Characteristic functions in cooperative differential games on networks

In the paper, a class of cooperative differential games on networks is considered. In such games, the new characteristic function is introduced based on the possibility of stopping interaction by players outside the coalition in each time instant or imposing sanction on players from the coalition. This gives the real possibility for the computation of characteristic function. Thus, the characteristic function is evaluated along the cooperative trajectory. It measures the worth of coalitions under the process of cooperation instead of under minimax confrontation or the Nash non-cooperative stance. The approach essentially simplifies the construction of the characteristic function and cooperative solutions such as the Shapley value, Core, $ \tau $-value and others. Also, it is proved that the proposed characteristic function is convex, time consistent, and as a result, the Shapley value belongs to the Core and is time consistent. Also, a modification of the dynamic game on networks, namely, dynamic network game with partner sets is considered. In this case, payoffs of a given player depend on his actions and the actions of the players from his partner set. Using previous ideas, the special type of characteristic function is introduced, and cooperative solutions are proposed.