Distributed Solutions to Complex Societal Problems Reunion Workshop

We study a Hamilton-Jacobi equation on the Wasserstein space on graphs, in the presence of linear operators which include the discrete individual noise operator. Under appropriate conditions, slightly different from the ones covered by the classical theory, we prove a comparison principle, which allows to apply standard arguments for a well posedness theory.
(This talk is based on a joint work with C. Mou and A. Swiech).

We study the optimal stopping problem of McKean-Vlasov diffusions when the criterion is a function of the law of the stopped process. A remarkable new feature in this setting is that the stopping time also impacts the dynamics of the stopped process through the dependence of the coefficients on the law. The mean field stopping problem is introduced in weak formulation in terms of the joint marginal law of the stopped underlying process and the survival process. Using the dynamic programming approach, we provide a characterization of the value function as the unique viscosity solution of the corresponding dynamic programming equation on the Wasserstein space. Under additional smoothness condition, we provide a verification result which characterizes the nature of optimal stopping policies, highlighting the crucial need to randomized stopping. Finally, we establish the convergence of the finite population multiple optimal stopping problem to the corresponding mean field optimal stopping limit. 

We study the long-time regime of the prediction problem in both full information and adversarial bandit feedback setting. We show that with full information, the problem leads to second order parabolic partial differential equations in the Euclidian space. We exhibit solvable cases for this equation. In the adversarial bandit feedback setting, we show that the problem leads to a second order equation in the Wasserstein space. Based on joint works with Ibrahim Ekren and Xin Zhang.

Transport equations are used to model complex systems in a variety of applications, including mean field games. Such equations often involve a drift depending on the solution or other parts of the system in a nonlinear way. Such nonlinear transport equations can be understood by developing a theory for transport equations with irregular drifts. In this talk, I will outline the well-posedness theory for transport equations with certain classes of drifts for which the divergence has a one-sided bound, yielding contractive or expansive behavior, depending on the direction in which the equation is posed. The analysis requires studying the relationship between the transport and continuity equations and the the associated ODE flow. The theory is then used to discuss certain nonlinear transport equations arising in the study of finite state-space mean field games.

In this talk I will discuss an ongoing joint work with Ludovic Tangpi, in which we obtain quantitative convergence results for a class of mean field games with common noise and controlled volatility. The basic strategy we employ is the one introduced recently by Laurière and Tangpi – roughly speaking, we use a synchronous coupling argument to prove a “forward-backward propagation of chaos” result for the FBSDEs which characterize the (open-loop) equilibria of the N-player and mean field games. Unlike in earlier works which have adopted this strategy, we do not require smallness conditions, and instead rely on monotonicity. In particular, displacement monotonicity of the coupling functions and a monotonicity condition on the Hamiltonian allow us to establish a (uniform in N) stability estimate for the N-player FBSDEs, which implies the convergence result. The arguments are relatively simple, and flexible enough to yield similar results in the setting of mean field control and infinite horizon (discounted) mean field games.

I will introduce a new variant of the optimal transport problem in which the target measure is stochastic. I will show that, under suitable assumptions on the target process, the value of this problem satisfies a HJB equation on the set of probability measures. I will then give a general comparison principle for such type of equations and prove how we can characterize the value of such a problem by means of the theory of viscosity solutions.

In this talk, we present new findings on the regularity of first-order mean field games systems with a local coupling. We focus on systems where the initial density is a compactly supported function on the real line. Our results show that the solution is smooth in regions where the density is strictly positive and that the density itself is globally continuous. Additionally, the speed of propagation is determined by the behavior of the cost function $f(m)$ for small densities $m$. When the coupling is entropic, $f(m)=log(m)$, we demonstrate that the support of the density propagates with infinite speed. On the other hand, when $f(m)=m^{theta}$ with $theta >0$, we prove that the speed of propagation is finite. In this case, we establish that under a natural non-degeneracy assumption, the free boundary $partial {m>0}$ is strictly convex and enjoys $C^{1,1}$ regularity. Our methods are based on the analysis of a new elliptic equation satisfied by the flow of optimal trajectories.

In this talk we present two new monotonicity conditions that could serve as sufficient conditions for uniqueness of Nash equilibria in mean field games. Here by uniqueness we mean unconditional uniqueness that is independent of the length of the time horizon, the regularity of the starting distribution of the agents, or the regularization effect of a non-degenerate idiosyncratic noise. Through a rich class of simple examples we show that these new conditions are not only in dichotomy with each other, but also with the two widely studied monotonicity conditions in the literature, the Lasry-Lions monotonicity and displacement monotonicity conditions. The talk is based on joint works with P.J. Graber (Baylor University, TX).

We propose new global solution algorithms for continuous time heterogeneous agent economies with aggregate shocks. We first approximate the state space so the master equation becomes a high, but finite, dimensional partial differential equation. We then approximate the value function using neural networks and solve the master equation using deep learning tools. The main advantage of this technique is that it allows us to find global solutions to high dimensional, non-linear problems. We demonstrate our algorithms by solving two canonical models in the macroeconomics literature: the Aiyagari (1994) model and the Krusell and Smith (1998) model. Joint work with Zhouzhou Gu, Sebastian Merkel and Jonathan Payne.

I will discuss some results on the long time behaviour in Mean Field Games, first under the assumption that the coupling is mildly nonmonotone in the usual Lasry-Lions sense, then in the displacement monotone setting. Finally, I will present recent results on the local uniqueness of solutions to an infinite horizon mean field game inspired by the Kuramoto model.Based on joint works with A. Cesaroni, A. Porretta and A. R. Meszaros.