Abstract
The Shapley value (Shapley in Ann Math Stud 2:28, 1953) is one of the most prominent one-point solution concepts in cooperative game theory that divides revenues (or cost, power) that can be obtained by cooperation of players in the game. The Shapley value is mathematically characterized by properties that have appealing real-world interpretations and hence its use in practical settings is easily justified. The down part is that its computational complexity increases exponentially with the number of players in the game. Therefore, in practical problems that consist of more than 25 players the calculation of the Shapley value is usually too time expensive. Among others the Shapley value is applied in the analysis of terrorist networks (cf. Lindelauf et al. in Eur J Oper Res 229(1):230–238, 2013) which generally extend beyond the size of 25 players. In this paper we therefore present a new method to approximate the Shapley value by refining the random sampling method introduced by Castro et al. (Comput Oper Res 36(5):1726–1730, 2009). We show that our method outperforms the random sampling method, reducing the average error in the Shapley value approximation by almost 30%. Moreover, our new method enables us to analyze the extended WTC 9/11 network of Krebs (Connections 24(3):43–52, 2002) that consists of 69 members. This in contrast to the restricted WTC 9/11 network considered in Lindelauf et al. (2013), that only considered the operational cells consisting of the 19 hijackers that conducted the attack.
Highlights
Cooperative game theory with transferable utilities studies situations in which players can work together and create additional revenues instead of the situation in which each player acts on its own
The Shapley value is defined as the average of all n! marginal vectors in a cooperative game consisting of n players
See for instance Fatima et al (2008) who present an algorithm that has time complexity linear in the number of players. They show that their algorithm outperforms the following approximation methods: Monte Carlo simulation (Mann and Shapley 1960), multi-linear extension (MLE) (Owen 1972), modified MLE (Leech 2003) and random permutation (Zlotkin and Rosenschein 1994)
Summary
Cooperative game theory with transferable utilities studies situations in which players can work together and create additional revenues (or costs reductions) instead of the situation in which each player acts on its own. There are many solution concepts in cooperative game theory, each satisfying its own set of properties. The most prominent one-point solution concept satisfying intuitive properties that are considered as fair in many situations in practice is the Shapley value (Shapley 1953). The properties symmetry (i.e., two players that are symmetric in a game should receive an equal share), dummy (i.e., a player that does not contribute in a game only receives its individual contribution) and monotonicity (i.e., if a game changes such that a set of players is rewarded more in each coalition they participate, these players should receive at least the same as allocated in the original game) are examples of such properties (cf Shapley 1953; Young 1985) that are satisfied
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
