My Math Forum How to Apply the Shapley Value Formula?

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 December 14th, 2012, 11:31 AM #1 Member   Joined: Nov 2009 Posts: 90 Thanks: 0 How to Apply the Shapley Value Formula? Trivia: This game theoretical formula is named after 2012 Nobel laureate Lloyd Shapley. The Shapley value is given by $\phi_i (v)= \sum_{S \subseteq N \setminus \left \{ i \right \}} \frac{\left| S \right|! (\left|N\right|-\left|S\right|-1)!}{\left|N\right|!}~ (v(S\cup\left \{ i \right \})-v(S))$ where $S$ is a coalition in the grand coalition $N=\left \{1,2,3,...,i,...n \right \}$ and $v(S)$ is the value generated by coalition $S$. How do we apply this formula to a numerical example? Peter, Joe and Glenn are producing grunks and can choose to cooperate if they like. Their profit from producing grunks are $v(\left\{Peter\right\})=100$ $v(\left\{Joe\right\})=150$ $v(\left\{Glenn\right\})=200$ $v(\left\{Peter,Joe\right\})=300$ $v(\left\{Peter,Glenn\right\})=380$ $v(\left\{Joe,Glenn\right\})=420$ $v(\left\{Peter,Joe,Glenn\right\})=580$. Using the formula for the Shapley value we can now obtain a 3-tuple representing an allocation of profit $(p,j,g)$ such that $p+j+g=580$. This is one of many potential solutions for this cooperative game. How can we apply the formula to find the solution? Apparently, it's something about imagining the players joining the game in different orders and thereby getting an average marginal value for each player. I don't see how to iterate in a numerical example. I understand that there are $\left| N \right|!=n!=3 \times 2 \times 1 = 6$ different orders in which all players can enter the grand coalition, but then I'm kinda lost. Any help is appreciated!

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# shapley value excel

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