Two-level cooperation in the game of pollution cost reduction

Cooperative differential games are one of most actual parts of the theory of games. They describe well the conflict-controlled processes in management and economics. The solution of differential game is a cooperative agreement, and the selected principle of optimality, according to which the received payoff is distributed. Unfortunately the initially selected cooperative solution often loses its optimality over time. Then the question arose about the time consistency of the cooperative solutions or dynamic stability. The concept of dynamic stability was formalized by L. A. Petrosyan. Cooperative solution is dynamically stable, if the selected principle of optimality keeps its optimality throughout the gameplay. For dynamic stability is necessary to carry out the regularization of the chosen principle of optimality. L. A. Petrosyan proposed to use the redistribution of received payoff in accordance with the “imputation distribution procedure”. In some cases in differential games coalitional solutions are studied in which the coalitions act as individual players and play with each other in a non-cooperative game, and payoff of each coalition is distributed among its members in accordance with some principle of optimality. Besides studied models where the coalitions act as individual players, but they can also cooperate to maximize joint payoff. In this case, the total payoff is distributed between coalitions in accordance with the selected principle of optimality, and then the payoff of each coalition is distributed between its members according to perhaps other principle of optimality. Such cooperation is called two-level cooperation. To solve models of two-level cooperation which requires at both levels of the cooperation it is necessary to determine the characteristic function and imputation distribution procedure. In this paper we consider a model of two-level cooperation in the game of pollution cost reduction. Participants of game are enterprises whose production harms the environment. The player's payoff is cost of compensation for damage from emissions. The aim of enterprises is minimization of costs and they can join in coalitions to minimize total costs and their redistribution. Coalitions can also cooperate. At the first (lower) level, enterprises form coalitions. At the second (top) level, coalitions, acting as individual players, form a one grand coalition to minimize total costs. The resulting top-level payoff is distributed between coalitions-participants. As a principle of optimality the dynamic Shapley value is selected. Then each coalition distributes the resulting share of payoff among its participants. In this paper we follow the model described in [8], and the specificity is to features of the characteristic function construction.