Differential game model with discretized solution for the use of limited water resources

Abstract

It is indispensable for the sustainable water supply of the human society to distribute the available water resources with maximal efficiency among the different consumers while assuring the needed time for recharge. The theoretically established, effective tool for the solution of this allocation problem is mathematical modelling with special regard to the methods of game theory.In the present paper, a differential game is proposed to describe the consumption and the (natural) recharge of a limited water resource. Then the general course of the solution of the allocation problem is given aiming at maximizing the players’ yields through the discretization of the strategy sets. In case of two consumers, the solution is given in more details along with several practical examples. The Pareto optimality of the non-cooperative solution (Nash equilibrium) is checked and the Pareto optimal strategy pair, which provides maximal sum payoff for the players, is suggested for cooperation (after a possible re-distribution of the payoffs). This cooperative solution generally provides higher payoff for each player than if they consumed according to conflicting non-cooperative behaviour. In addition, examples with three consumers are provided similarly.