Patience, Fish Wars, rarity value & Allee effects

In a Small Fish War two agents interacting on a body of water have essentially two options: they can fish with restraint or without. Fishing with restraint is not harmful; shing without yields a higher immediate catch, but may induce lower future catches. Inspired by recent work in biology, we introduce into this setting rarity value and Allee effects. Rarity value means that extreme scarcity of the sh may affect its unit pro t 'explosively'. An Allee effect implies that if the population size or density falls below a so-called Allee threshold, then only negative growth rates can occur from then on. We examine equilibrium behavior of the agents under the limiting average reward criterion and the sustainability of the common-pool resource system. Assuming fixed prices at fi rst, we show that patience on the part of the agents is bene cial to both sustainable high catches and fish stocks. An Allee effect can not influence the set of equilibrium rewards unless the Allee threshold is (unrealistically) high. A price mechanism reflecting effects of the resource's scarcity, is then imposed. We obtain a rather subtle picture of what may occur. Patience may be detrimental to the sustainability of a high fish stock and it may be compatible with equilibrium behavior to exhaust the resource almost completely. However, this result does not hold in general but it depends on complex relations between the Allee threshold, the dynamics in the (interactive) resource and price systems, and the actual scarcity caused if the agents show no restraint.