This paper is concerned with the open-loop linear quadratic (LQ) Stackelberg game of the mean-field stochastic systems in finite horizon. By means of two generalized differential Riccati equations (GDREs), the follower first solves a mean-field stochastic LQ optimal control problem. Then the leader turns to solve an optimization problem for a linear mean-field forward-backward stochastic differential equation (MF-FBSDE). By introducing new state and costate variables, we present a sufficient condition for the existence and uniqueness of the Stackelberg strategy in terms of the solvability of some Riccati equations and a convexity condition. Furthermore, it is shown that the open-loop Stackelberg equilibrium admits a feedback representation involving the new state and its mean. Finally, two examples are given to show the effectiveness of the proposed results.

This study is concerned with the necessary and sufficient conditions for the existence of Pareto solutions in the infinite horizon cooperative difference game. Based on the assumption about the Lagrange multipliers, utilising the equivalent characterisation of the Pareto optimality, the necessary conditions for the existence of the Pareto solutions are put forward. Furthermore, two conditions are presented to guarantee that zero does not belong to the Lagrange multiplier set. In addition, it is shown that the necessary conditions are also sufficient under certain convexity assumptions and a transversality condition. Next, the indefinite linear quadratic case is discussed. For a fixed initial state, under the condition of controllability, the necessary conditions are put forward. In addition, the necessary conditions, the convexity condition on the weighted sum cost functional as well as a transversality condition provide the sufficient conditions for a control to be Pareto optimal. For an arbitrary initial state, if the system is stabilisable, then the solvability of the related algebraic Riccati equation provides a sufficient condition under which all Pareto optimal strategies are obtained by the weighted sum minimisation method.