On Nash-Stackelberg-Nash Games under Decision-Dependent Uncertainties: Model and Equilibrium

TL;DR

This paper introduces a new framework for analyzing hierarchical Nash-Stackelberg-Nash games with decision-dependent uncertainties, establishing equilibrium existence and illustrating the impact of uncertainties on strategic interactions.

Contribution

It formulates N-S-N games under decision-dependent uncertainties, defines a new equilibrium concept, and proves its existence using fixed-point theory.

Findings

Equilibrium exists under decision-dependent uncertainties.

Uncertainty strategies influence players' equilibrium decisions.

Illustrative example demonstrates the impact of DDUs on game outcomes.

Abstract

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.