Transboundary water allocation in critical scarcity conditions: a stochastic bankruptcy approach

A common problem in water resource allocation is to design a stable and feasible mechanism of water sharing in critical scarcity conditions. The task becomes very challenging when the water demand exceeds the available water resources reserves. To address this pervasive allocation problem related to transboundary rivers, the bankruptcy method is used. The bankruptcy method distributes water among riparian states when their total demand exceeds the total available water. This paper describes a new methodology for the allocation of scarce water resources in a complex system using a stochastic game theory which is an extension of bankruptcy theory. The authors have also proposed ‘weighted bankruptcy’ approach that can be used under a stochastic setting. The weighted bankruptcy approach favors agents with ‘high agricultural productivity’. The bankruptcy rules have been applied in the water resource system in four critical scarcity scenarios. The available water is allocated under the simple and weighted bankruptcy rules. The results showed that under all four scenarios, the weighted bankruptcy rules favor the agents which have a high agricultural productivity. The stochastic bankruptcy approach under the simple and the weighted bankruptcy rules can provide important strategic information for better management and sustainable sharing of water resources. This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY 4.0), which permits copying, adaptation and redistribution, provided the original work is properly cited (http://creativecommons.org/licenses/by/4.0/). doi: 10.2166/aqua.2020.014 om http://iwaponline.com/aqua/article-pdf/69/3/224/823616/jws0690224.pdf er 2021 Shahmir Janjua (corresponding author) Ishtiaq Hassan Department of Civil Engineering, Capital University of Science and Technology, Islamabad, 46000, Pakistan E-mail: shahmir.janjua88@gmail.com


INTRODUCTION
Management and allocation of water in scarce conditions is a common problem in water resource management (Kanakoudis ; Kanakoudis et al. , ). This problem of water allocation can be analyzed using bankruptcy game (BG) technique, which is a branch of cooperative games theory (CGT) (Young ). The problem of bankruptcy arises when some agents have claims on the available goods or assets, but their total claim is greater than the total available assets. The assets must be divided among the claimants or agents in such a way that each claimant or agent might receive a non-negative amount that cannot be greater than its claim. There are numerous Several political disputes have been caused throughout the world due to the non-equitable distribution of water resources, the increasing consumption of resources, and the scarcity of water resources (Homer-Dixon ). There is a total of 148 transboundary river basins that are shared among 148 countries (De Stefano et al. ). During the past 50 years, 43 political or military acts related to shared water resources have taken place around the world (Wolf ). Due to factors such as climate change, growing crop production, increasing populations, and soil degradation, freshwater has become a source of conflict among riparian states. Hence, one of the main challenges in transboundary river management concerns how we can allocate the limited and shared available water among riparian states when it is not sufficient to satisfy the claims of all riparian states. Therefore, 'equitable' and 'reasonable' water resource reallocation faces the question of which criteria and mechanisms should be considered for this 'equitable' and 'reasonable' reallocation (Mianabadi et al. ). A variety of climatic, socioeconomic, environmental, geographical, historical, and political factors and conditions can affect water resources and, consequently, different types of water conflicts (Delli Priscoli & Wolf ). In one category, there are two different approaches to addressing water conflicts: first, conflicts over international water resources (Yoffe et al. ) such as aquifers, lakes, and rivers that are shared between two or more countries; second, conflicts over internal water resources shared between states, provinces, cities, or different groups within a specific country (Toset et al. ). In another category, there are two different types of water conflicts: conflicts over the quality (Perry & Vanderklein ) and conflicts over the availability (Ping & Ping ) of water resources. These conflicts can have different degrees in terms of intensity and seriousness. The complexity of water conflicts calls for accurate investigations for such conflicts to be effectively resolved. Shared water can be a source of both cooperation and conflict among riparians.
The main problem arises when the total demand of riparian countries or provinces is more than the total available water. In quantitative conflict resolution, the equitable allocation of water among riparians is a complex process at both the national and international scales (Jarkeh et al. ).
To manage conflicts and allocate resources, the bankruptcy method is widely used. This method is applicable when the total claims (C) exceed the total resources or assets (E). Bankruptcy theory has been applied to problems related to the allocation of resources. Grundel et al. () used this method for multipurpose resource allocation situations. Ansink & Marchiori () used it for water resource management. In addition, Beard () provided a detailed review of the connection between river sharing and the bankruptcy literature. The frequent application of the bankruptcy method reveals that it is a popular tool for resolving conflicts and for achieving agreement on water resource allocation problems. Bankruptcy theory can be used in resource allocation and dispute resolution when the total available resources are less than the total demand or claims of riparian countries or provinces ( A methodology for water allocation using the bankruptcy games is described in this paper for water resources under scarcity. A new methodology is also developed which considers the priorities in the allocation of water. This novelty is presented by the consideration of the user's priorities, which are settled by 'high agricultural productivity': the user with high agricultural productivity will be given preference in water allocation. First, the water distribution among the provinces of Pakistan is done under four different scenarios using the bankruptcy rules. Second, we apply an allocation procedure that includes claimants' priority by considering the agricultural productivity of the users.

