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

Abstract

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.