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. We call it a "moving free boundary" problem to highlight the difference between the standard control problem versus the stochastic games. We also show our NE strategies can be reformulated as controlled rank-dependent stochastic differential equations. Our result gives an economic interpretation that sharing will have lower cost than dividing with pooling yielding the lowest cost.
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