RenataRochellyde
Mesquita Cavalcante1✉Emailrenatarochellymc@gmail.com LeandroChavesRêgo2Emailleandro@dema.ufc.br
SamiriaMariaOliveirada
Silva1Emailsamiriamaria@ufc.br 1Department of Hydraulic and Environmental EngineeringFederal University of Ceará (UFC)Pici Campus UFC block 713FortalezaCearáBrazil
2Department of Statistics and Applied MathematicsFederal University of Ceará (UFC)Pici Campus UFC block 910FortalezaCearáBrazil
Renata Rochelly de Mesquita Cavalcante1*, Leandro Chaves Rêgo2 and Samiria Maria Oliveira da Silva3
1* Department of Hydraulic and Environmental Engineering, Federal University of Ceará (UFC), Pici Campus UFC block 713, Fortaleza, Ceará, Brazil.
2 Department of Statistics and Applied Mathematics, Federal University of Ceará (UFC), Pici Campus UFC block 910, Fortaleza, Ceará, Brazil
3 Department of Hydraulic and Environmental Engineering, Federal University of Ceará (UFC), Pici Campus UFC block 713, Fortaleza, Ceará, Brazil.
*Corresponding author(s). E-mail(s): renatarochellymc@gmail.com;
Contributing authors: leandro@dema.ufc.br; samiriamaria@ufc.br;
Renata Rochelly: Writing – original draft; Writing – review & editing; Formal analysis. Leandro Chaves Rêgo: Supervision; Review; Validation. Samiria Maria Oliveira da Silva: Supervision; Conceptualization; Methodology; Investigation; Data curation; Visualization; Project administration.
ABSTRACT
The growing pressure on water resources, intensified by factors such as climate change, increasing agricultural demand, and urban expansion, has amplified the occurrence of conflicts in semi-arid regions. This study analyzes water allocation conflicts in Ceará, Brazil, focusing on the inter-regional transfer between the Middle Jaguaribe River Basin and the Metropolitan Region of Fortaleza. The research employed a three-stage methodological approach: (i) identification and classification of conflicts documented in the minutes of River Basin Committees (2004–2021); (ii) selection of a real conflict for analysis; and (iii) modeling with the Graph Model for Conflict Resolution (GMCR+). A total of 111 conflicts were identified, with water allocation being the most recurrent category. The selected case refers to the Castanhão reservoir, the main source of the transfer. The modeling included three decision makers — irrigators, the Water Resources Management Company (COGERH), and urban users — resulting in 256 possible states, of which 112 were feasible. The analysis revealed one equilibrium state (state 106), robust to variations in preferences, characterized by the prioritization of human supply, the adoption of alternative sources, and the rejection of irregular withdrawals. The results demonstrate that even under structural scarcity, stable solutions can emerge through adaptive and participatory governance frameworks. By integrating a decision-support model into an empirical governance context, this study contributes to a broader understanding of how analytical tools like GMCR + can inform conflict resolution and cooperative water management in drought-prone regions globally.
KEYWORDS:
Water Resources
Conflicts
Game Theory
Management
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1. INTRODUCTION
Water within society is revealed to be important both in terms of availability and in relation to its social significance. Regarding the availability of surface freshwater, only about 2.5% of the planet’s total water is available for consumption and use. Furthermore, there is an issue of poor distribution of water resources, where some regions have high population density but low water availability (Ribeiro, 2019).
In addition to these factors, population and economic growth, as well as the intensification of agricultural production, drive an increase in water consumption. The Food and Agriculture Organization of the United Nations – FAO (2022) estimates that global water demand grows annually by about 0.8%. Of this demand, human supply accounts for approximately 12%, industry for 16%, and agriculture for 72%, making it the largest global water consumer.
Parallel to the growing demand for water, climate change has generated uncertainty regarding water availability, resulting in and/or exacerbating water stress in several regions. Water scarcity caused by pressure on natural resources and by the effects of climate change is considered one of the greatest global risks. It is estimated that more than 2 billion people live in regions of high water stress, approximately 1 billion lack access to safe drinking water, and around 3.4 million people die annually from the use/ingestion of contaminated water (Tzanakakis et al., 2020).
The global prevalence of conflicts and crises related to water is already substantial, highlighting the potential of scarce resources as conflict inducers (Homer-Dixon, 1994). In some contexts, water stress may not act as the determining factor in triggering conflicts, but it functions as a catalyst in such situations (Mohammadinezhad & Ahmadvand, 2020; Unfried et al., 2022).
Considering the scarcity scenario, Homer-Dixon (1994) points out that in places where water is a scarce resource, competition for this limited supply can lead to conflicts. This is particularly relevant in arid and semi-arid regions, where high water stress prevails and water is vital for development, making conflicts over access to and control of water more likely to intensify (Gleick, 1993).
