Abstract

Water scarcity is a critical global issue, specifically in regions with complex water dependency dynamics. This study analyzes the efficiency of two models, the Bankruptcy Theory and the Nash Bargaining Solution (NBS), for equitable water distribution in Anbar Province, Iraq, from 2019 to 2040. The research compares these models under varying water availability conditions to address water distribution across agricultural, industrial, and domestic sectors. The Bankruptcy model allocated a uniform water supply across the sectors, whereas the NBS prioritized agriculture's domain demand. Over time, both models showed a declining trend in fulfillment rates for all sectors due to an increasing water shortage. The study employed the Mean Absolute Error (MAE) and Reliability Index (RI) to determine stability, with values of 0.037 and 0.076 for the Bankruptcy model, and 0.035 and 0.073 for NBS, respectively, highlighting a slightly higher reliability for the NBS. This research signifies the necessity for strategic management that integrates forecasting and stakeholder participation to execute balanced and sustainable water distribution.

1 INTRODUCTION

Water is a crucial resource worldwide that has become complicated to address because of growing population, industrialization, and agriculture. Effective water management requires equitable and sustainable distribution, which addresses both immediate and future demands. This issue is more severe in the Arab countries as water scarcity increases with rising water demands (Alodah 2023; Hussein and Al-Ajarma 2023; Yan et al. 2018). Numerous researchers have utilized models to address these challenges by improving water allocation. Feng (2021) proposed a dynamic model that impartially allocated water to have maximum economic benefits and satisfy the needs of different sectors.

Iraq is experiencing increasing water challenges. Internal management issues and dam projects in Turkey have caused a reduction in the annual water flow of the Euphrates River to downstream countries (Al-Ansari et al. 2018a). Al-Anbar province receives only 17% of the total annual water diverted at the Hasiba station (Hamad 2015). Recent WEAP software projections showed a gradual decrease in the Euphrates River's water from 2,333 MCM in 2020 to 1,735 MCM in 2040, even though demand increased from 2,727 MCM to 4,135 MCM (Al-Fahdawi and Satam 2022). These trends illustrate the gap in effective resource allocation across agricultural, industrial, and domestic sectors highlighting traditional water usage practices and the increasing population. The growing water scarcity threatens the region's future and stresses on groundwater because of no strategic management and updated dam infrastructure (Ejaz et al. 2024). Effective water allocation techniques are mandatory to address these issues and secure a sustainable future for Al-Anbar province.

The challenges of water scarcity and floods are being managed with various dynamic models for water resource allocation (Mianabadi et al. 2014; Mahmood et al. 2022). These models range from simulation and optimization strategies to address intricate schemes like water rights and game theory. A one-to-many Rubinstein bargaining water resources allocation conceptual model (OMRBCM) has been proposed recently to resolve transboundary conflicts, implementing equity and efficiency rules (Far and Ashofteh 2024). The game theory conflict resolution model was proposed by Wolf (1999) and applied to resolve water-sharing conflicts (Fang et al. 1993). The game-theory methodologies, including NBS, SL-Nash Bargaining, two-person cooperative game, Bankruptcy, and ordinal games, have attained significance by introducing solutions to potential conflicts and accommodating stakeholders’ interests, thus providing an optimum strategic plan. (Forgó and Fülöp 2008; Nash 1953; Gudmundsson et al. 2018; Safari et al. 2014; Varouchakis et al. 2018; Nafarzadegan et al. 2018).

Wickramage et al. (2020) defined the equitable allocation of water from the Missouri River, which is the longest river in North America, flowing through seven U.S. states including Montana, North Dakota, South Dakota, Nebraska, Iowa, Kansas, and Missouri. The river plays a pivotal role in supplying water for agricultural, industrial, and domestic needs across this region. Li and Ju (2023) applied three Bankruptcy rules to proffer an efficient and balanced resource allocation strategy. The optimization models of Bankruptcy, proportional, adjusted rule, and constrained equal loss were utilized for the water distribution of the Hirmand River in Iran. (Shahraki et al. 2024). The Nash Bargaining model was employed in the water distribution and management of the Mahabad Dam in West Azerbaijan, Iran (Sharifazari et al. 2021). It concludes that game-theoretical techniques can balance conflicts among stakeholders and proffer strategic and fair water distribution solutions. Shahraki et al. (2024) and Babamiri and Dinpashoh (2024) implemented the Bankruptcy theory for water distribution in an arid environment, focusing on resolving water scarcity issues. Qin et al. (2019) and Khalid et al. (2024) integrated these models in transboundary river basins, exemplifying their applicability in managing water demands under scarcity.

