An evolutionary bargaining framework for allocating water and reclaimed wastewater in agricultural regions

The drastic increase in water consumption – as a result of rapid population growth, as well as industrial and agricultural developments, along with insufficient water resources of acceptable quality – has made efficient and equitable water allocation a global challenge. A problem of water resource allocation is that it usually involves multiple stakeholders, often with conflicting preferences and utilities. Thus, modeling the negotiation and social choice processes between stakeholders helps to eliminate or reduce the conflicts that arise. Negotiation processes have been modeled in different areas such as the allocation of irrigation water based on regional cooperation (Dinar et al. 1992); water quality management in river-reservoir systems (Kerachian & Karamouz 2006, 2007); surface and ground water quality management in urban areas (Kerachian et al. 2010; Estalaki et al. 2016; Ahmadi et al. 2020); allocating costs of best management practices (BMPs) among landowners (Skardi et al. 2013); conjunctive use of surface and ground water resources (Rafipour-Langeroudi et al. 2014; Parsapour-Moghaddam et al. 2015); resolving conflicts among water users considering environmental issues (Safari et al. 2014), urban runoff quality management (Ghodsi et al. 2016a, 2016b), dealing with conflicts regarding water pollution (Medeiros et al. 2017), allocating water resource over transboundary rivers (Madani et al. 2014; Qin et al. 2019), water and reclaimed wastewater allocation (Mahjouri & Pourmand 2017; Pourmand & Mahjouri 2018; Pourmand et al. 2020), optimizing contamination sensors in water distribution systems (Naserizade et al. 2021) and trading inter-sectoral groundwater (Zolfagharipoor et al. 2021).

Over the past decade, there have been several studies on modeling negotiation among stakeholders in water management problems. Mahjouri & Bizhani-Manzar (2013) simulated the negotiation process among stakeholders in a waste load allocation problem using the concept of Fallback Bargaining. They used Unanimity Fallback Bargaining (UFB) and Fallback Bargaining with Impasse (FBI) to the maximum dissatisfaction of bargainers in the negotiation process and to obtain a compromise set, and compared the results of the two techniques. Degefu et al. (2016) combined the asymmetric Nash bargaining solution concept and bankruptcy theory in a water allocation framework to allocate water in transboundary river basins facing water scarcity. They applied their proposed framework to the Nile river basin as well as a hypothetical transboundary river basin under water scarcity. They compared their allocation outcomes with those obtained from the classical allocation rules of bankruptcy. In a later study, Degefu et al. (2018) proposed a three-stage framework for water and welfare allocation in water-scarce transboundary river basins by combining the bargaining theory with resource allocation and bankruptcy games.

He et al. (2018) proposed a framework for cooperative water and benefit allocation using four solution concepts of crisp and fuzzy games. They applied Fallback bargaining to reach a consensus over the four solution concepts. Nasiri-Gheidari et al. (2018) proposed a deterministic multi-objective bargaining methodology (DMOBM) and a robust multi-objective bargaining methodology (RMOBM), to incorporate the major uncertainties in water allocation problems with the objectives of maximizing benefit and minimizing pollution. They applied their methodology to a case study of large-scale inter-basin water transfer and showed that Nash equilibrium gave a fairly narrow range of solutions. Tarebari et al. (2018) proposed a methodology for resolving conflicts among stakeholders using non-symmetric Nash bargaining, in a multi-objective optimization problem of finding the optimum degree of utility functions considering economic, social, and environmental aspects.

Tian et al. (2019) used Sperner's lemma to resolve the conflicts among objectives and multi-regions in a water resources allocation approach. They first developed a multi-objective optimization model with two objectives: maximizing the economic benefits and minimizing the pollutants to obtain a Pareto front. Then, they used the total amount of water and envy-free constraints to achieve acceptable allocation schemes out of the Pareto front solutions. By applying their proposed methodology to a case study of the Hanjiang river basin in China, they showed that the obtained water allocation scheme was a Nash equilibrium. Xue et al. (2019) proposed a monthly allocation model based on fallback bargaining and the water footprint concept for allocating water resources and pollutant loads. They designed several schemes for initial water allocation based on typical bankruptcy rules and optimized those schemes by applying fallback bargaining considering sustainability.

