Uniqueness Versus Indeterminacy in the Tragedy of the Commons: A ‘Geometric’ Approach

This paper characterizes continuous Markov perfect equilibria as smooth connections between an ‘initial’, i.e., at the origin of the state space, and an ‘end’ manifold that result from patching with the boundary solution. The major result is that multiple equilibria require a non-monotonic initial manifold. This necessary condition for multiple equilibria can be tested without (or prior to) solving the Hamilton-Jacobi-Bellman equation. Application to a familiar dynamic tragedy of the commons with nonlinear instead of linear-quadratic utilities shows that the elasticity of marginal utility is the crucial property: If this elasticity is (everywhere) greater than $\frac{n-1}{n}$ , n=number of polluters, then the Nash equilibrium is unique. Assuming the opposite inequality (globally) implies that no saddle-point equilibrium exists. Therefore, the ‘focal’ point equilibrium is gone and all conceivable boundary conditions determine a corresponding equilibrium, e.g. ‘anything goes’ for power utility functions.