This study focussed on the Euphrates River Basin (ERB), a transboundary water resource shared by three neighboring countries. Turkey is dominant in its ERB share and its negotiations because of its political influence, military strength, and strong economy. The other downstream countries, Syria and Iraq, received comparatively less water share and bargaining power due to lack of political stability and limited military power. Metrics for measuring water satisfaction show a notable difference: 96.30% of Turkey’s water demand is met, 84.23% in Syria, and only 40.88% of water needs are fulfilled in Iraq (Qin et al. 2019).

Although several studies (Li and Ju 2023; Shahraki et al. 2024) have employed the Bankruptcy or Nash Bargaining models in specific regions, this study put forward the comparative analyses of these models in semi-arid conditions, providing a more equitable and sustainable resource distribution strategy under conditions of extreme scarcity. Traditional models, like optimization techniques and dynamic models, primarily focus on maximizing economic efficiency without adequately addressing the equivalent in resource distribution, particularly in the context of severe scarcity. The aim of this study is the strategic allocation of water in Iraq’s Anbar province among the agricultural, industrial, and domestic sectors, considering their varying demands and growth patterns. The fair and effective allocation of fluctuating water demands led to the selection and employment of the Nash Bargaining and the Bankruptcy theory. Nash Bargaining ensures equitable allocation by considering the claims and demands among stakeholders, and the Bankruptcy theory provides a framework for distributing limited resources when total demand exceeds available supply. The innovation of this study lies in its comparative application of these two models. By integrating Nash Bargaining and the Bankruptcy theory, this research develops a water allocation strategy that resolves conflicts and promotes long-term sustainability. Using these models, the main aim is to propose a more sophisticated and efficient water allocation system that addresses management and sharing conflicts, both current and future challenges in resource distribution. These models are implemented in Anbar province as they provide a balanced approach that accomplishes each sector's claims and can be employed in complex systems. Importantly, these techniques are based on the claims or demands of each sector involved.

Although existing models primarily focus on maximizing economic efficiency, they fail to address equitable resource distribution in severe scarcity contexts. The innovation of the study lies in comparing these two models to achieve a fair and efficient resource allocation framework, addressing the ongoing and future water management issues of the region. This model could serve as a blueprint for similar regions facing water distribution challenges. The research emphasizes the importance of strategic water resource management to maintain future water supply and mitigate the risk of floods. The preference for these strategies necessitates monitoring in response to environmental changes, water availability, and actual usage. Furthermore, Iraq's development is greatly influenced by unresolved issues regarding the quality and quantity of water (Sharifazari et al. 2021; Qin et al. 2019). Water resource management authorities and local government organizations are responsible for modifying and adapting water management strategies into practice. This study emphasizes that unresolved water disputes can lead to political and economic tensions in Iraq, affecting the nation’s development. Employing the Bankruptcy and Nash Bargaining models to distribute water in water-deficit regions is recommended. 

2 Study Area

Al-Anbar Province, commonly referred to as Dulaim or Ramadi, is the largest province in Iraq, encompassing nearly one-third of the nation's total land area, at approximately 137,808 km² (Muhie Hussein 2010). Geographically, it is situated on the western side of Iraq, sharing borders with Syria, Jordan, and Saudi Arabia (Figure 1). Despite its vast land expanse, 90% of which lies in the western desert, the province is home to a population spread across approximately 1.66 million dunams (1660 km²) of land. (Hamad 2015; Abdullah and Al-Ansari 2021).

Figure 1 Map of study area with land use/land cover classifications.

The province's coordinates stretch between 34° 24' 54" to 34° 11' 54" latitude from the north and 40° 28' 12" to 41° 25'48" longitude (Sulaiman et al. 2021; Noon et al. 2021; Pena et al. 2016). Central to the province's geography is the Euphrates River, which makes its entry into Iraq from the western side, specifically at Husaybah in Al-Anbar province. This river plays a pivotal role in the province's drinking, agricultural, and industrial needs. However, Al-Anbar faces significant environmental challenges. Despite its size and resource-rich rivers, it grapples with severe desertification. This has recently emerged as a pressing threat to the province (Mahmood 2021).

The larger geographical context features the Tigris-Euphrates River system, which encompasses a drainage area of 235,000 km², making it the most prominent drainage basin in Southwest Asia (Yaseen et al. 2018; Al-Ansari et al. 2018b; Abduljaleel 2020). Originating from the eastern Turkish highlands, which rise to altitudes of 4,500 m near Lake Van and the Black Sea, the Euphrates River flows 2,700 km before draining into the Arabian Gulf (Issa et al. 2014). Interestingly, 35% of its pathway lies within Iraq, 25% in Syria, and the majority (40%) within Turkey's boundaries. Specific to Al-Anbar, around 450 km or 43% of the river's total length in Iraq, flows within its territories (Sulaiman et al. 2019). Al-Anbar province's economy mainly depends on agriculture, especially near the Euphrates River, where major crops like barley and wheat are cultivated. Although the industrial sector is not a major contributor to the economy of the state, industries like cement and phosphate production still depend on river water. The population primarily relies on the river water for domestic usage. Economic stability in the region is closely tied to effective water management and addressing environmental challenges.

