We consider a dynamic game in which multiple ride-hailing companies, each comprised of a large number of drivers, are competing over a shared traffic infrastructure to minimize individual teams’ total travel time. In realistic scenarios where the underlying traffic systems are described by nonlinear, stochastic, and high-dimensional dynamical systems, analyzing such a game is a challenging task. In this talk, we discuss two novel mathematical frameworks that offer powerful tools for such an analysis. As the first framework, we introduce the class of linearly-solvable mean-field games (MFGs). This is a special class of the MFGs where an equilibrium can be found simply by solving a linear system. This is in contrast to the conventional MFG framework where coupled Hamilton-Jacobi-Bellman and Fokker-Planck-Kolmogorov equations must be analyzed. Traffic congestion mitigation mechanism based on linearly-solvable MFG is discussed. In the second framework, we discuss Kappen’s path-integral control and its generalization to dynamic games. We demonstrate that a Nash equilibrium among multiple ride-hailing companies in a stochastic game with nonlinear dynamics can found numerically by forward-in-time Monte-Carlo sampling.