We consider a one-dimensional model that describes the depinning of an elastic string of particles in a strongly pinning, phase-disordered periodic environment under a slowly increasing force. The evolution towards depinning occurs by the triggering of avalanches in regions of activity which are at first isolated, but later grow and merge. For large system sizes the dynamically critical behavior is dominated by the coagulation of these active regions. Our analysis and numerical simulations show that the evolution of the sizes of active regions is well described by a Smoluchowski coagulation equation, allowing us to predict correlation lengths and avalanche sizes in terms of certain moments of the size distribution.

When a mean field game satisfies certain monotonicity conditions, the mean field equilibrium is unique and the corresponding value function satisfies the so called master equation. In general, however, there can be multiple equilibriums which typically lead to different values. In this paper we study the set of values over all mean field equilibriums, which we call the set value of the game. We shall establish two main properties of the set value: (i) the dynamic programming principle; (ii) the convergence of the set values of the corresponding N-player games. We emphasize that the set value is very sensitive to the choice of the admissible controls. For the dynamic programming principle, one needs to use closed loop controls (instead of open loop controls). For the convergence, one has to restrict to the same type of equilibriums for the N-player game and for the mean field game. We shall investigate three cases, two in finite state space models and the other in a diffusion model.

Building upon the dynamic programming principle for set valued functions arising from many applications, in this paper we propose a new notion of set valued PDEs. The key component in the theory is a set valued Itô formula, characterizing the flows on the boundary surface of the dynamic sets. In the contexts of multivariate control problems, we establish the wellposedness of the set valued HJB equations, which extends the standard HJB equations in the scalar case to the multivariate case. As an application, a moving scalarization for certain time inconsistent problems is constructed by using the classical solution of the set valued HJB equation.

This work provides a unified framework for exploring games. In existing literature, strategies of players are typically assigned scalar values, and the concept of Nash equilibrium is used to identify compatible strategies. However, this approach lacks the internal structure of a player, thereby failing to accurately model observed behaviors in reality. To address this limitation, we propose to characterize players by their learning algorithms, and as their estimations intrinsically induce a distribution over strategies, we introduced the notion of equilibrium in terms of characterizing the recurrent behaviors of the learning algorithms. This approach allows for a more nuanced understanding of players, and brings the focus to the challenge of learning that players face. While our explorations in discrete games, mean-field games, and reinforcement learning demonstrate the framework's broad applicability, they also set the stage for future research aimed at specific applications.