This river system's significance extends beyond its hydrological role. It covers five major countries: Iraq, Turkey, Iran, Syria, and Saudi Arabia, with the southern region known as Mesopotamia. Historically, its shared waters have necessitated international collaboration, leading to bilateral agreements, such as those between Syria-Turkey and Syria-Iraq, aiming for equitable water division. Al-Anbar's geographical significance, enhanced by the Euphrates River's presence, underlines its vital role in Iraq's landscape. Yet the challenges of desertification and shared water resources highlight the delicate balance of sustaining its environment and population.

3 Methodology

This study utilized the Bankruptcy and Nash Bargaining methods to navigate conflict resolution and achieve optimal water allocation among various users and reservoir operators. While several computational techniques have applied Nash equilibrium in non-cooperative games (Ahmadi and Moreno 2013), the Bankruptcy model is one cooperative game theory tool that examines how each agent's claim to a resource can be addressed when the resource is insufficient to meet all claims. Specifically, this research integrates the principles of Nash Bargaining and the Bankruptcy theory, which are often used together in economic scenarios where demand exceeds available resources. These integrated approaches have previously been applied to natural gas or resource distribution (Janjua et al. 2022; Wickramage et al. 2020) and have proven effective for addressing transboundary or intra-state water challenges (Muratoglu et al. 2022; Janjua and Hassan 2020; Sharifazari et al. 2021).

The methodology of this study focused on the equitable distribution of water among three principal sectors: industry, irrigation, and domestic. In this study, the terms agents, stakeholders, players, and sectors refer to the same water-claiming entities involved in the allocation process. Central to water distribution theory is the identification of a decision point that maximizes the mutual benefit from a predetermined disagreement point. In this study, the objective function is based on the Bankruptcy and Nash product, derived by multiplying the differences between the utility function and the disagreement point across all sectors. This objective was carefully designed to ensure fair freshwater allocation in Anbar province, recognizing agriculture, industry, and domestic sectors as critical stakeholders. Furthermore, the study incorporates a model that defines precise allocation points during disagreements, thereby strengthening its potential to deliver both equitable and enforceable water allocations within Anbar province.

3.1 Bankruptcy theory

A river bankruptcy problem is a specific type of distribution issue where scarce water resources need to be allocated among competing agents, but the available supply is insufficient to meet all their claims. A river bankruptcy problem arises when total available water E is insufficient to satisfy the claims of n sectors. Each sector i (i = 1,2,…n) has a claim (c_i) and receives an allocation (x_i). In this study, the agriculture, industrial, and domestic sectors are modeled as decision-making agents (or stakeholders) that claim a share of the available water resource. Each agent's allocated water amount is denoted as x_i. This problem is relevant when the demands of stakeholders exceed the available resources. For effective resource distribution, the following three conditions must be met:

• Pareto Efficiency: The sum of all allocated resources should equal the total available water ().
• Demand Limitation: The allocated resources should not exceed the claims of the agents.
• Non-Negative Allocation: Each allocation must be non-negative ().

According to Ansink and Weikard (2012), bankruptcy allocation rules are straightforward for water distribution and are easy for planners to understand and implement.

Proportional rule

The Proportional Rule (PRO) ensures that each agent receives a proportional share of its claim. According to Madani et al. (2014), this percentage is determined by dividing the total available resources (E) by the sum of all agents' claims (Khalid et al. 2024). The calculated proportion is then multiplied by each agent's claim (c_i) to determine their specific allotment (). Equation 1 outlines the PRO allocation rule.

(1)

Where:

β	=	,
E	=	total available resources,
C	=	sum of all agents' claims,
xiPRO	=	amount of water allocated to the agent i according to the PRO rule, and
ci	=	claim made by an agent i (Li and Ju 2023).

By using this rule, each agent receives an equal share of the available resources based on the proportion of their claim.

3.2 Nash Bargaining theory

The Nash Bargaining theory, introduced by John Nash in the 1950s, provides a unique and equitable solution to complex resource distribution challenges (Nash Jr 1950). It is praised for its principles of feasibility and Pareto optimality, making it a preferred method for allocating shared resources fairly. Numerous studies, including those by Beygi et al. (2014), Nehra and Caplan (2022), Kampas and White (2003), and Adler et al. (2009), have demonstrated the theory's wide-ranging applicability. A notable application is the Indus River water distribution among Pakistan's provinces (Janjua et al. 2020).

