Games with exhaustible resources

We study N-player continuous-time Cournot games in an oligopoly where firms choose production quantities. These are nonzero-sum differential games, whose value functions may be characterized by systems of nonlinear Hamilton-Jacobi PDEs. When resources such as oil are in finite supply, exhaustibility enters as a boundary condition for the PDEs. We analyze the problem when there is an alternative, but expensive, technology (for example, solar power for energy production) and give an asymptotic approximation in the limit of small exhaustibility. We illustrate the two-player problem by numerical solutions and discuss the impact of limited oil reserves on production and oil prices in the duopoly case.