Stackelberg-Nash-Pareto Games: A (p,q)-Mean Equilibrium Framework for Hierarchical Decision-Making

Abstract

Real-world decision-making often unfolds within hierarchical structures, involving parallel cooperative and competitive behaviors. To model such scenarios, this paper introduces a class of hierarchical games with coalition structures at both the leader and follower levels, termed Stackelberg-Nash-Pareto (S-N-P) games. In this framework, individual and collective interests coexist inherently within a strategically coupled hierarchy. To reconcile individual and collective rationality, a $(p,q)$ -mean S-N-P equilibrium equipped with two tunable parameters and an averaging process is proposed, allowing flexible adjustment of the emphasis on individual interests and resolving uncertainties at the follower level. Subsequently, by converting the original S-N-P game into a leader-auxiliary game, the question of equilibrium existence is addressed by solving a system of linear equations. In turn, two considerably simpler methods for verifying the equilibrium are developed, based on a game that satisfies the aforementioned linear conditions. Finally, the theoretical results are validated through a smart-grid demand-side management simulation. Note to Practitioners—This paper has great practical applications in the demand-side management of smart grids. Considering the issue of sequential market entry in electricity markets, we adopt a hierarchical game framework to characterize the smart grids. On this basis, we propose an equilibrium point that balances both individual and collective interests, offering significantly greater practical value to practitioners compared to related works focusing on either individual or collective interests. Then, we search for the existence of the equilibrium point through both direct and indirect approaches, which simplifies the process of identifying equilibrium points in practical applications. This paper shows a novel perspective for the analysis and optimization of demand-side management in smart grids and also offers valuable ideas for analogous engineering challenges like resource allocation.