# Bayesian formulation of the Shapley Value

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 .

Thanks !

• I think you have to convert you Bayesian game into a game of characteristic function form, then you can compute the Shapely value. How to do that for normal form games can be figured out by Aumann, R. J. (1961), A Survey on Cooperative Games without Side Payments, in M. Shubik (ed), Essays in Mathematical Economics in Honor of Oskar Morgenstern (pp. 3-27), Princeton University Press. – Holger I. Meinhardt Mar 15 '16 at 12:06
• thanks a lot @HolgerI.Meinhardt ! – vphenix Mar 15 '16 at 16:29