On the core and Shapley value for regular polynomial games

Considering some classes of polynomial cooperative games, we describe the integral representation of the Shapley values and the support functions of their cores. Also, we analyze the relationship between the Shapley values and the polar forms of homogeneous polynomial games. The found formula for the support function of the core of a convex game is applied for the dual description of the Harsanyi sets of finite cooperative games. The main peculiarity of the proposed approach to the study of optimal solutions of game theory is a systematic use of the extensions of polynomial set functions to the corresponding measures on symmetric powers of the initial measure spaces.