Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion

Abstract

Abstract-Here, we consider one-dimensional firstorder stationary mean-field games with congestion. These games arise when crowds face difficulty moving in highdensity regions. We look at both monotone decreasing and increasing interactions and construct explicit solutions using the current formulation. We observe new phenomena such as discontinuities, unhappiness traps and the nonexistence of solutions.

Weakly coupled mean-field game systems

“…In that reference and also in [21], various examples of first-order MFGs are shown to have a vanishing density. The oscillation of H(0, x) plays an essential role in these examples.…”

The current method for stationary mean-field games on networks

“…Unfortunately, prior methods are not valid for first-order MFG, where a distinct set of phenomena that includes the loss of smoothness for Hamilton-Jacobi (HJ) equations and lack of continuity for the value function at the vertices can occur. Because first-order MFG on networks are coupled systems of first-order MFG, we use the current method from [GNP17], [GNP16] to construct a novel approach for stationary MFG on networks.…”

One-dimensional, non-local, first-order stationary mean-field games with congestion: A Fourier approach

“…Following [18], [17] we call (1.6) the current formulation of (1.1). There are two possibilities: j = 0 and j = 0.…”