I wanted to know if there is a Bayesian formulation of the Shapley value in cooperative Games. I'm not sure if my problem really fits the definition of Bayesian games so here is the problem :
For a cooperative game $<N,v>$ with $N$ players and a payoff function $v$, I want to compute the Shapley value for each player knowing that the function $v$ does not have a unique value for each coalition. To be more clear, I'm not in the non-deterministic situation, but rather in $k$ different scenarios (let's say $k$ parallel universes) where, in each different universe, each coalition has a fixed payoff value. I have a prior on being in one of those $k$ situations (or universes) and I want to have a unique Shapley value that captures all possible scenarios .