BANKRUPTCY PROBLEM AND COOPERATIVE GAME THEORY
The bankruptcy methods are used in economics when the available asset (resource) is not sufficient to satisfy the claim of creditors (stakeholders). When total available resources are below the aggregate demand, the expected use of each user needs to be reduced by some amount, which can be calculated using different bankruptcy methods Synthetically, to define a cooperative game problem, the following definitions are needed: N ¼ (1, 2, …, n) is the set of players participating in the game S ∈ N is a 'coalition', and for S ¼ N we have the so-called 'grand coalition' v(i) represents the minimum cost or maximum benefit con-  Regarding a benefit-sharing game, to obtain a fair solution, CGT exploits three fundamental principles (in the case of a cost game the inequality signs are the opposite).
The efficiency principle, which guarantees the total sharing of the grand coalition benefit among all the participants of the game: The rationality principle, for which no user (or coalition) can be assigned less than its standalone benefit (i.e., opportunity benefit): The marginality principle, which states no user should be charged more than its marginal benefit from being included in a coalition: The sharing of total available goods is ensured by statement (1); the incentives for the voluntary cooperation is ensured by statement (2) while statement (3) provides the consideration for equity. Conditions (2) and (3) become equivalent as condition (1) is ensured.
Two different definitions are given in CGT for the game problem solution. The first is given by the set of admissible solutions: the so-called 'core' that is the set of all allocations x ∈ R N , such that (1) and (2), or equivalently (3), hold for all S of N (Young & Okada ). The second type is given by a single allocation, which individuates only one solution, and this is more similar to the classic idea of the solution to a problem.

BANKRUPTCY RULES
As described above, the bankruptcy methods are used in economics when the available stock is not sufficient to satisfy the claim of the claimants or creditors. Based on this, the assumption can be made that the total water resources are not sufficient to satisfy the demand of the claimants, therefore, these rules of bankruptcy can be used for The set N of claimants is of the form {1, 2, …, n}. Each claimant i ∈ N advances one claim di on the estate E with A division rule f (E, d) associated with the bankruptcy problem gives a solution as a vector such that: Xi di In Equation (4), x i represents the amount of the estate E assigned to the ith claimant.
The cooperative game, related to the bankruptcy problem, is defined by the characteristic function as given in Equation (5): where v E,D (S) in Equation (5) denotes the minimal amount that the coalition S , N will receive once the claims of the creditors outside S have been fully compensated.
In this situation, the solution to the bankruptcy problem and the related cooperative game is the same. Hereafter, we For a resource allocation problem, we have: where Equations (6) and (7) are the contribution and claims of the agents, respectively. Equation (8) states that the assets are fully allocated. Equation (9) states that the allocation cannot exceed its claims and can never be negative.

Proportional rule (PRO):
The proportional rule (PRO) is given by Equation (10): where C is the total amount of claims and E is the total assets.

Constrained equal award (CEA) rule:
This rule is given by Equation (11): CEA assigns each agent an equal share λ of E, except that no creditor receives more than his or her claim.
3. Constrained equal losses (CEL) rule: As per Equation (12), this rule is defined as: CEL allocates each claimant a share of the asset such that their losses in comparison with their claims (λ) are equal, constrained to no claimant receiving a negative allocation.

The Talmud rule:
The Talmud rule is derived by combining the CEL and CEA rule and is given by Equation (13): 5. Piniles' rule: For each c i , Piniles' rule is calculated as given by Equation In the above bankruptcy rules, the 'estate' amount E to be divided is the available water for users; moreover, different priorities or rights of the claimants are also considered by using the riparians' agricultural productivity.

WEIGHTED BANKRUPTCY RULE: METHODOLOGY DEVELOPMENT
A novel weighted bankruptcy mechanism has also been developed and the above-defined bankruptcy rules have also been applied that include the riparians' priority considering the agricultural productivity of each riparian: higher agricultural productivity produces higher user priority. The method will encourage the riparians to increase their agricultural productivity, which is very essential considering the scarcity of water in future. A simple method to define agricultural productivity is given by crop production per acre feet of water (US$). Therefore, the claimants 'weights' are included, and the BG allocation has been modified. These weights are evaluated considering the agricultural productivity: higher weight w i is given to the riparian with high crop productivity; the weighted demands will be consequently defined as: All the bankruptcy rules defined above will then be applied again considering the weighted water demands. The weighted water requests are considered as the claims of the agents or riparians. If any riparian or claimant receives a greater water allocation than its original request, the assignment will be equal to its original request and available surplus will be shared among the other users using the same rule.

SOLUTION FRAMEWORK
After defining the objective, the fundamental principles of water sharing as stated in Equations (1)

).
Thus, there is no proper mechanism of water distribution when the total volume falls short or when the demands of the provinces exceed the total available water.
Also, Khyber Pakhtunkhwa (KPK) and Baluchistan have not yet developed their irrigation system properly, therefore, they always get more water than they can use.

