Solving Nash equilibria in nonlinear differential games for common-pool resources

Many resources are provided by ecological systems that are vulnerable to a sudden big loss of ecosystem services when exceeding a certain level of pollution. This leads to non-convexities in managing ecological systems. An ecological system is often also a common-pool resource and therefore vulnerable to suboptimal use resulting from non-cooperative behavior. An analysis requires methods to derive cooperative and non-cooperative solutions in managing these types of ecological systems. Such a game is a differential game that has two well-defined non-cooperative solutions: the open-loop and feedback Nash equilibria. This paper provides new numerical methods for solving open-loop and feedback Nash equilibria, for one-dimensional and multiple-dimensional systems. The methods are applied to the lake game, which is the classical example of these types of problems. Especially, the two-dimensional feedback Nash equilibria for the lake game are a novelty of this paper. Such a Nash equilibrium can be close to the cooperative solution which has important policy implications.