Time-Consistent Equilibria in Common Access Resource Games with Asymmetric Players Under Partial Cooperation

Given a differential game, if agents have different time preference rates, cooperative (Pareto optimum) solutions obtained by applying Pontryagin’s maximum principle become time inconsistent. We derive a set of dynamic programming equations in continuous time whose solutions are time-consistent equilibria for problems in which agents differ in their utility functions and also in their time preference rates. The solution assumes cooperation between agents at every time. Since coalitions at different times have different time preferences, equilibrium policies are calculated by looking for Markov (subgame perfect) equilibria in a (noncooperative) sequential game. The results are applied to the study of a cake-eating problem describing the management of a common property exhaustible natural resource. The extension of the results to a simple common property renewable natural resource model in infinite horizon is also discussed.