Water allocation using game theory under climate change impact ( case study : Zarinehrood )

The combined effects of climate change and growing water demand due to population growth, industrial and agricultural developments cause an increase in water scarcity and the subsequent environmental crisis in river basins, which results in conflicts over the property rights and allocation agreements. Thus, an integrated, sustainable and efficient water allocation considering changes in water resources due to climate change and change of users’ demands is necessary. In this study, the drainage basin of Zarinehrood was chosen to evaluate the function of selective methods. Assessing climate change impact scenarios of the Fifth IPCC reports, e.g., RCP2.6, RCP4.5, RCP6.0 and RCP8.5, have been used. For downscaling outputs of GCMs an artificial neural network (ANN) and for bias correction a quantile mapping (QM) method have been used. Using a bargaining game and the Nash bargaining solution (NBS) with two methods, one symmetric and two AHP methods, the water available for users was allocated. Results indicate an overall increase in temperature and precipitation in the basin. In bargaining game solutions, AHP provided better utilities for players than the symmetric method. These results show that with water management programs and use of a cooperative bargaining game, water allocation can be done in an efficient way.


INTRODUCTION
Water allocation is central to the management of water resources. Due to geographically and temporally unevenly distributed precipitation (Al Radif ), rapidly increasing water demands driven by the world population, effects of climate change on river flow and other stresses, and degradation of the water environment, there are increasing scarcities of water resources in many countries. Conflicts often arise when different water users (including the environment) compete for limited water supply. The need to establish appropriate water allocation methodologies and associated management institutions and policies is recognized by researchers, water planners, and governments.
Many studies have been carried out in this domain, but there are still many obstacles to reaching equitable, efficient, and sustainable water allocations (Dinar et al. ; Syme et al. ). Alongside these conflicts, there are also indications that recent climate changes have already affected many physical and biological systems. The impact of these effects includes extreme occurrence of flooding and droughts.
Various methods and models have been used in water resources allocation, including simulation methods, optimization methods, water rights, game theory, and complex adaptive systems (Di Nardo et al. ). The water resources allocation problem usually involves various rational decision-maker interactions, and water resources allocation needs to consider multiple objectives (such as economic, social, environmental, etc.), which yields multiobjective decision-making problems. This kind of allocation model solves water resources allocation problems via optimization approaches, which reflect the indirect interaction between decision-makers, but ignore the direct interaction between decision-makers, making them impractical in realworld applications (Madani ). Game theory is a theory of decision-making and equilibrium during the process of direct interaction between decision-makers (Loáiciga ). Therefore, water resources allocation based on game theory is a promising method for reducing this deficiency. Moreover, compared with traditional water resources allocation, which only focuses on the interests of the whole society, using the game theory to study the conflict of water resources allocation allows full consideration of the influence of all decision-makers. It is recognized that there are different interests among decision-makers in the process of water resources allocation, and game theory can be used to maximize the benefits of all water users while achieving the rational allocation of water resources. Therefore, using the game theory to study the conflict of water resources allocation is more practical. In recent years, water resources allocation based on game theory has been studied and extended. Carraro et al. () systematically expounded the application of non-cooperative negotiation theory in water resources conflict. Parrachino () applied cooperative game theory to water resource issues, and their results showed that cooperation over scarce water resources was possible under various physical conditions and institutional arrangements. Madani () demonstrated that the application of game theory in the field of water resources can be divided into five parts, i.e., water or benefit allocation among water users, groundwater management, transboundary water allocation, water quality management, and other types of water resources management. Dinar et al. () divided the application of game theory in the conflict of water resources allocation into three aspects: (1) the application of non-cooperative negotiation theory in water resources allocation conflict; (2)  The impacts that climate change may have on water availability are largely affected by water allocations and the countermeasures undertaken (IPCC ). Climate change is believed to cause changes both in water quantity and water quality. The prospect of these changes will help decision-makers formulate mitigation and adaptation strategies to effectively deal with the impacts posed by climate change. A variety of general circulation models (GCMs) has been developed to project climate over longterm horizons under pre-determined greenhouse gas emission scenarios. Their projections often provide the source of data used for assessing impacts of climate change in various fields, such as agriculture, water resources, and the environment. The GCM outputs are coarse in resolution (a horizontal resolution of GCM is generally about 300 km) and have the statistical characteristics of an average area rather than of a point quantity (Osborn & Hulme ).
However, hydrological models often require regional and to utilize GCM outputs in regional studies, two approaches, namely, dynamic downscaling, such as using regional circulation models (RCMs), and statistical downscaling, have been developed. Murphy () indicated that there is no clear difference in the performance level between these two techniques in terms of downscaling monthly climate data. Arnbjerg-Nielsen & Fleischer () studied the impact of climate change and identified suitable adaptation strategies due to flooding posed by climate change from an economic perspective. Two types of models, namely, deterministic (i.e., conceptual and physically based) models and data-driven or statistical models (e.g., artificial neural networks (ANNs)) have been implemented to evaluate the impacts of climate change on water resources.
As an alternative approach to deterministic models, the ANN approach has been employed in various fields including hydrology and water resources (Govindaraju ).
Some of the advantages of using a data-driven modeling approach include needing less data and less extensive user expertise and knowledge into physical processes. In  The objective of this paper is to analyze water allocation in Zarinehrood river basins, Iran, considering the impact of climate change on its hydrologic parameters. Assessing climate change impact, an ANN for downscaling outputs of GCMs and for bias correction a quantile mapping (QM) method have been used. Then, considering some assumptions for predicting future demands, a water resources management program has been used to assess the water available for allocating to users. Using a bargaining game and the Nash bargaining solution (NBS) with two methods, one symmetric and the other AHP method, the water available for users has been allocated.