This study aims to implement the Nash Bargaining theory from a theoretical concept to a practical tool for sustainable resource management. It builds on previous research (Sulaiman et al. 2021; Noon et al. 2021; Al-Fahdawi and Satam 2022) to analyze current and future water demands in Anbar province and propose a balanced water distribution model. By offering a robust framework for resolving global water-sharing disputes, this study showcases the Nash Bargaining theory as an effective instrument for managing scarce resources (Figure 2).

Figure 2 Steps involved in the Nash Bargaining solution and Bankruptcy theory for water allocation.

The Nash Bargaining solution offers a structured framework for distributing a limited resource, such as water, among multiple stakeholders or sectors. In the scenario under discussion, the key components of the Nash Bargaining equation are represented by the variables n, c, x, and E. Here, n indicates the total number of sectors or stakeholders involved, indexed by i =1,2,…n. Each sector has a specific quantity of water it claims, represented by c_i and the amount of water each sector receives is denoted by x_i. Meanwhile, E encapsulates the total amount of water available for distribution among all sectors.

For more accuracy, boundedness of claims, and rationality, the vector x = x₁, … x_n is constituted of real values when delineating the distribution of the total water E among the n sectors. This model ensures that allocations are both equitable and self-enforceable within a defined area. The theory further incorporates the Nash Bargaining weights and the allocation points in case of disagreements,

Equation 2 establishes the minimum water allocation for each sector. In this equation, λ_i stands for the threshold water requirement for each sector. Interestingly, sectors like industry, domestic, and agriculture have individual minimum water requirements, represented as I₁, I₂, … I_n. Researchers often consider this minimal allocation as the fundamental water right for each sector. It's the bare minimum quantity of water that a sector is prepared to accept.

(2)

However, it's crucial to note that any claim exceeding the available water resources is deemed irrational. Consequently, after the initial allocation of the minimum required water to each sector, the total water distributed should always remain within the bounds of the available resource, as depicted by the constraint (Equation 3):

(3)

Diving deeper into the theoretical realm, the Nash Bargaining theory delineates both the upper core and the lower core of the problem in the optimization approach. Harsanyi (1963) generalized the Nash Bargaining theory for equitable resource distribution, be it natural gas or water. This represents the negotiation's essence, where each sector's allocated water minus other sectors' claims, adjusted by the sector's bargaining weight, determines the outcome.

Weight Constraints: The sum of all bargaining weights should be unity (Equation 4):

(4)

Allocation Bounds: Every sector's allocation should be between its minimum requirement and its claim (Equation 5):

(5)

Total Allocation Limit: Total distributed water shouldn't surpass available resources (Equation 6):

(6)

Turning our attention to the specific context of the Euphrates River in Anbar province, the water allocation challenge among the three predominant sectors—industry, irrigation, and domestic—can be formulated using the Nash Bargaining model. In this context, the claims of each sector are delineated as the upper core, while the disagreement points are defined as the lower core bounds. The water allocation problem is formulated as the maximization of the weighted Nash product, denoted by N, which represents the combined benefit of all sectors relative to their disagreement points. The solution of Equation 7 provides the optimal water allocations for the domestic, industrial, and agricultural sectors under feasibility and individual rationality constraints.

(7)

In Equation 7, variables such as x_p, x_I, and x_A represent the water allocated to the population, industry, and agriculture sectors, respectively, with their sum equal to the total available water (Janjua et al. 2020). Similarly, the corresponding water claims of these sectors are represented by c_P, c_I, and c_A.

The model is, however, constrained by certain limitations. Each sector's allocated water share should be at least its minimum requirement (lower core) but should not surpass its claim (upper core). Moreover, the cumulative water allocated across all sectors should not exceed the available resources (Janjua et al. 2022). By considering these constraints, the Euphrates River's water gets equitably distributed among the three sectors in Anbar province through the Nash Bargaining approach. It's commendable how the Nash Bargaining theory, with its intrinsic ability to discern dispute points and adapt allocations based on shifting demand and supply, emerges as a preferred solution to such resource allocation challenges.

Bargaining weights

In Iraq, water distribution among three vital sectors is determined using the Nash Bargaining theory, as illustrated in Equation 7, with the allocated water volumes expressed in million cubic meters (MCM). This method addresses agent disagreements by leveraging Nash Bargaining weights to deliver an optimal resource allocation solution (Chu et al. 2024; Fallahnejad et al. 2024). It is imperative to underscore that each sector possesses distinct water needs. While domestic water needs are driven by community essentials, the water demands of agricultural and industrial sectors echo the country's developmental momentum. Water demand intricacies evolve with time. Each sector's water demand is driven by distinct factors: domestic needs stem from population growth, while agricultural and industrial demands mirror developmental trends. By employing the Nash Bargaining equation, water demands have been discerned, contrasting homogeneous weights against distinct bargaining weights for each sector. A comprehensive account of current and anticipated water availability is encapsulated in Table 1.

