TWO-LEVEL COOPERATION IN TECHNOLOGICAL ALLIANCE DIFFERENTIAL GAMES
The paper is devoted to two-level cooperation in diﬀerential games. Cooperative diﬀerential games are currently one of most important parts of game theory. They mathematically describe the conﬂict-controlled processes in management and economics. The solution of a diﬀerential game is a cooperative agreement, and the selected principle of optimality, according to which the received payoﬀ is distributed. Studies showed that initially selected cooperative solution often loses its optimality over time. Therefore, the question arose about the stability of the co-operative solutions. The issue of dynamic stability or time consistency is considered. This concept was formalized by L. A. Petrosyan. Cooperative solution is dynamically stable if the principle of optimality selected early in the game stays consistent throughout the gameplay. For dynamic stability it is necessary at each moment of time to carry out the regularization of the chosen principle of optimality. For this regularization L. A. Petrosyan proposed to use the redistribution of received payoﬀ in accordance with the “imputation distribution procedure”. In some cases coalitional solutions in diﬀerential games are studied, where the coalitions act as individual players. Coalitions play with each other in a noncooperative game, and payoﬀ of each coalition is distributed among its members in accordance with some principle of optimality. In this paper we began to investigate the model in which participants form coalitions, acting as individual players, but the coalitions may also cooperate to increase joint payoﬀ. In this case, the coalition play their cooperative game, maximizing overall beneﬁts and distributing them among themselves according to some principle of optimality. Then, payoﬀ of each coalition is distributed among its members as well according to maybe a diﬀerent principle of optimality. Such cooperation is called two-level cooperation. Principles of proﬁt distribution between coalitions and within coalition may be diﬀerent. To solve such models which is required at both levels of the cooperation it is necessary to build the characteristic function and imputation distribution procedure. This paper describes a model of a two-level cooperation on the example of a technological alliance diﬀerential game. Participants in the game are the ﬁrms with the technology that brings proﬁt. On the ﬁrst (lower) level ﬁrms form coalitions to increase joint proﬁt. On the second (upper) level coalitions act as individual players and also form the one grand coalition to increase the joint proﬁt. The resulting top-level payoﬀ is distributed between coalitions-participants. Thus, each coalition party may get more than it would receive by playing individually. Then each coalition distributes the resulting share of payoﬀ among its member ﬁrms. This article also presented a stable cooperative solution in this model. For its implementation at every level of cooperation we build the characteristic function and prove its superadditivity. As a sharing the dynamic Shapley value is selected. The results are illustrated by a quantitative example.