Tag - Mean-field game theory

David Ambrose: Some existence results for mean-field games

3rd April, 2023

When considering N-player differential games, making the approximation that there are instead infinitely many agents leads to the mean-field games system of PDEs. This system has two unknowns, the probability distribution of the players, and the value function being optimized by a representative agent. One of these satisfies a forward parabolic equation and the other satisfies a backward parabolic equation. The forward parabolic equation comes with initial data while terminal data (at a fixed time T > 0) is specified for the backward parabolic equation. We will describe some existence results for this coupled forward-backward system, without assuming that the non-linearity (the Hamiltonian) has any special structures such as convexity or monotonicity. Results presented will including treating a specific system which has been given as a model of household savings and wealth.

Diogo Gomes: Hessian Riemannian flows in mean-field games

15th February, 2023

Hessian Riemannian flows are a powerful tool for the construction of numerical schemes for monotone mean-field games that have their origin in constrained optimization problems. In this talk, we discuss the general construction of these flows for monotone mean-field games, their existence and regularity properties, and their asymptotic convergence.

Julián David Gutiérrez Pineda: Machine learning architectures for mean-field games models of price formation

29th September, 2022

In this talk, we approach the solution of mean-field game systems arising in price formation models employing machine learning. We use a min-max characterization of the optimal control and price variables. We guarantee the convergence of the training algorithm using first-order conditions of the underlying optimal control problem. Numerical results for linear-quadratic models illustrate our results.

Julian Barreiro-Gomez: The role of stochastic differential games of mean-field type in smart cities applications

13th June, 2022

This brief talk aims to show how the stochastic differential games contribute to the optimal solution of large-scale engineering problems emerging in smart cities where several dynamical interactions occur, e.g., the water distribution system, the crowd management, the traffic flow, power systems, among many others. We show that the general simplest problem statement leads to a complex PIDE system involving a backward Hamilton-Jacobi-Bellman equation coupled with a forward Fokker-Plank-Kolmogorov equation. Then, we discuss how this complexity can be handled for specific cases pursuing to develop real implementation. As an example, we focus on the crowd evacuation problem. Finally, future directions we are currently working on involving machine learning and stability are presented.