Shapley Value
Hi,
I have the following exercise and I am not sure about the calculation, if you could help me out, that would be awesome.
Consider the following characteristic function game: N = {1, 2, 3} and v({1}) = 0, v({2}) = 0, v({3}) = 0,
V({1,2}) = 40, v({1,3}) = 0, v({2,3}) = 50, v({1,2,3}) = 50. Find the core and the Shapley value.
Thanks in advance.
X
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Jaap de Jonge Editor, Netherlands


What is Shapley Value?
The Shapley value is a solution model in cooperative game theory named after Lloyd Shapley, who first introduced and axiomatized it in 1953^{1}. It assigns a unique distribution among the participants of a total surplus generated by the coalition of all participants in any cooperative game. The Shapley value is characterized by a collection of desirable properties.
A cooperative setup is typically as follows: a coalition of participants cooperates, and obtains a certain overall gain (total benefits) from that cooperation. Because some participants may contribute more to the coalition than others, or some participants may possess different bargaining power, a particular final distribution of generated surplus among the players will arise in any particular game.
In other words it predicts, considering the importance of each participant to the overall cooperation, what payoff each participant can reasonably expect.
The formula is quite complex and can be found in Wikipedia.
Note that Shapley distributes the total gains to the players, assuming that they all collaborate, publicly share their contributions, and that a "fair" distribution has to be reached. As always, reality could be quite different from these assumptions...
^{1}L.S. Shapley (1953). A value for nperson games. In H. W. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games, volume II. Annals of Mathematics Studies 28. Princeton University Press, pp.307–317.

