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. (C) 2022 Elsevier Ltd. All rights reserved.

Automated summary

This paper discusses a class of two-stage hierarchical games with multiple leaders and followers, known as Nash-Stackelberg-Nash (N-S-N) games, under decision-dependent uncertainties (DDUs). By formulating the N-S-N games with DDUs and establishing a rigorous definition for the equilibrium of the game, the existence of the equilibrium of N-S-N games under DDUs is proved using Kakutani's fixed-point theorem. An illustrative example is provided to demonstrate the impact of DDUs on the equilibrium of N-S-N games.