Pareto Cooperative Differential Game

Pareto cooperative differential game is a typical representative of coop-erative game,where all players negotiate with each other and make at least one person better without making anyone worse.It is widely used in finance,economics,engineering,political science and many other fields.To our best knowledge,there are more literatures studying Pareto cooperative differential game driven by forward linear stochastic system with the technique of weighted sum minimization.Pareto solution depends on the weight α,and Pareto fron-tier is obtained by changing the value of α.In fact,the players prefer to get a unique best solution in the game,so the optimization of weights has a certain practical meaning.In addition,when the players set goals in advance and hope to realize the ultimate interest in a cooperative way,we prefer to describe this behavior in terms of backward stochastic differential equation(BSDE)rather than the traditional forward stochastic differential equation(FSDE).Because the terminal state of BSDE is given in advance,which can exactly represent the case.Therefore,the main research are as follows:(1)It’s the first time that we study the optimization problem of Pareto solution driven by FSDE,in which the state and index are non-homogeneous.With the help of Nash bargaining theory,we successfully transform the bar-gaining game solution into the optimization problem with Nash product,which is regarded as objective function in the sequel.Then we obtain the most ap-propriate weight.(2)It’s the first time that we study Pareto cooperative differential game driven by BSDE.By an equivalent representation of each player’s admissible control set,we transform the multi-objective optimal control problem into n single objective optimal control problems with constraints.Based on stochastic maximum principle and Ekeland’s variational principle,we derived necessary and sufficient conditions for the Pareto optimality in the game.In addition,the results are applied to a linear quadratic case.According to the theory of linear backward stochastic optimal control,Pareto effective strategy and the corresponding Pareto solution are obtained.