Another serious problem in the Water Apportionment
Accord is that the water allocations are fixed which creates a quantified entitlement. Fixed water allocations' mechanisms can lead to water allocations which are unacceptable for the provinces, especially in the uncertainty, droughts, and the stochastic nature of river flow. The Water Apportionment Accord between the provinces of Pakistan was signed almost 28 years ago. Since then, the water demands of the provinces have changed due to the increase in population and the irrigated area. Therefore, the gap between the water supply and water demand has increased considerably in Pakistan. Several features and attributes of the Indus River disputes are described in the following section.

Surface water diversions
As shown in Figure 2, the Indus River, which is composed of

Agricultural water requirements for Pakistan
In this study, the water requirements (in feet) for various crops were taken from the Planning Commission Report whereas the cropped area was taken from the Agricultural Statistics of Pakistan.
The total agriculture water requirements or demands for   (), 'the demand of water to meet net crop needs would be 154.5 km 3 by 2020'. The estimates in these two reports suggest that our calculation of the agricultural water requirements of 157.25 km 3 is reliable.

The canal/irrigation water supplies for Indus River
Various researchers have different views regarding the flows of the Indus River and its tributaries. According to

BANKRUPTCY GAMES APPROACH APPLIED IN WATER RESOURCE ALLOCATION IN THE PROVINCES OF PAKISTAN
This paper aims to show the distribution of limited amount of water available to satisfy the riparians or claimants using the BG procedures. The case of Indus River system is considered. Four different scenarios are developed for the water allocation, as shown in Table 3. First the distribution of water for all four scenarios is done using the five bankruptcy rules given earlier, then, a novel allocation procedure is applied which includes claimants' agricultural productivity. More agricultural productivity would result in higher priority in water allocation.
The core solutions of the cooperative game are defined by the principles of efficiency, rationality, and marginality, as defined in Equations (1)-(3). The lower bound for each user is given by the rationality principle and the upper bound is given by the marginality principle. Table 4 summarizes the results, x(i) represents the water allocation for the ith user expressed in Mm 3 /year; the parenthesis value represents the assigned percentage in the four deficit scenarios with respect to the available resource. In Table 4, the upper and lower bounds of water allocation inside the core can be considered as limits of 'feasible values' that could be accepted by each user.  (Table 4) and they belong to the set of Scenario-1 (Canal diversion: 5 in 10) Scenario-2 (Canal diversion: 8 in 10) 0(0%) x(B) 9.42 (7.5%) 0(0%) x(B) 9.42 (7.5%) 0(0%) x(P) 8.28 (6.6%) 0(0%) x(P) 8.28 (6.6%) Scenario-3 (Canal diversion: minimum) Scenario-4 (Canal diversion: 5 in 10) (15% increase in water demands) 0(0%) x(B) 9.42 (7.5%) 0(0%) x(B) 9.42 (7.5%) 0(0%) x(P) 8.28 (6.6%) 0(0%) x(P) 8.28 (6.6%)   Due to the complex nature of transboundary water allocation, we cannot be certain that the simple bankruptcy rules and the weighted bankruptcy rules will be able to solve all the issues related to shared water resource allocation. Water sharing is viewed differently by the people living in different regions, hence their appreciation of the resource and the values attributed to the various functions of the water as a result of cultural, climatic, and economic circumstances.

CONCLUSIONS
The water scarcity issue can be a cause of conflict among riparian countries, states, or provinces. This paper examined the utility of bankruptcy rules in addressing the supplydemand gap in shared rivers. Five bankruptcy rules were used in this study to resolve the conflict between the provinces of Pakistan over the allocation of water. Apart from water scarcity, the uncertain and stochastic nature of rivers and the increasing water demands due to climate change makes the water sharing mechanism more complex and challenging. The water allocation mechanism described in this study uses the bankruptcy game (BG) technique, which is a branch of cooperative game theory. This approach can be a useful tool for decision-making when it comes to the sharing of water resources under critical scarce conditions for complex water resource systems having competing water demands. Using the five bankruptcy rules, the water distribution was done among the four provinces of Pakistan under four different critical scenarios. Also, the bankruptcy rules were applied again, which included the water allocation priorities favoring the users which have a higher agricultural productivity. From the results obtained, it can be seen that the simple bankruptcy rules and weighted bankruptcy rules can be important tools for decision-makers to allocate water in critical scarcity and uncertain conditions between the different water users. Although appropriate vision is provided by the allocation rules for the conflict management of transboundary water resources, the distribution of water among riparians can be a complex problem that cannot be solved only by mathematical methods; therefore, water diplomacy and negotiation between the provinces of Pakistan are suggested, which would help them to develop a consensus and reach an agreement. This method can help policymakers to facilitate negotiation in managing conflict and dispute over water resources allocation problems. It is a tool to create more options that may assist riparian countries when negotiations are tedious. However, some fine-tuning may still be necessary. Further studies may address the limitations of this study and consider some additional influential factors such as the impacts of climate change, reliable relative weights of states, and socio-political aspects of the basin as well as the effects of external powers.