Stochastic Games for Fuel Followers Problem: N vs MFG

In this paper we formulate and analyze an $N$-player stochastic game of the classical fuel follower problem and its Mean Field Game (MFG) counterpart. For the $N$-player game, we obtain the Nash Equilibrium (NE) explicitly by deriving and analyzing a system of Hamilton--Jacobi--Bellman (HJB) equations, and by establishing the existence of a unique strong solution to the associated Skorokhod problem on an unbounded polyhedron with an oblique reflection. For the MFG, we derive a bang-bang type NE under some mild technical conditions and by the viscosity solution approach. We also show that this solution is an $\epsilon$-NE to the $N$-player game, with $\epsilon =O(\frac{1}{\sqrt{N}})$. The $N$-player game and the MFG differ in that the NE for the former is state dependent while the NE for the latter is threshold-type bang-bang policy where the threshold is state independent. Our analysis shows that the NE for a stationary MFG may not be the NE for the corresponding MFG.