Solving Nash Equilibria in Nonlinear Differential Games for Common-Pool Resources

TL;DR

This paper introduces new numerical methods to compute open-loop and feedback Nash equilibria in nonlinear differential games modeling common-pool resources, with a focus on ecological systems vulnerable to tipping points.

Contribution

It presents novel numerical techniques for deriving both open-loop and feedback Nash equilibria in one- and two-dimensional dynamical systems, including the first computation of two-dimensional feedback equilibria.

Findings

Two-dimensional feedback Nash equilibria are successfully computed.

Feedback Nash equilibria closely approximate cooperative solutions.

Methods are demonstrated on the classical lake game.

Abstract

Many resources are provided by an ecological system that is vulnerable to tipping when exceeding a certain level of pollution, with a sudden big loss of ecosystem services. An ecological system is usually 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 for managing a dynamical system with tipping points. Such a game is a differential game which has two well-defined non-cooperative solutions, the open-loop and feedback Nash equilibria. This paper provides new numerical methods for deriving open-loop and feedback Nash equilibria, for one-dimensional and two-dimensional dynamical systems. The methods are applied to the lake game, which is the classical example for these types of problems. Especially, two-dimensional feedback Nash equilibria are a novelty of this paper. This Nash equilibrium is close to the cooperative solution which has important policy implications.