Scientific

The purpose of this work is to introduce a notion of weak solution to the master equation of a potential mean field game and to prove that existence and uniqueness hold under quite general assumptions. Remarkably, this is shown to hold true without any monotonicity constraint on the coefficients. The key point is to interpret the master equation in a conservative sense and then to adapt to the infinite dimensional setting earlier arguments for hyperbolic systems deriving from a HJB equation. Here, the master equation is indeed regarded as an infinite dimensional system set on the space of probability measures. To make the analysis easier, we assume that the coefficients are periodic and accordingly that the probability measures are defined on the torus. This allows to represent probability measures through their Fourier coefficients. Most of the analysis then consists in rewriting the master equation and the corresponding HJB equation for the mean field control problem lying above the mean field game as PDEs set on the Fourier coefficients themselves.

Joint work with A. Cecchin (Ecole Polytechnique, France)

Modeling and understanding crowd evacuation dynamics has been a long-standing problem. Most realistic models involve nonlinear effects to capture individual velocity decrease with crowd density increase. This leads to useful but essentially intractable partial differential equation-based models. We consider here for tractability purposes, a class of linear quadratic large-scale evacuation games where velocity can be improved through crowd avoidance. This is simulated in the agent cost functions through negative costs which accrue when agents drift away from variously defined population mean trajectories, in a multi-exit situation. The presence of negative cost components generically induces a finite escape time phenomenon if the time horizon is not adequately bounded. We formulate two types of models for which we provide sufficient time horizon upper bounds for agent cost convergence and establish existence of limiting mean field game equilibria as well as their ε-Nash property. This is joint work with Noureddine Toumi and Jérôme Le Ny.

This talk will be a fairly high-level one, addressing various current issues in mean field games (MFGs), the underlying challenges primarily with regard to robustness, learning, and incentivization, and paths toward their resolution. Among these are: (i) use of multi-agent reinforcement learning for the computation of mean-field equilibrium (MFE) with state samples drawn from an unmixed Markov chain, and studying the performance of the associated (actor-critic) algorithms; (ii) adversarial MFGs on multi-graphs where agents interact with their neighbors, with such interactions propagating from neighborhoods to the entire network, and with an adversary counteracting the consensus formation process among the agents; and (iii) MFGs with a decision hierarchy, where the agent at the top of the hierarchy (leader) aims at designing incentive strategies (as in mechanism design) to induce a high population of agents at the lower level (followers) to act rationally toward a globally optimal solution in spite of their non-cooperative behavior. The talk will also identify several fruitful directions of research in this domain.

This talk is an overview of a recent and ongoing line of work on large sparse networks of interacting diffusion processes. Each process is associated with a vertex in a graph and interacts only with its neighbors. When the graph is complete and the size grows to infinity, the system is well-approximated by its mean field limit, which describes the behavior of one typical process. For general graphs, however, the mean field approximation can fail, most dramatically when the graph is sparse. Nevertheless, if the underlying graph is locally tree-like (as is the case for many canonical sparse random graph models), we show that a single process and its nearest neighbors are characterized by an autonomous evolution which we call the "local dynamics." This can be viewed as a sparse counterpart of the usual McKean-Vlasov equation. The structure of the local dynamics depend heavily on the symmetries of the underlying graph and the conditional independence structure of the solution process. In the time-stationary case, the local dynamics take a particular tractable form. Based on joint works with Kavita Ramanan, Ruoyu Wu, and Jiacheng Zhang.

The study of large populations of interacting agents connected via a non-trivial network of connections represents a field of growing interest in Applied Mathematics. The relatively recent theory of graphons turns out to be well-adapted to model the emergence of complex networks and has been applied in several contexts by now, including mean-field systems. In this talk, we discuss how some of the key properties of graphon objects, e.g., exchangeability and labelling, are related to the study of interacting particle systems. Hopefully, this will shed some light on the behavior of systems described by partial differential equations and graphons, as the Graphon Mean-Field Game equations.