Polar Representation of Shapley Value: Nonatomic Polynomial Games

The paper deals with polar representation formula for the Shapley value, established in (Vasil’ev, 1998). Below, we propose a new, simplified proof of the formula for nonatomic polynomial games. This proof relies on the coincidence of generalized Owen extension and multiplicative Aumann-Shapley expansion for polynomial games belonging to $pNA$ (Vasil’ev, 2009). The coincidence mentioned makes it possible to calculate Aumann-Shapley expansion in a straightforward manner, and to complete new proof of the polar representation formula for nonatomic case by exploiting the generalized Owen integral formula, established in (Aumann and Shapley, 1974).