Table 1 Future water demand and availability analysis in Anbar Governate.

Year	Domestic (MCM)	Industrial (MCM)	Agriculture (MCM)	Total Demand (MCM)	Available Water (MCM)	Difference (MCM)
2019	253.26	169.60	2304.77	2727.65	2333.66	-393.99
2020	258.4	173.84	2450.08	2882.33	2318.15	-564.18
2025	285.72	196.69	2704.62	3187.04	2172.52	-1014.51
2030	315.94	222.53	2959.16	3497.64	2026.89	-1470.75
2035	349.36	251.78	3213.70	3814.84	1881.26	-1933.58
2040	386.32	284.86	3468.24	4139.43	1735.63	-2403.79

3.3 Mean Absolute Error and reliability evaluation

Integrated water resource management meets the needs of all agents involved. The performance of the developed Nash Bargaining and Bankruptcy theory is applied for water distribution. A reliability index (RI) and mean absolute error (MAE) are often employed to assess the variance in the degree of satisfaction of stakeholders in comparing these solutions. To compare these solutions, the MAE and reliability index are often employed to evaluate the variance in the satisfactory degree of stakeholders.

The MAE indicates the precision and accuracy of the model; the value of MAE determines which technique corresponds effectively to the stakeholder's claims. A MAE value closer to zero is more accurate, while a larger value shows error or variance in approach (Hussain and Khan 2020).

Mean Absolute Error (MAE) value is defined (Alakbar and Burgan 2024) in Equation 8:

(8)

Where:

	=	predicted/allocated water supply (PRO and Nash Bargaining values),
	=	observed/actual water demand, and
T	=	total number of years.

Here, t = 1,2,…T denote the time index (years), where T is the total number of years considered, and let I = 1,2,…n denote the water-using sectors. The traditional water system reliability index measures the degree to which water supply meets user requirements (Zarei et al. 2019). The RI reflects the mean satisfactory degree of each stakeholder's rationality (Yu et al. 2022). A lower RI indicates less variance in supply satisfaction, implying a more reliable system.

Reliability Index (RI) Calculation: The RI for the i^th stakeholder (Yu et al. 2022) is defined in Equation 9 as:

(9)

Where:

	=	Reliability Index for the ith stakeholder,
	=	water supply-to-demand ratio of player i during the tth period, and
	=	mean water supply-to-demand ratio for player i over all periods.

4 Results and Discussion

The analysis for the year 2019, as depicted in the accompanying figure, underscores the distribution patterns of water among the three pivotal sectors in Anbar Province: agricultural, industry, and domestic. The agricultural sector exhibits the most pronounced demand for water (Figure 3). This graph is consistent with global trends, where agriculture typically consumes the major share of available water resources.

Figure 3 Water demand distribution in Anbar Governate, 2019.

Table 1 presented the total water demands with its availability. Moreover, it anticipated water demands for the years 2020, 2025, 2030, 2035, and 2040. The results emphasized the overarching influence of the agricultural sector in Anbar's water demand landscape. The industrial and domestic sectors, while currently consuming less water compared to agriculture, exhibit their unique demands and growth trajectories. Their specific allocations, both current and forecasted, are pivotal for ensuring balanced regional development and sustainability.

In 2019, the Bankruptcy model facilitated a high-water consumption rate for all sectors. Specifically, agriculture received 85.57% of its required water, industry got 85.55%, and the domestic sector received 85.39%. These allocations were quite close to meeting the total demands in each sector, indicating a relatively abundant water supply at the start of the period (Figure 4).

Figure 4 Future water demand and availability in Anbar Governate.

As time progressed to 2020 and beyond, each sector began experiencing tighter water allocations. By 2025, the agricultural sector's allocation was at 1,843.67 MCM, representing about 80% fulfillment of its demand. Similarly, the industrial and domestic sectors saw slight decreases in fulfillment percentages to levels around 80%, showcasing the theory's application in progressively tightening resource scenarios.

4.1 Bankruptcy theory

The Bankruptcy theory has been employed to analyze the distribution of water resources among the agricultural, industrial, and domestic sectors in Prince Anbar from 2019 to 2040. This method is used to manage the equitable distribution of insufficient resources where demand exceeds supply. In 2019, the Bankruptcy model allocated water quite generously across all sectors. The agricultural sector was allocated 1,971.87 MCM, nearly meeting its demand of 2,304.77 MCM. The industrial sector received 145.18 MCM against a demand of 163.61 MCM, and the domestic sector was allocated 216.68 MCM, closely approaching its demand of 255.26 MCM (Table 2). This initial allocation shows a high fulfillment rate, indicating a relatively balanced approach towards all sectors.

