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(k(2)n) and the upper bound on its approximation error is O(k(2)/root 2). (c) 2008 Elsevier B.V. All rights reserved,