We study in this paper three aspects of Mean Field Games. The first one is the case when the dynamics of each player depend on the strategies of the other players. The second one concerns the modeling of "noise" in discrete space models and the formulation of the Master Equation in this case. Finally, we show how Mean Field Games reduce to agent based models when the intertemporal preference rate goes to infinity, i.e. when the anticipation of the players vanishes. 2

This paper is interested in the problem of optimal stopping in a mean field game context. The notion of mixed solution is introduced to solve the system of partial differential equations which models this kind of problem. This notion emphasizes the fact that Nash equilibria of the game are in mixed strategies. Existence and uniqueness of such solutions are proved under general assumptions for both stationary and evolutive problems.

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We present results of existence, regularity and uniqueness of solutions of the master equation associated with the mean field planning problem in the finite state space case, in the presence of a common noise. The results hold under monotonicity assumptions, which are used crucially in the different proofs of the paper. We also make a link with the trajectories induced by the solution of the master equation and start a discussion on the case of boundary conditions. Contents 1.2. Preliminary results 2. Planning problem master equation in R d 2.1. Statement of the problem 2.2. Properties of the Yosida approximation 2.3. The limit master equation 2.4. Links with the induced trajectories 3. A comment on the case of a restricted domain 3.1. Main differences with the previous case 3.2. The half space case with vanishing conditions

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We propose a plausible mechanism for the short-term dynamics of the oil market based on the interaction of a cartel, a fringe of competitive producers, and a crowd of capacity-constrained physical arbitrageurs that store the resource. The model leads to a system of two coupled nonlinear partial differential equations, with a new type of boundary conditions that play a key role and translate the fact that when storage is either full or empty, the cartel has enhanced strategic power. We propose a finite difference scheme and report numerical simulations. The latter result in apparently surprising facts: 1) the optimal control of the cartel (i.e., its level of production) is a discontinuous function of the state variables; 2) the optimal trajectories (in the state variables) are cycles which take place around the discontinuity line. These patterns help explain remarkable price swings in oil prices in 2015 and 2020.

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