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 .
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 that it depends on both the changes of the system and the control strategies of other players. Finally, we reformulate NE strategies in the form of controlled rank-dependent stochastic differential equations .
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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
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,
D
the sense
R
state of the players
changes of the system
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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
Finally, we reformulate NE strategies in the form of
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. Our result gives an economic interpretation that sharing will have lower cost than dividing with pooling yielding the lowest cost