Li et al. (2020) developed a model based on asymmetric Nash bargaining to reach a feasible and efficient solution to the problem of water allocation in transboundary basins. They investigated the performance of their model by applying it to a problem of water resource allocation in a transboundary river basin between China, Thailand, and Myanmar. Naghdi et al. (2021) integrated a simulation–optimization technique based on the system dynamics and the Nash bargaining theory for optimal allocations of surface and ground water resources to different sectors. Also, Bahrini et al. (2021) used the Nash solution concept along with several definitions of rationality and stability, and obtained the socially optimal solutions based on the outcomes of the graph model. They added a new analytical step to the equilibria in the Graph model and applied rules of social choice including the Borda count, Plurality voting, Median voting, Copeland, Simpson, and Fallback bargaining.

Nehra & Caplan (2022) extended the Nash bargaining method in a framework for interregional water allocation. They applied their framework in the case of the Bear River Development Project (BRDP) in Utah and showed that the Nash bargaining solution is not generally a feasible solution for sharing surplus water produced as a result of implementing the BRDP. Moghaddam et al. (2022) developed a model based on non-cooperative game theory by extending the asymmetric Nash bargaining solution for addressing conflicts and aquifer restoration in arid regions considering sustainability and socioeconomic indicators. Through the results of sensitivity analysis, they showed that by increasing the power of water managers, the strategies for groundwater stress reduction become the equilibrium point.

Based on the review of past studies, evolutionary bargaining methods have not been used in water and wastewater allocation. These algorithms do not necessarily converge to a single solution. Thus, in this paper, by combining evolutionary bargaining and social choice methods, a new framework is presented that can provide a single solution to the bargaining problem. In this paper, a new framework for allocating water and treated wastewater to water demands in agricultural regions is developed by coupling bargaining methods and social choice rules. Based on this framework, two methods of evolutionary bargaining coupled with Borda count (BBC) and evolutionary bargaining coupled with pairwise voting (BPV) are proposed and their results are analyzed. The BBC and BPV are intended to find the best scenario for resolving conflicts among stakeholders in the challenging problem of allocating water and reclaimed wastewater in agricultural regions. After proposing some candidate scenarios, non-dominated scenarios are determined. Then, using a social choice technique, each stakeholder sorts the scenarios and chooses their most desirable scenario out of the non-dominated scenarios, regardless of the utilities of other stakeholders. The scenarios selected by all stakeholders can provide them an estimate of other stakeholders' expected utilities. This enables each stakeholder in the next step of bargaining to suggest a scenario that improves their own utility and gives minimum acceptable utilities to other stakeholders. This evolutionary bargaining process does not necessarily converge to a unique solution. If the bargaining process provides more than one scenario, the most preferred scenario is selected between the remaining scenarios based on the social choice methods. The proposed framework is applied to the Varamin plain, in the south-east of Tehran, the capital city of Iran, and the results obtained from the BBC and BPV models are analyzed. In the next section, details of the methodology are given. Then, the results of applying the proposed framework for allocating groundwater and reclaimed wastewater in the agricultural region of the Varamin plain are provided and discussed.

Important data and information related to the study area need to be collected, which include information regarding the population, water demands, hydro-climatologic data such as temperature, precipitation and evaporation, the number of available water resources and their quality, wastewater collection system and reclaimed wastewater consumption, as well as urban, industrial, and agricultural developing plans. Moreover, information regarding the preferences and priorities to be used to form the utility functions of the main stakeholders is gathered.

For sustainable management of water and reclaimed wastewater, the interactions of water users and stakeholders who are in charge of water supply or conservation need to be modeled. Hence, it is necessary that the utilities of all parties are taken into consideration. Usually, water consuming sectors incline to scenarios that allocate more water to agricultural or industrial demands, while water suppliers or distributors tend to choose scenarios that conserve the quantity and quality of water. This raises conflicts among stakeholders and makes decision-making a challenging task. Here, the utility functions of all stakeholders are derived based on the information collected in the previous step, along with the information gathered from questionnaires and experts' judgment.

In this step, several scenarios of groundwater and reclaimed wastewater management are defined based on the utilities of the key stakeholders and domestic, industrial, and agricultural water demands. These scenarios are especially defined to supply a large portion of water demands from treated wastewater rather than fresh water, and stabilize the groundwater table. In the following section, an algorithm is introduced to eliminate scenarios that are less likely to be chosen, considering the utilities of the stakeholders.

