I am trying to understand the Shapley values formula.
I have gained an understanding of the formula through the marginal contributions explanation provided by Wikipedia
My confusion comes in understanding how the coalition values are found $v{\{S}\}$. If 3 parties A, B, C would be thinking about going into business with each other they would each know their own independent contributions (i.e $v{\{A}\}$,$v{\{B}\}$,$v{\{C}\}$) but how would it be possible to get the marginal contributions of all parties in order to calculate a Shapley value before actually going into business together?
This question might be basic for the topic but I appreciate any help!
Thank you!
There is no unique way to get the coalitional values of a TU-game. In general, there are two basic approaches to get a TU-game. The first is to define a proper TU-game by yourself or to use one of the games that are defined in the literature to capture an underlying bargaining problem like an airport game, a bankruptcy game, a minimum cost spanning game, etc. To get on overview of some game types have a look into the book by
M. Maschler, E. Solan and Sh. Zamir (2013)
The second approach is to start with an economic situation from which a normal form game can be derived. Then one can derive the $\alpha$-, $\beta$-TU-games by the approach described in
R. J. Aumann. A Survey on Cooperative Games without Side Payments. In M. Shubik, editor, Essays in Mathematical Economics in Honor of Oskar Morgenstern, pages 3–27, Princeton, 1961. Princeton University Press.
or the $\gamma$-, $\delta$-TU-games based on a partition function approach as it was described by
S. Hart and M. Kurz. Endogenous Formation of Coalitions. Econometrica, 51(4): 1047–1064, 1983.
Thus, if you have got the TU-game, then one can easily apply the formula of the Shapley value to get a distribution of the proceeds of mutual cooperation. A bargaining outcome if you want, since a Tu-game can be interpreted as indicating a bargaining situation.