Before revision: In this paper , we propose and analyze a class of stochastic N-player games where the original one player game is a two-dimensional stochastic control problem. Three different constraints are considered: pooling, dividing and sharing. The controlled dynamics under the Nash Equilibrium is derived via solving a system of free boundary problems where the boundary is "moving" in the sense that it depends on both the state of the players and the control strategies of other players.
After revision: In this paper , we propose and analyze a class of stochastic N-player games that includes finite fuel stochastic games as a special case. We first derive sufficient conditions for the Nash equilibrium (NE) in the form of a verification theorem, which reveals an essential game component regarding the interaction among players. It is an analytical representation of the conditional optimality condition for NEs, largely missing in the existing literature on stochastic games. The derivation of NEs involves first solving a multi-dimensional free boundary problem and then a Skorokhod problem, where the boundary is "moving" in the sense that it depends on both the state of the players and the control strategies of other players.