METHODOLOGY Artificial neural network
An ANN is a computational tool based on the biological processes of the human brain (Sudheer et al. ). Its capability to predict output variables by using a series of interconnected nodes that recognize relations between input and output variables makes ANN models powerful tools for hydrologic analyses (Mutlu et al. ). Model inputs are weighted and passed to internal nodes in hidden layers, which develop functions for output ( Figure 1). There can be one or more of these hidden layers between the input and output. Compared with conventional rainfall-runoff models, ANN models require fewer parameters but still provide reliable results in hydrological forecasting (Riad et al. ). The complexity of physical processes involved in the conventional hydrologic models has triggered the increasing use of ANN models (Rezaeianzadeh  ), widely adopted in a broad spectrum of disciplines.
LM can be thought of as a combination of steepest descent and the Gauss-Newton method.
As noted above, the LM algorithm is a variant of the Gauss-Newton method and was designed to approach second-order training speed without having to compute the Hessian matrix (Hagan & Menhaj ). Typically, for the learning of feed-forward neural networks, a sum of squares is used as the performance function. (2) for downscaling from coarse climate model scales to finer observed scales. In this study, quantile is applied as the bias correction step of a larger downscaling framework.
The QM for precipitation preserves model-projected relative changes in quantiles, while at the same time correcting systematic biases in quantiles of a modeled series with respect to observed values.

Quantile mapping
Quantile mapping equates cumulative distribution functions (CDFs) F o,h and F m,h of, respectively, observed data x o,h , denoted by the subscript o, and modeled data x m,h , denoted by the subscript m, in a historical period, denoted by the subscript h. This leads to the following transfer function: for bias correction of where x m,h and x m,p (t) are, respectively, estimates of the long-term modeled mean over the historical period and at time t in the projected period p.

Bargaining game
There is a high risk of water conflicts in the allocation of The only solution that satisfies the following maximization condition is the Nash bargaining solution:

Bargaining power calculation
Each player's weight will be obtained by the AHP method, • Defining the unstructured problem and stating the objectives and outcomes clearly.
• Decomposing the complex problem into a hierarchical structure with decision elements (criteria and alternatives).
• Employing pairwise comparisons among decision elements and forming comparison matrices.
• Using the eigenvalue method to estimate the relative weights of decision elements.
• Checking the consistency property of matrices to ensure that the judgments of decision-makers are consistent.
• Aggregating the relative weights of decision elements to obtain an overall rating for the alternatives.
The weights gained by AHP will be implemented as subjective weights in the entropy method. This measure of uncertainty is given by Shannon & Weaver () as: The entropy of the set of project outcomes of attribute j is in which, E j is the entropy of attribute j, m is the number of alternatives, P i is the probability of the i-th alternative that is preferred by the decision-maker. Where k is a constant defined as it guarantees that 0 E j 1.
The degree of diversification of information provided by the outcomes of attribute j can be defined as: then the weights of attributes can be obtained by Then the mean value of weights that is the outcome from each matrix is considered as the weights of each attribute.
For a better understanding of this research's steps and progress, a flowchart is presented in the Appendix, Figure A1.

