This thesis consists of three main parts. In the first part, we discuss first-order stationary mean-field games (MFGs) on networks.
We derive the mathematical formulation of first-order MFGs on networks with particular emphasis on the conditions at the vertices both for the Hamilton-Jacobi equation and for the transport equation. Then, we review the current method, which, for the stationary case, allows us to convert the MFG into a system of algebraic equations and inequalities. Finally, we discuss in more detail the travel cost and its properties.
In the second part, we discuss the Wardrop equilibrium model on networks with flow-dependent costs and its connection with stationary MFGs. First, we build the Wardrop model on networks. Second, we show how to convert the MFG model into a Wardrop model. Next, we recover the MFG solution from the Wardrop solution. Finally, we study the calibration of MFGs with Wardrop travel cost problems.
In the third part, we explain the algorithm for solving the algebraic system associated with the MFG numerically, then, we present some examples and numerical results.
Fatimah Al Saleh received her Bachelor of Science in Mathematics from King Faisal University. In 2016, she earned her MSc degree in Applied Mathematics from King Abdullah University of Science and Technology (KAUST). Currently, she is pursuing her Ph.D. degree focusing on first-order stationary mean-field games on networks and the connection with Wardrop equilibria. She has worked as a faculty in King Faisal University since 2012.