In this step, non-dominated scenarios out of the candidate scenarios of water and reclaimed wastewater allocation are obtained using an optimization model with objective functions based on the stakeholders' utilities. A pairwise comparison of all scenarios is made according to the amount of utilities gained by the stakeholders as a result of each scenario. Based on this, non-dominated scenarios are chosen. For example, comparing given scenarios of 1 and 2, if the values of some of the objective functions improve as a result of implementing the second scenario and the other ones do not worsen, the first scenario is defeated. If using the second scenario, the values of some of the objective functions improve and some others worsen, and neither of these two scenarios is dominated by the other. If there is a third scenario that, compared to the first two scenarios, improves at least one objective function and worsens none of them, the first and second scenarios are defeated by the third scenario. On the other hand, if in this sense, none of the scenarios defeats the first two scenarios, they are called non-dominated. The selected non-dominated scenarios are used in the bargaining process through the following steps.

In this paper, we develop a bargaining approach, in which voting methods are used during the bargaining process. In the bargaining process, first, every stakeholder offers their most desirable scenario, which may only improve their own utilities, without considering the impact of implementing the selected scenario on other stakeholders' utilities. Then, the values of the utilities of stakeholders corresponding to the proposed scenarios of the first step are averaged.

This value is considered the expected value for the utilities (EV). Estimating the EV is done at the end of every step of bargaining. Thus, at the beginning of the next step, stakeholders have an estimation of the minimum acceptable utility. This can be used for choosing a relatively socially acceptable scenario to be offered in the next step. Next, every stakeholder offers a scenario that does not worsen the utility of other stakeholders, based on the average utility (expected value of their utility) obtained in the previous step, while improving one's own utility (compared with the expected value of one's utility provided in the previous step). The mentioned procedure continues until the stakeholders can unanimously agree on one scenario.

In the BBC framework, each stakeholder offers scenarios that can at least provide themselves and other stakeholders with the minimum acceptable utility. These scenarios are entered into the proposed algorithm of the Borda count method. According to the proposed framework, the values of the objective functions of each stakeholder are determined based on the scenario they offered. Then, stakeholders score the scenarios according to the values of their objective functions. The worst-case scenario scores zero points, which gives the stakeholders the lowest level of utility. The second worst-case scenario scores one point, which gives the stakeholders a low level of utility. Therefore, in the end, the stakeholders give points to the best-case scenario (⁠ is the number of scenarios proposed by a stakeholder). Then, the final score given to each proposed scenario of every stakeholder is obtained by calculating the sum of scores that all stakeholders have given to that particular scenario. The scenario which gains the highest score will finally be selected. The explained procedure is done for all stakeholders until the stakeholders agree on one scenario.

In the BPV framework, each stakeholder considers the scenarios that provide themselves and other stakeholders with at least a minimum acceptable utility estimated previously. According to the proposed framework, these scenarios were previously offered and are entered into the algorithm of pairwise voting. In the first step of the BPV, the values of objective functions (stakeholders' utility) assigned to the stakeholders are determined according to the offered scenarios. Then, these scenarios are compared in pairs based on the values of utilities. The pairwise comparison process is done based on the number of stakeholders who agree with each of the compared scenarios. The final score given to each proposed scenario of every stakeholder is obtained by calculating the total score that all stakeholders give to that scenario in the pairwise comparisons of that scenario with the other proposed scenarios of that stakeholder. The scenario with the highest score will be selected as the best scenario. The process is repeated for all stakeholders so that all stakeholders choose their best scenario. Then, the minimum acceptable utility of the stakeholders (the expected value of utilities (EV)) is calculated. In the next step, stakeholders will propose the scenarios based on the value of the EV. This means that they only offer scenarios that at least provide the minimum acceptable utility of all. The explained procedure will continue until the stakeholders agree on one scenario.

The scenario which has the highest value of t is selected.

The soil in the Varamin plain is mostly fertile and is composed of the alluvial deposits of the Jajrud River. This has turned this region into a major agricultural center. The total population in the Varamin plain is approximately 90,000. More than three-quarters of the employment in this region is related to agricultural activities, while industrial activities on the plain are very limited. Annually, 854,766 t of agricultural products are produced in this region. The main cultivated crops are wheat and barley. Nearly 38,000 ha out of 80,000 ha of agricultural land in this plain are under cultivation. The low rate of precipitation on this plain has made it necessary to utilize water from the Jajrud river basin along with pumping wells and qanats for irrigation purposes.

