Shapley value of homogeneous cooperative games

The paper gives a description of the integral representation of the Shapley value for polynomial cooperative games. This representation obtained using the so-called Shapley functional. The relationship between the proposed version of the Shapley value and the polar forms of homogeneous polynomial games is analyzed for both a finite and an infinite number of participants. Special attention is paid to certain classes of homogeneous cooperative games generated by products of non-atomic measures. A distinctive feature of the approach proposed is the systematic use of extensions of polynomial set functions to the corresponding measures on symmetric powers of the original measurable spaces.