On Nash–Stackelberg–Nash games under decision-dependent uncertainties: Model and equilibrium

In this paper, we discuss a class of two-stage hierarchical games with multiple leaders and followers, which is called Nash–Stackelberg–Nash (N–S–N) games. Particularly, we consider N–S–N games under decision-dependent uncertainties (DDUs). DDUs refer to the uncertainties that are affected by the strategies of decision-makers and have been rarely addressed in game equilibrium analysis. In this paper, we first formulate the N–S–N games with DDUs of complete ignorance, where the interactions between the players and DDUs are characterized by uncertainty sets that depend parametrically on the players’ strategies. Then, a rigorous definition for the equilibrium of the game is established by consolidating generalized Nash equilibrium and Pareto-Nash equilibrium. Afterward, we prove the existence of the equilibrium of N–S–N games under DDUs by applying Kakutani’s fixed-point theorem. Finally, an illustrative example is provided to show the impact of DDUs on the equilibrium of N–S–N games.