Based on the study of the Varamin region, the main stakeholders include residential, industrial, and agricultural sectors (ASs) (denoted respectively as RS, IS, and AS) as well as Tehran Regional Water Company (TRWC), Tehran Province Water and Wastewater Company (TPWWC), Ministry of Health (MOH), Tehran Municipality (TM), and Governorate (GO). Each of these stakeholders' decisions and actions directly or indirectly impact other stakeholders along with the quantity and quality of water resources.

For selecting a socially optimum scenario, first, the scenarios are ranked by each stakeholder based on their own preferences. Some experts as representatives of these stakeholders are asked to determine their criteria and the importance of each criterion used for ranking the scenarios (which their main features, costs, and benefits have been discussed with them). These criteria can be aggregated using group decision-making techniques such as analytical hierarchical process (AHP) for obtaining the matrices containing the weights of the scenarios. The following section provides information on the key stakeholders and their utility functions.

The term is an index regarding water supply shortage to the AS in year y. ⁠, ⁠, and are, respectively, the amount of water demand of the AS, the average amount of groundwater allocated to the AS, and the average amount of allocated water from surface water resources to the AS in year y.

In Equation (19), and are the amount of water allocated to the RS and the amount of water demand of the RS in year y, respectively. In Equation (20), is an index regarding water supply shortage to the IS in year y. The terms ⁠, ⁠, and are the average amount of surface water allocated to the IS, the average amount of groundwater allocated to the IS, and the amount of water demand of the IS in year y, respectively.

The components of the above equations were described in the previous sections.

The results of applying the proposed bargaining methodology for the allocation of water and reclaimed wastewater as well as aquifer stabilization in the study area are presented in this section. The proposed bargaining algorithm has been coded in the MATLAB environment. First, we assess the effect of executing the 12 introduced scenarios in terms of providing the utilities of stakeholders. Based on Table 2 and experts' judgment, comparisons between the scenarios can be made by the stakeholders. Then, a Pareto-optimal front of non-dominated solutions is obtained. Here, since the number of options and respected objective functions are limited, finding non-dominated scenarios does not require executing an optimization model. Thus, out of the 12 candidate scenarios, scenarios Sc. 1, Sc. 5, Sc. 6, and Sc. 8 are selected as non-dominated ones. These scenarios, while improving the utilities of some stakeholders, do not worsen the utilities of others.

Table 3 shows the preferred scenarios and the values of the stakeholders' utility functions using the BBC. At the end of each step of bargaining, the expected value of the utility of each stakeholder based on the selected scenario in that step is calculated. This value is used as the minimum acceptable utility of each stakeholder in the next step (Table 3). According to the results of the BBC, two different scenarios of Sc. 8 and Sc. 6 are chosen. Since no consensus on a preferred scenario is reached at this step, bargaining enters the next step.

In the BPV, each of the stakeholders selects a scenario that provides its minimum acceptable utility obtained in the previous step of bargaining. The stakeholders rank the scenarios from zero to n − 1, based on how desirable the scenarios may be to other stakeholders. In the end, the total score given to every scenario in question is the criterion used for choosing the final scenario. Based on the results of this step of bargaining, the selected scenarios are similar to the ones selected in the previous step (Sc. 6 and Sc. 8). Since an agreement has not been reached, the social choice method of pairwise voting is used to select a final scenario between Sc. 6 and Sc. 8. Here, Sc. 8 and Sc. 6 comparatively do better than the other scenarios for most of the stakeholders. Thus, the results of the two methods of BPV and BBC are not substantially different. The results of the second approach (BPV) are shown in Table 4. Here, since the outcome of the bargaining process is not a single scenario, the selection between the two remaining scenarios will be finalized in the following step. In the next step, social choice methods are used to converge the results of the bargaining process and select between the remaining scenarios (Sc. 6 and Sc. 8).

Based on Figure 1, the remaining scenarios are compared, using the social choice methods following the bargaining process, to obtain a final solution. To do so, the preference matrix of the scenarios is formed by the stakeholders. This matrix has eight rows and two columns (⁠⁠) since there are two scenarios and eight stakeholders (Table 5).