Table 2 Yearly comparison of actual water demand and Bankruptcy allocation across sectors (2019–2040).

Year	Variable (sectors)	Agriculture		Industrial		Domestic
		(MCM)	(%)	(MCM)	(%)	(MCM)	(%)
2019	PRO water allocation	1971.87	85.56	145.11	85.56	216.68	85.56
	Actual water demand	2304.78	100	169.60	100	253.26	100
2020	PRO water allocation	1970.51	80.43	139.82	80.43	207.67	80.43
	Actual water demand	2450.08	100	173.84	100	258.41	100
2025	PRO water allocation	1843.67	68.17	134.08	68.17	194.77	68.17
	Actual water demand	2704.62	100	196.69	100	285.72	100
2030	PRO water allocation	1714.84	57.95	128.96	57.95	183.09	57.95
	Actual water demand	2959.16	100	222.53	100	315.94	100
2035	PRO water allocation	1584.81	49.31	124.16	49.31	172.28	49.31
	Actual water demand	3213.70	100	251.78	100	349.36	100
2040	PRO water allocation	1454.21	41.92	119.44	41.92	161.98	41.92
	Actual water demand	3468.24	100	284.87	100	386.32	100

By 2020, the model continued to allocate water in a manner that allowed most sectors to receive a significant portion of their demands, albeit with a slight reduction compared to 2019. The agricultural sector received 1970.51 MCM, the industrial sector 139.82 MCM, and the domestic sector 134.08 MCM (Table 2). These allocations represent a minor decrease, maintaining the proportional distribution but signaling the beginning of tightening resources.

The trend of restricting water allocations becomes more evident in 2025. The agricultural sector’s allocation reduced slightly to 1,843.67 MCM, the industrial sector to 134.08 MCM, and the domestic sector saw a slight decrease to 194.77 MCM. Although these figures still represent substantial accomplishment of sector demands, they reflect the increasing pressure on water resources. By 2030, the trend of decreasing allocations continued, with agricultural allocation at 1,714.84 MCM, industrial at 128.96 MCM, and domestic at 183.09 MCM. This pattern shows an incremental but consistent reduction in water allocations, aligning with the growing demand and diminishing water supplies. In the Bankruptcy model, the water has been allocated in equal proportions. As the graphs represent the same percentage of water allocation regarding their demand, water is distributed equally among all sectors.

In 2035 and 2040, the water allocations reflect severe constraints. By 2035, the agriculture, industrial, and domestic sectors were allocated 1,584.82 MCM, 124.16 MCM, and 172.24 MCM, respectively. The situation becomes even more critical by 2040, with further reductions to 1,454.26 MCM for agriculture, 119.49 MCM for industry, and 161.98 MCM for domestic use. These allocations significantly drop to around 41.92% of the demands by 2040, underscoring the acute water scarcity faced by all sectors (Figure 5).

Figure 5 Comparison of annual Bankruptcy-based water allocation to actual demand from 2019 to 2040.

The consistent decline in water allocations from 2019 to 2040 highlights the application of the Bankruptcy theory in a practical setting. Li and Ju (2023) proposed a refined approach for solving Bankruptcy problems, which could potentially enhance the efficiency of water distribution in similar scenarios. Their method could provide a strategy for allocating resources, ensuring that even as supplies dwindle, the most critical needs are prioritized efficiently.

4.2 Nash Bargaining solution

Utilizing the Nash Bargaining solution (NBS) framework, the dynamics of water distribution for Anbar Province have been critically analyzed, emphasizing both the anticipated future supply and actual demand. The data indicates a dynamic distribution approach, where the agricultural, industrial, and domestic sectors have varied significantly in the allocation of water resources.

In 2019, the Nash Bargaining solution revealed that the agricultural sector was close to fully satisfied with 92.07% (2,122 MCM) of its total water demand being met. Meanwhile, both the industrial and domestic sectors only received about half of their respective demands (85 MCM, 127 MCM) (Table 3). This trend of a higher allocation towards agriculture while maintaining nearly 50% allocation to the other sectors persisted into 2020, though agricultural fulfillment slightly decreased to approximately 85.79%. By 2025, there was a noticeable decrease in water fulfillment across all sectors with agriculture receiving just 71.4% of its demand. The consistent pattern of providing half of the demanded water to the industrial and domestic sectors continued unchanged, underscoring a potential prioritization policy favoring agricultural needs over others. The year 2030 showed a further decline in agricultural water fulfillment to 59.41%, while the other sectors continued to receive approximately 50% of their water demand.

