Dissertation
Equilibria and bifurcation theory for mean-field games
Doctor of Philosophy (Ph.D.), Drexel University
Jun 2023
DOI:
https://doi.org/10.17918/00001649
Abstract
To represent the interaction of N rational competitors traditionally, a coupled system of N differential equations must be solved simultaneously, yielding the equilibrium strategy for each player. This approach becomes impractical as N grows, prompting the adoption of a mean-field approach, in which we assume N is large enough that the dynamics of the competition may be suitably represented distributionally in the continuum case. The trajectory of each player through the state space is then driven by a dynamic control, together with an adapted Brownian motion, reducing the computation from N differential equations to two Partial Differetial Equations (PDE's), a Hamilton-Jacobi-Bellman equation governing the evolution in time of a utility function, and a Fokker-Planck (Forward Kolmogorov) equation governing the evolutio n in time of the distribution of players, coupled by means of the nonlinear Hamiltonian. In this thesis, we use the Implicit Function Theorem and Bifurcation Theory to obtain nontrivial equilibria of various mean-field games, and we go on to demonstrate the use of Schauder's fixed-point theorem to prove the existence of low-regularity time-bound solutions of a congestion-type mean-field game.
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Details
- Title
- Equilibria and bifurcation theory for mean-field games
- Creators
- Luke Candler Brown
- Contributors
- David M. Ambrose (Advisor)
- Awarding Institution
- Drexel University
- Degree Awarded
- Doctor of Philosophy (Ph.D.)
- Publisher
- Drexel University; Philadelphia, Pennsylvania
- Number of pages
- xx, 110 pages
- Resource Type
- Dissertation
- Language
- English
- Academic Unit
- College of Arts and Sciences; Drexel University; Mathematics
- Other Identifier
- 991020668800504721