The current method for stationary mean-field games on networks

Abstract

We discuss first-order stationary mean-field games (MFG) on networks. These models arise in traffic and pedestrian flows. First, we address the mathematical formulation of first-order MFG on networks, including junction conditions for the Hamilton-Jacobi (HJ) equation and transmission conditions for the transport equation. Then, using the current method, we convert the MFG into a system of algebraic equations and inequalities. For critical congestion models, we show how to solve this system by linear programming.

Discrete approximation of stationary Mean Field Games

“…Combining the estimate |D(η τ * φ)(x n )| ≤ ∥Dφ∥ ∞ with fact (iii) in subsection 2.2, we conclude that there exists R > 0 such that ∥q n ∥ ≤ R for every n ∈ N. Thus, by extracting a further subsequence if necessary, we have that q n converges to some q ∈ R d . By considering the limit n → +∞ in (29), we obtain ∆φ(x 0 ) + ⟨Dφ(x 0 ), q⟩ − L(x 0 , q) ≥ ρ + F(x 0 , µ).…”

“…MFGs theory was introduced in the mathematics community by J-M. Lasry and P-L. Lions in [37,38] and independently in the engineering community by M. Huang, P. Caines, and R. Malhamé in [34,33]. This theory has expanded tremendously and has found applications in population dynamics [36,5,29], economics [4,30,10], finance [17,19], and engineering [22,35], to name just a few.…”

First-order mean-field games on networks and Wardrop equilibrium

“…Thus, it does not describe the microstructure in the edge. Hence, in [GMAS19] authors introduced a mean-field game (MFG) model on undirected networks that attempts to model and address these matters.…”

“…The relation between Wardrop equilibrium and MFGs on networks was first observed in [GMAS19]. However, the precise correspondence between these two models was not established there.…”

Discrete Approximation Of Stationary Mean Field Games

“…MFGs theory was introduced in the mathematics community by J-M. Lasry and P-L. Lions in [37,38] and independently in the engineering community by M. Huang, P. Caines, and R. Malhamé in [34,33]. This theory has expanded tremendously and has found applications in population dynamics [36,5,29], economics [4,30,28], finance [16,18], engineering [21,35], to name just a few.…”