Table 3 Annual comparison of actual water demand and Nash Bargaining allocation across sectors (2019–2040).

Year	Variable (sectors)	Agriculture		Industrial		Domestic
		(MCM)	(%)	(MCM)	(%)	(MCM)	(%)
2019	NBS water allocation	2122	92.07	85	50.12	127	50.15
	Actual water demand	2304.78	100	169.61	100	253.26	100
2020	NBS water allocation	2102	85.79	87	50.05	129	49.92
	Actual water demand	2450.08	100	173.84	100	258.41	100
2025	NBS water allocation	1931	71.40	98	49.82	143	50.05
	Actual water demand	2704.62	100	196.69	100	285.72	100
2030	NBS water allocation	1758	59.41	111	49.88	158	50.01
	Actual water demand	2959.16	100	222.54	100	315.94	100
2035	NBS water allocation	1607	50.00	126	50.04	149	42.65
	Actual water demand	3213.70	100	251.78	100	349.36	100
2040	NBS water allocation	1400	40.37	142	49.85	193	49.96
	Actual water demand	3468.24	100	284.87	100	386.32	100

In 2035, the pattern shifted significantly with agriculture receiving a fulfillment at 50% of its demand. This return to a lower agricultural allocation might reflect a response to specific economic, environmental, or social needs. However, industrial demands were still confined to 50%, indicating a consistent policy towards this sector despite the decreased allocation to agriculture. By 2040, there was a sudden drop in the percentage of agricultural demand being met, with only 40.37% of the water needs fulfilled (Figure 6). This severe reduction could signal acute water scarcity issues or a strategic reallocation towards more critical or efficient uses. The domestic sector saw a slight increase in allocation by 2040, indicating a possible shift in priorities towards essential human needs. Khalid et al. (2024) analyzed the water availability in the Kabul River Basin until 2040 and presented a Nash Bargaining solution for its distribution among Pakistan and Afghanistan. Sohrabi et al. (2022) utilized an asymmetric model that achieved high fulfillment rates for the industrial and agricultural sectors, with up to 94% and 96% of their demands met, respectively. Additionally, they managed to meet 94.6% of potable water needs. This compares favorably to the outcomes observed in Anbar Province, where the percentages, particularly for agriculture, were much lower by 2040. The contrast in these outcomes underlines the complexity and challenges during water distribution, especially under conditions of varying resource availability and demand pressures.

Figure 6 Annual comparison of NBS-based water allocation and actual demand from 2019 to 2040.

By optimizing the weighted Nash product, various models can yield significantly different results in water allocation strategies (Sharifazari et al. 2021; Khalid et al. 2024). These divergences emphasize the need for a precise selection of bargaining weights and strategies to balance equity, efficiency, and sustainability in water distribution. The NBS under heterogeneous weights like in the Anbar Province analysis, offers a balanced yet realistic approach to water allocation. However, the declining allocation percentages significantly reflect the growing challenges of water scarcity over time. This scenario demands continuous implementation of water management strategies to ensure that the distribution not only addresses each sector's needs but also considers long-term sustainability and resource conservation.

4.3 MAE and reliability analysis of Bankruptcy and Nash Bargaining theory

Bankruptcy and Nash Bargaining models have been applied for equitable water distribution resources in Anbar Province. Both approaches offer unique ways to manage this distribution. To assess and compare the efficiency of these two approaches, the Mean Absolute Error (MAE) and the Reliability Index (RI) were calculated for agriculture, industrial, and domestic sectors using Equations 10 and 11. The MAE assures a better technique for water allocation, while the RI measures the variance in the water supply-to-demand ratio across each agent (Yu et al. 2022; Yaseen et al. 2016). A lower value of RI and MAE indicates better reliability, less variance, and more consistent water supply compared to demand.

Table 4 highlights the MAE values for water allocation using the Bankruptcy and Nash Bargaining techniques across three sectors. In the agriculture sector, the MAE is 0.064 in the Bankruptcy approach, with a slightly lower value of 0.06 in Nash Bargaining. For the industrial sector, the MAE values are very low in both techniques, with 0.005 under Bankruptcy and 0.004 under Nash Bargaining, showcasing minimal allocation errors in this sector. While in the domestic sector, MAE value is 0.008 for the Bankruptcy approach and a slightly improved 0.007 value for Nash Bargaining. The total MAE is 0.077 under the Bankruptcy approach, which reduces to 0.073 with the Nash Bargaining technique. This lower MAE in the Nash Bargaining approach indicates that it provides a slightly better accuracy and balanced water distribution system overall compared to the Bankruptcy method (Hussain and Khan 2020).

