Stackelberg-Nash solution to the mean-field stochastic difference game with two followers☆

This paper studies the Stackelberg-Nash strategy for the discrete-time mean-field stochastic systems with two followers. First, utilizing two pairs of generalized difference Riccati equations, a non-zero sum mean-field stochastic difference game is solved for two followers and the Nash equilibrium solution is obtained. Then, the leader settles the optimization problem of a mean-field forward-backward stochastic difference equation. Necessary and sufficient condition for the solvability is put forward. Moreover, feedback representation of Stackelberg-Nash strategy is obtained by introducing new state and costate variables. Finally, two examples are given to verify the effectiveness of the proposed results.