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Shapley value
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== Generalization to coalitions == The Shapley value only assigns values to the individual agents. It has been generalized<ref>{{cite journal |last1=Grabisch |first1=Michel |last2=Roubens |first2=Marc |title=An axiomatic approach to the concept of interaction among players in cooperative games |journal=International Journal of Game Theory |date=1999 |volume=28 |issue=4 |pages=547β565 |doi=10.1007/s001820050125|s2cid=18033890 }}</ref> to apply to a group of agents ''C'' as, : <math> \varphi_C(v) = \sum_{T \subseteq N \setminus C} \frac{(n - |T| - |C|)! \; |T|!}{(n - |C| + 1)!} \sum_{S \subseteq C} (-1)^{|C| - |S|} v( S \cup T ) \; . </math> In terms of the synergy function <math>w</math> above, this reads<ref name="Grabisch Representations"/><ref name="Grabisch Representations 2"/> : <math> \varphi_C(v) = \sum_{C \subseteq T \subseteq N} \frac{w(T)}{|T| - |C| + 1} </math> where the sum goes over all subsets <math>T</math> of <math>N</math> that contain <math>C</math>. This formula suggests the interpretation that the Shapley value of a coalition is to be thought of as the standard Shapley value of a single player, if the coalition <math>C</math> is treated like a single player.
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