The stability of the Nash equilibrium in a single-leader multi-follower game with strategic adjustment costs

In a single-leader multi-follower game (SLMFG), leaders or followers often incur additional costs while revising their decisions, such as suppliers versus retailers, central government versus local government, and conglomerates versus subsidiaries; however, these costs are usually ignored. To focus on this issue, using quadratic cost functions, these strategic adjustment costs are introduced into a dynamic SLMFG with adaptive expectations, thereby yielding a new dynamic model. Furthermore, we obtain some necessary and sufficient conditions for local asymptotic stability of the SLMFG. We also prove that both strategic adjustment costs and the leader have increased the stable region of the original Nash equilibrium. By comparing interval increments, the leader and strategy adjustment cost contribute equally to the Nash equilibrium, which, presented in the paper, develop corresponding conclusions to previous research.