: 60b35f5c-4e5b-43e5-93e1-c3eddbcca70b

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Date deposited: 27 May 2008 15:07
Last modified: 02 Dec 2019 21:03

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Contributors

Author: Shaheen S. Fatima
Author: Michael Wooldridge
Author: Nicholas R. Jennings

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