Two-person cooperative uncertain differential game with transferable payoffs

Uncertain differential game models conflicts and interests among players in the context of an uncertain dynamic system. However, cooperative behavior in uncertain differential game is an unexplored terrain. This paper proposes a spectrum of a cooperative uncertain differential game with transferrable payoffs. First, group rationality is achieved by maximizing the coalitional payoff through an uncertain optimal control method. Second, the concept of imputation is introduced as a solution, and a stability condition called subgame consistency condition for an imputation is proposed as well. Furthermore, the derivation of payoff distribution procedure for subgame consistent imputation is discussed. In addition, two optimal principles, i.e., Nash bargaining solution and Shapley value, are extended to this kind of model and are proved to be subgame consistent imputations, and their payoff distribution procedures are derived analytically. Finally, a two-person resource extraction game is studied for illustrating purpose.