Therefore, again Sc. 8, which is preferred by five stakeholders compared to Sc. 6, which is preferred by four stakeholders, is selected as the best scenario in bargaining combined with the Borda count method. In this scenario, a great portion of agricultural water demand is supplied from the Varamin channel and the WWTPs in the south. In addition, according to Table 2, in terms of groundwater table drawdown, scenario 8 is one of the best scenarios. In both scenarios, the groundwater discharge from the Varamin aquifer is lower than that of most other scenarios. Also, scenarios 6 and 8 are both the best among all scenarios in terms of impact on land subsidence. Both these scenarios supply more than 75% of residential water demands from surface water resources rather than the Varamin aquifer. All stakeholders, their utility functions, and even purposed scenarios have a realistic basis. In the proposed methodology, the stakeholders bargain over a series of scenarios. These scenarios provide better conditions for all stakeholders than the existing situation. The results of the social choice model show that the stakeholders have the most agreement regarding the choice of scenario 8. Therefore, the condition of individual rationality has been met. Generally, scenario 8 in terms of most of the important criteria, such as nitrate concentration in the groundwater, land subsidence, and supplying water demands perform satisfactorily compared to the other scenarios. It seems that according to the characteristics of the study area and the interests of the stakeholders, the superior scenario (Sc. 8) is reasonable and the stakeholders of the region agree on it. Also, the performance and efficiency of this bargaining model for very dry conditions have been checked and its results seem reasonable. To evaluate the efficiency of the proposed methodology, its results have been compared with those obtained from the model developed by Parsapour-Moghaddam et al. (2015) and the current condition of the system. Comparing the results of the proposed method with the bargaining algorithm presented by Parsapour-Moghaddam et al. (2015) shows that their algorithm does not converge to a single solution, while this limitation has been removed in the method proposed in the present paper. The results also show that the selected scenario outperforms the current water and wastewater allocation policy in the study area. Also, the performance and efficiency of this bargaining model for very dry and wet conditions have been checked and the results seem reasonable.

It should also be mentioned that there is no water market in the study area. Currently, water and wastewater allocation in the study area is done by the Tehran Regional Water Board Company. Therefore, there is no need to establish a new organization or company for applying the results of this study. The proposed scenarios are different only in terms of water allocation to different types of demands from different water resources. Therefore, no institutional change seems necessary.

The process of bargaining among competing stakeholders, on the problem of allocating water and reclaimed wastewater in agricultural regions, was modeled in a new framework that utilized bargaining methods and social choice techniques. To find the best scenario, an evolutionary bargaining method was coupled with Borda count (BBC) and pairwise voting (BPV) in a new framework. In this framework, if more than one scenario resulted, social choice methods were to be applied after the bargaining process, to reach a final solution. To show its applicability and effectiveness, this framework was applied to the case study of Varamin plain in the south-east of Tehran. This region with a high rate of urban and agricultural growth is facing complicated issues in terms of managing quantity and quality of its water resources. Thus, the aim was to find an optimum and socially acceptable scenario of water allocation to supply the water demands of different stakeholders from groundwater and reclaimed wastewater, which could help to stabilize the groundwater table and improve its water quality. The results showed that the proposed framework can be successfully applied for finding a feasible and socially acceptable scenario in allocating water and reclaimed wastewater to stakeholders in agricultural regions. This scenario supplied most of the water demands from reclaimed wastewater rather than freshwater resources. Also, the final scenario involved relatively low groundwater withdrawal by the water users in agricultural and urban sectors, which could lead to a more stabilized groundwater condition, while achieving an acceptable degree of consensus among the stakeholders.

One of the drawbacks of the current research is that it does not consider the impact of climate change on the involved variables, particularly, the available water resources. Considering the uncertainties related to the available water resources as a result of climate change is recommended for extending the current research. Also, to continue this research, the uncertainties related to the preferences of the stakeholders and the ranking of the scenarios can be taken into account. In addition, in future studies, one may obtain real-time water allocation policies by extending the current framework.

The authors declare that this manuscript and the results obtained from this research have not been published or submitted elsewhere and this research is in compliance with ethical standards. Also, the authors declare that all three participants of the work have contributed in preparing the manuscript for submission and their names are mentioned as the authors of the paper.

All authors confirm their co-authorship.

The authors give their permission to publish.

M.H. conceptualized the study, performed methodology, did software analysis, and prepared codes. N.M. supervised the study, performed the methodology, validated the results, reviewed and edited the manuscript. N.F. prepared figures and tables, did some validations, wrote and prepared the manuscript.

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.