Sören Christensen: Competition Vs. Cooperation - A Class of Explicitly Solvable Mean Field Impulse Problems
Time: Wed 2020-02-05 15.15
Lecturer: Sören Christensen, Kiel University
We discuss a class of explicitly solvable mean field problems/games with a clear economic interpretation. More precisely, we consider long term average impulse control problems with underlying general one-dimensional diffusion processes motivated by optimal harvesting problems in natural resource management. We extend these classical stochastic Faustmann models by allowing the prices to depend on the wood supply on the market using a mean-field structure. This means that the reward of an agent \(J(R,S)\) depends on her own impulse strategy \(R\) and the expected volume of wood of the other agents based on their strategy \(S\). The mean field game formulation now consists of finding an equilibrium strategy \(R^*\). This corresponds to a competitive market situation. We prove that, under natural conditions, there exists an equilibrium strategy of threshold type and characterize the threshold explicitly. If the agents cooperate with each other, we are faced with the mean field problem of maximizing \(J(R,R)\) over all strategies \(R\). Using a Lagrange-type argument, we prove that the optimizer of this non-standard impulse control problem is of threshold-type as well and characterize the optimal threshold. Furthermore, some ecological and economic implications are discussed. The findings are illustrated with explicit examples.