Table 4 Comparative MAE analysis for Bankruptcy and Nash Bargaining models.

Demand sectors	MAE (Bankruptcy)	MAE (Nash Bargaining)
Agriculture	0.064	0.060
Industrial	0.005	0.004
Domestic	0.008	0.007
Total MAE	0.077	0.073

The agriculture sector shows an RI value of 0.036 using the Bankruptcy approach and 0.034 with the Nash Bargaining solution (Table 5). Whereas in the industrial sector, the RI values are very low for both approaches. The domestic sector shows an RI value of 0.00075 (Bankruptcy) and 0.0007 (Nash). Concluding the results across all sectors, the total RI for the Bankruptcy solution is 0.037, while for Nash Bargaining, it's 0.035. The lower total RI value in the Nash Bargaining approach suggests a more reliable water distribution system with less variance across all sectors. Overall, the Nash Bargaining solution provides a marginally more reliable water distribution than the Bankruptcy approach in Anbar Province.

Table 5 Comparative RI Analysis for Bankruptcy and Nash Bargaining models.

Demand sectors	RI Index (Bankruptcy)	RI Index (Nash Bargaining)
Agriculture	0.036	0.034
Industrial	0.000002	0.00000127
Domestic	0.00075	0.0007
Total RI	0.037	0.035

5 Policy Implications and Recommendations

The agriculture sector constantly receives higher water share because it is important for food security and the country’s economy, but with decreasing water availability, this prioritization may become unsustainable. However, water use can be decreased by growing crops resistant to drought and implementing more effective watering techniques, such as drip irrigation. Moreover, ensuring transboundary water-sharing treaties with neighbouring countries like Turkey and Syria is crucial for securing a more equivalent water share. Involving local stakeholders, including farmers and industrial users in decision-making, guarantees that the water allocations accurately reflect actual needs. Better forecasting of water demand is also needed for future water resource management, while considering climate change and population growth. Further, to overcome the issue of water shortage in Anbar Province, it is necessary to develop a water infrastructure by proposing reservoirs and adopting water-saving technologies in agriculture and industry. Improving water management policies by incorporating models like game theories can promote equitable allocation. To properly plan for future water demands, public awareness initiatives, incentives for sustainable practices, and improved forecasts should all be promoted.

6 Conclusion

Bankruptcy and Nash Bargaining models were used for water distribution in Anbar Province to propose an effective water distribution solution among the agricultural, domestic, and industrial sectors. The water availability, demand, and management have been calculated for 2019–2040. The Bankruptcy model distributed equal water share across all sectors, meeting 85% of the demand in 2019. It allocated 1,454 MCM, 119.4 MCM and 161.9 MCM to the agricultural, industrial, and domestic sectors, which is only 42% of the demand of all sectors by 2040. In 2019, the Nash Bargaining solution indicated that the agricultural sector received 92% of its total water demand. On the other hand, both the industrial and domestic sectors were allocated only 50% of their respective demands. By 2040, the model showed a significant decline in water allocation for agriculture at only 40.37% of demand, with an increase of 50% in water share to the domestic and industrial sectors. In this solution, water allocation is accomplished considering the rise in growth rate or demand rate in each sector. The models utilized in the study are based on predictions of water availability and demand that assume average or static conditions over an extended period (2019–2040). Static models have limited flexibility in addressing real-time changes and cannot consider abrupt changes or gradual shifts in these variables. Adaptive models that can incorporate real-time data on water levels, consumption trends, and shifting industry demands should be investigated in future research. Dynamic simulation models or machine learning algorithms could provide revised projections and more adaptable water allocation strategies. The Reliability Index (RI) for the agricultural sector showed marginal variation between the models. The Nash Bargaining model has the potential to manage the varying conditions of supply and demand more efficiently. The resulting RI value of 0.036 under Bankruptcy and 0.034 for the Nash solution in the agricultural sector indicate a more stable water distribution under the Nash model. To improve the management of water resources in Anbar Province, resilient management strategies should be adopted to reassess water distribution with changing environmental conditions. Immediate steps should include adopting efficient irrigation practices to reduce water consumption in agriculture. Iraq must prioritize negotiations with Turkey and Syria to secure a fair share of the Euphrates River water. Stakeholder involvement along with the latest technology can lead to more resilient, adaptable, and equitable water management plans that cope with varying shifting socioeconomic and environmental circumstances. Aligning water distribution policies with actual demands would require strengthening methodologies and incorporating stakeholders, which will ensure sustainable resource management besides equity. Furthermore, future water management should incorporate stakeholder participation and improved forecasting models to adjust allocation strategies based on evolving demands.

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