So I'm trying to estimate a Shapley value in a game with uncertain payoffs. Specifically, imagine a game where the payoff function as as follows
(A) = 1 (B) = 2 (B,C) = 4
For instance, you can imagine this game representing Person A and Person B deciding whether to go into business by themselves or together.
Calculating the Shapley value for this game is very straightforward by listing all the permutations along with the marginal contribution of each person in the given permutation divided by the number of permutations:
[A, B] = [1, 3] [B, A] = [2, 2] [A, B] = [(1+2)/2, (3+2)/2] [A, B] = [1.5, 2.5]
So Person A should get \$1.5 and Person B should get \$2.5 in the grand coalition.
However, I am struggling to figure out how I would go about computing the Shapley value in the presence of uncertainty. For instance, imagine I change the payoff function to use confidence intervals as follows:
(A) = 1 ± 0.2 (B) = 2 ± 0.6 (B,C) = 4 ± 0.5
This is important as I am trying to compute the Shapley value in an real life setting with experimental data.
What will the Shapley value be in this case WITH uncertainty?