The tragedy of the commons: a Mean-Field Game approach to the reversal of travelling waves

Abstract The goal of this paper is to investigate an instance of the tragedy of the commons in spatially distributed harvesting games. The model we choose is that of a fishes’ population governed by a parabolic bistable equation and that fishermen harvest. We assume that, when no fisherman is present, the fishes’ population is invading (mathematically, there is an invading travelling front). Is it possible that fishermen, when each of them is acting selfishly in his or her own best interest, might lead to a reversal of the travelling wave and, consequently, to an extinction of the global population? To answer this question, we model the behaviour of individual fishermen using a Mean Field Game approach, and we show that the answer is yes. Up to our knowledge, this is the first theoretical work proving that a pool of common resources might go extinct due to the uncoordinated action of a large number of selfish and greedy agents. Additionally, we prove that adding coordination may result in higher outcome for each individual agent, while still guaranteeing the survival of the fishes’ population.