Mean-Field Games are games with a continuum of players that incorporate the time-dimension through a control-theoretic approach. Recently, simpler approaches relying on the Best Reply Strategy have been proposed. They assume that the agents navigate their strategies towards their goal by taking the direction of steepest descent of their cost function (i.e. the opposite of the utility function). In this paper, we explore the link between Mean-Field Games and the Best Reply Strategy approach. This is done by introducing a Model Predictive Control framework, which consists of setting the Mean-Field Game over a short time interval which recedes as time moves on. We show that the Model Predictive Control offers a compromise between a possibly unrealistic Mean-Field Game approach and the sub-optimal Best Reply Strategy.

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We introduce a parallel machine scheduling problem in which the processing times of jobs are not given in advance but are determined by a system of linear constraints. The objective is to minimize the makespan, i.e., the maximum job completion time among all feasible choices. This novel problem is motivated by various real-world application scenarios. We discuss the computational complexity and algorithms for various settings of this problem. In particular, we show that if there is only one machine with an arbitrary number of linear constraints, or there is an arbitrary number of machines with no more than two linear constraints, or both the number of machines and the number of linear constraints are fixed constants, then the problem is polynomial-time solvable via solving a series of linear programming problems. If both the number of machines and the number of constraints are inputs of the problem instance, then the problem is NP-Hard. We further propose several approximation algorithms for the latter case.

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We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be formulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the linear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.