How to calculate Shapley Value with uncertainty? 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?
It sounds like you may have a quadratic loss function, but I am guessing a little.  If the players have played enough to be experienced in how the game works, then I would simply use the sample mean reward as my calculation point.

You should read up on the theory behind loss functions.  The players are making estimates, certain types of estimates are the natural minimization of certain loss functions.  Most gambling type behavior results from quadratic loss, though there are particular exceptions.  Equity securities center of location is the mode because there can be no such thing as a mean return.  Equity securities behaviors should result from an all-or-nothing loss function.  Physicians are another example where there is an asymmetric loss.

If a physician fails to diagnose you with a disease, then they lost the revenue from treatment and face malpractice claims.  If they misdiagnose you as having a disease but you do not, then they still collect the medical billings and may not face malpractice claims, especially if they do not reveal it was a misdiagnosis.  A diagnosis is a form of gamble.

Take a look at the losses experienced from failing to take a particular type of behavior.