CASE STUDY
In this study, the drainage basin of Zarinehrood was chosen to evaluate the function of selective methods. The drainage basin of Zarinehrood, with an area of 1,100 square kilometers, is the largest sub-basin of Urmia which is located in the north-west of Iran, and a valuable water resource that supplies water needs such as drinking, industrial, agricultural, and environmental, and makes 40% inflow to Urmia Lake. and future data from 2018 to 2050 predicted in the following steps. Because raw data reduces the speed and accuracy of PNN, at first, inputs were standardized by the following formula: where X n is standardized parameter, and i, min and max, respectively, are row, minimum, and maximum of the parameter in the series.
After downscaling, a QM method for bias correction was used. In this step, difference in quantiles of observation and simulated historical data in the form of polynomial function was applied on downscaled RCP data to minimize the overall errors. In order to distribute the individual station's parameters to the whole basin, the Thiesson polygon method was used; in this method every station is in the middle of a polygon that overall covered the basin, and parameters were calculated by the following formula where A 1 , A 2 , …, A n are polygon area and P 1 , P 2 , …, P n are parameters related to the central station.
Then, a rainfall-runoff model based on SCS method and using precipitation of climate change scenarios was developed. Next, using time series and historical data an ARMA(p,q) model for forecasting evaporation was created.
Then, considering some assumptions for predicting future demands and using SOP (standard operating policy) method for reservoirs in the area, a water resources management program was developed to assess the water available for allocating to users. The water allocation among users is based on meeting the drinking, environmental, and industrial demand and the remainder for agriculture demand.
For allocation of available water among users, game theory concepts regarding consideration of their interactions is used. The set of players for the game consisted of West Azarbaijan, East Azarbaijan, Kordestan and because of the sensitive situation of Urmia Lake it has been considered as the fourth player of the game. A bargaining game with the NBS in two ways, one symmetric and two using the AHP method, was created. In this game, optimization game terms are as follows: s.t: where WA, EA, KRD, and URL indices are, respectively, West Azarbaijan, East Azarbaijan, and Kordestan as players, f is utility function, w is weight of each player, R t is available water in each period of time, D is demand of each player, D min and D dr are minimum and drinking demand.
The bargaining weights of players were determined by AHP method and Shannon entropy. First, considering each demand of every player, an effective factor was calculated for each of the demands. Then, according to the effective factors, an overall importance weight for each player was determined and used to apply to NBS optimization.

RESULTS AND DISCUSSION
In this paper, a water allocation considering the impacts of climate change was studied.
Based on the Fifth IPCC report, an ANN model for downscaling and a QM model for bias correction impacts of climate change on Zarinehrood basin for precipitation, minimum and maximum temperature in a period of 33 years, were predicted. An output example of the process of QM correction, presented in Figure 2, shows that the accuracy percentage trend of simulated data gets closer to observation data. As for downscaling, three evaluation methods, agreement index (AI), root mean square error (RMSE), and Pearson correlation coefficient (r) were used, with results for five synoptic stations presented in Table 1. Correlation between observation and simulated data is between 0.6 and 0.8, which shows that in this interval the algorithm simulated data in a linear way and the reset of simulation is non-linear. The AI index is approximately near 1 which shows an acceptable connection between sets of data. Using symmetric NBS and asymmetric NBS based on the AHP method, the results of players' utility function are represented in Table 4.

CONCLUSION
In this study, water allocation under the effects of climate change based on game theory was implemented for the Zarinehrood River basin located in the north-west part of Iran. The main objective is to work and evaluate selected methods for achieving this goal.
For the first phase of the study, which is assessing climate change impacts, models were developed on the Considering the complexity and systemics of water resources allocation to establish a bargaining power     (Svejnar ). Hence, this work has made efforts to find water allocation methods that are fair, efficient, and sustainable. When the minimum survival water demand is considered, the disagreement points are more reasonable than when the minimum survival water demand is not considered. This method could avoid the unreasonable phenomenon in which there are disagreement points below the minimum water supply, or zero. The proposed disagreement points can guarantee basic water demands are met. In the process of water resources allocation, calculation of the bargaining weights using the AHP method can result in an efficient, equitable, and sustainable benefit among stakeholders, which could be more in line with actual water resources allocation. The results can be utilized as a basis for supporting decision-makers of a river basin to resolve social conflicts.
Regarding Zarinehrood basin, results show an increase in peak daily rainfall but reduction in overall precipitation which follows water deficiency. Also, an increase in temperature results in evaporation growth and this too follows with a reduction in available water. Allocation using bargaining power driven by the AHP method showed an equity in participants' utility which follows satisfaction of parties and a more stable union with a minimum chance of leaving the agreement.
In general, considering climate change's three parameters, the demands of each player, each demand's priority, and the lowest point of demand, and also taking into account Urmia Lake as a player due to its critical situation, this water allocation prediction was good and fair.
For improvement of this study, using a different rainfallrunoff model, multi-model climate change methods, and another assumption on demands are recommended.