Dynamic market games with time delays and their application to international fishing

Abstract

International fishing as a special dynamic game will be analyzed, which is a combination of classical population dynamics and oligopoly theory. The interaction of the countries or firms is through market rules assuming that all markets are open to all participants. In addition, all fishing parties base their activity on the existing common fish stock. The available fish stock and the beliefs of the participants on the fish stock are the state variables. Depending on the possible symmetry of the fishing parties and on their behavior several alternative models will be formed. The classical competitive model will be first formulated and examined, and two special cases will be introduced. First, when the countries, or firms, are identical, second, when one country, or firm, is significantly different than the others. Next, we will assume that a grand coalition is formed, and the total profit of the industry is maximized. Finally, the partially cooperative case will be examined, in which each participant's objective function contains a certain proportion of the profits of the others in addition to its own profits. In all cases, a detailed mathematical model will be constructed, the equilibrium will be computed and the modified population dynamics rule will be formulated. For each case, I will determine the number of positive equilibria, the stability of which will be analyzed first based on the assumption that each participant has instantaneous information on the fish stock. However, there is always a time lag due to information collection and implementation. Since the delay is uncertain, continuously distributed time lags will be assumed. Under this assumption, the dynamic system will be described by Volterra-type integro-differential equations. The asymptotical behavior of the state trajectory will be analyzed by using linearization. Conditions for the local asymptotical stability will be first derived, and in the case of instability, special bifurcations, especially the birth of limit cycles, will be studied. In illustrating the theoretical, analytic results, simple computer studies will be presented.