Analysis and computation of equilibria in multilevel games with finitely and infinitely many players

Thünen, Anna; Banda, Mapundi (Thesis advisor); Herty, Michael (Thesis advisor); Schwartz, Alexandra (Thesis advisor)

Aachen : RWTH Aachen University (2021)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2021

Abstract

This thesis deals with certain classes of multilevel games. Game theory is an extension to classical optimization that couples several optimization problems. Certain hierarchies within the game may be considered by introducing a multilevel structure. This may model time-delayed decisions or an imbalance in the possibility of individual players' influence. The players announcing their decision first are referred to as leaders and the others as followers. We distinguish the case with a single leader, the Stackelberg game, and the case of several leaders, the so-called Multi-Leader-Follower game. In the last decades multilevel games served as a tool in the analysis of systems of multiple competing interests and hierarchies. A prominent application is the analysis of electricity markets using multi-leader follower games. This applications usually involve the modeling of large populations of followers. The demand of all customers for electricity is represented by one single independent system operator. In fact, this currently provides a precise model for the current practice in energy markets. The development of multiscale modeling may be used for the analysis of large populations of interacting agents. Historically, multiscale modeling has its origin in statistical mechanics. For example, it connects the dynamical model of many individual gas molecules to equations describing the density of the gas. The simplifications made for the multilevel games with many followers are useful and valid in many respects, but in this thesis we would like to go further to elaborate the modeling of these populations in the context of multilevel games. In this thesis we consider these challenging features separately and therefore this work is divided into two parts: In the first part, we study the multi-leader-follower structure in games. In the second part we consider a Stackelberg game with possibly infinitely many followers. The first part bases on the work in (Herty et al 2020a) and (Steffensen, Thünen 2019):Models of optimization and game theory in finite dimensional spaces are introduced together with related fundamental theory. Furthermore, a numerical method is explained for each model. The existence and uniqueness of Nash equilibria for a class of quadratic multi-leader-follower games is analyzed. The problem is reformulated as Nash game and equilibria are characterized. For this we use a smoothing technique and prove convergence to a Nash equilibrium of the original multi-leader-follower game. Then a numerical method consistent with the developed theory is proposed. The second part is based on the work in (Herty et al. 2020b):Foundations of multiscale modeling, optimal control, and differential games are covered. We present models as well as methods here. In particular, a Stackelberg game is considered for which we derive the mean-field limit for the set of follower dynamics which yields an evolution equation of the density of followers. The succession of optimization and the derivation of the mean-field limit can be varied and we prove conditions for consistency. We conclude by developing a numerical method for this type of Stackelberg game.

Institutions

• Department of Mathematics [110000]
• Lehr- und Forschungsgebiet Mathematik [114620]