Contents lists available at ScienceDirect Journal of Hydrology journal homepage: www.elsevier.com/locate/jhydrol Research papers Differential game model with discretized solution for the use of limited water resources R. Kicsiny ⁎ , Z. Varga Department of Mathematics, Institute of Environmental Systems, Faculty of Mechanical Engineering, Szent István University, Páter K. u. 1., 2100 Gödöllő, Hungary ARTICLE INFO This manuscript was handled by G. Syme, Editor-in-Chief, with the assistance of Joseph H.A. Guillaume, Associate Editor Keywords: Water resource management Differential game model Discretized solution Nash equilibrium Pareto optimality Cooperation ABSTRACT It is indispensable for the sustainable water supply of the human society to distribute the available water re- sources 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-co- operative 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. 1. Introduction The peril of the decreasing water resources, which are usable for the human society, is well-known. The problem has reached a critical level, which must be handled locally and globally as well (Budapest Water Summit, 2016; Chen et al., 2018). The crisis is caused by the increasing consumption of the inhabitants, industry and agriculture, besides the pollution of the environment (Yang et al., 2018) and, within that, in- creasing pollution of water (Chen et al., 2017b). In the field of agri- culture, which is likely the largest water consumer, irrigation stands for a very significant amount that grows increasingly, partly because of accelerating global warming (Lauffenburger et al., 2018). Conse- quently, in addition to the development of environmental protection and economical water use, it is indispensable for the sustainable water supply of our human society to distribute the available water resources with maximal efficiency among the different consumers while assuring the needed time for (natural) recharge of the water (Ziolkowska and Peterson, 2017.). The theoretically established, effective tool for the solution of this allocation problem is mathematical modelling with special regard to the methods of game theory. For a general overview of the different fields of application of game theory in water resource management, we refer the Reader to the early work of Bogárdi and Szidarovszky (1976) and the recent works of Madani (2010) and Podimata and Yannopoulos (2015). A business simulation game is applied for irrigation water policy impact at farmers’ level in Bucholz et al. (2016). The results show that a water quota is able to reduce the irrigation applications while a water pricing scheme has no effect. Leader-follower (Stackelberg) games are applied for water resource management problems in Chen et al. (2017a), Kicsiny et al. (2014) and Kicsiny (2017), where some players (leaders) have priority with respect to time before other players (fol- lowers). A two-level game including one main game and four sub-games is given in Wei et al. (2010) to describe a water allocation problem in China. In Zanjanian et al. (2018), a water allocation problem among organizational stakeholders is solved with game theoretical tools, more particularly, with the so-called GMCR (graph model for conflict re- solution) method. The paper (Madani and Hooshyar, 2014) presents a game theoretical reinforcement learning method for social planners to optimize their policies in multi-operator multi-reservoir systems. Differential games are special games governed by differential equations. Accordingly, the players move continuously in time from an initial time point to a final one in such games. Negri (1989) models common property aquifers by means of a differential game. Negri deals with both open-loop solutions, where the players commit themselves at the beginning of the game to their strategies (pumping rates) in all future periods, and feedback solutions, where they can adapt their https://doi.org/10.1016/j.jhydrol.2018.12.029 Received 8 August 2018; Received in revised form 19 December 2018; Accepted 21 December 2018 ⁎ Corresponding author. E-mail address: kicsiny.Richard@gek.szie.hu (R. Kicsiny). Journal of Hydrology 569 (2019) 637–646 Available online 27 December 2018 0022-1694/ © 2018 Elsevier B.V. All rights reserved. T