Since water is an essential factor in the production of goods and services, and given its variability in stocks and flows within the hydrological cycle, water control and use become important social and political objectives (Gleick & Shimabuku, 2023). In this regard, climatic phenomena such as El Niño, with great intensity and duration, have been identified as possible drivers of increasing conflicts in some parts of the world (Karesdotter et al., 2023).
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Water conflicts can be resolved or mediated through different strategies, ranging from negotiation and mediation processes to legal proceedings (Korbéogo,
2020). However, regardless of the approach used, achieving a successful outcome requires an understanding of the nature of the conflict (UNESCO, 2023).
GMCR + is a decision support system based on the Graph Model for Conflict Resolution (GMCR), designed with a human-centered structure that allows intuitive and interactive manipulation of conflict models. These systems emphasize simplicity and flexibility by requiring relatively few input data, making them an important tool for decision-making support in conflict situations, as well as enabling continuous modifications and assessments. The main difference between GMCR + and earlier models is that this version operates on any platform (MAC, Windows, or Linux) and features a robust modular design that enables error tracking within any module, since the structures are independent.
The graph model simulates real-world problems, presenting complex issues in a simplified manner through a graphical structure. In addition, it allows for the simulation of preference rankings among stakeholders and, through matrix interpretation, enables more efficient stability calculations of conflict scenarios. Another advantage of the tool is the analysis of both individual definitions and coalitions formed by two or more decision makers (Kilgour, 2007; Hipel et al., 2014; Kinsara et al., 2015; Xu et al., 2018; Kilgour et al., 2021).
A
The use of GMCR in water conflict modeling can be observed in Hipel et al. (
2014), Chu et al. (
2015), Philpot et al. (
2016), Zanjanian
et al. (2018), Yang et al. (
2021), Alfjeri
et al. (2019), and Sabino et al. (
2023). In general, the identified studies address real situations of water conflicts involving issues of water allocation, upstream vs. downstream users, and obstruction of water access, such as in the case of the Carás Valley in Northeastern Brazil (Sabino et al.,
2023).
The modeled conflicts include sensitivity analyses to assess model robustness, the use of prioritization methods to rank decision makers’ preferences (Yang et al., 2021), preference valuation approaches to quantify and classify preferences (Zanjanian et al., 2018), and group modeling, adopting Simon’s (1977) intelligence phase in addition to the modeling and analysis phases already used in GMCR.
In this context, the research modeled the real conflict of inter-regional water transfer between the Middle Jaguaribe River Basin and the Metropolitan Region of Fortaleza, located in Northeastern Brazil, using the GMCR + decision support software proposed by Hipel et al. (2014). This conflict was identified through a documentary analysis of the minutes from River Basin Committee meetings and classified according to the typology proposed by Studart et al. (2021). This step is fundamental for the parameterization of the GMCR + model, which requires the structured establishment of individual preferences. The modeling sought to identify strategic equilibrium states among the decision makers involved and to evaluate unilateral transition pathways that could lead to an institutionally viable resolution of the conflict.
This article subsequently presents the study area, the Middle Jaguaribe River Basin and the Metropolitan Region of Fortaleza. The methodological steps focus on identifying and classifying conflicts based on RBC documents, defining the real conflict to be modeled using GMCR+, and analyzing possible scenarios and positions taken by each stakeholder involved. Finally, stability concepts are applied to assess a potential solution to the conflict.
2. INTER-REGIONAL WATER TRANSFER
The inter-regional water transfer takes place between the Middle Jaguaribe River Basin (donor basin) and the Metropolitan Region of Fortaleza (receiving basin).
The Middle Jaguaribe River Basin (BHMJ) is located in the eastern portion of the state of Ceará, and its eastern boundary borders the state of Rio Grande do Norte, in Northeastern Brazil (Santos & Silva, 2016). It covers 7.09% of the state’s territory, comprising 13 municipalities. This sub-basin is part of the Jaguaribe River Hydrographic Region, together with the sub-basins of the Upper Jaguaribe, Lower Jaguaribe, Salgado, and Banabuiú.
The Metropolitan Hydrographic Basins consist of 16 independent basins covering an area of 15,085 km² in the northeastern portion of the state. This area comprises 31 municipalities and includes the largest water-consuming center in Ceará, the Metropolitan Region of Fortaleza (MRF) (Ceará, 2009). This region was the first metropolitan region established in Ceará in 1973, through Complementary Law No. 14 (Brazil, 1973). According to IPECE (2018), the region had a population of 4,074,730 inhabitants, corresponding to 44.9% of Ceará’s population, within a territorial area of 7,440 km². It also concentrated the largest GDP in the state in 2016, with a value of R$ 22,242, reflecting a nominal growth of 33.4% between 2012 and 2016 (IPECE, 2018).
The transfer between these regions is carried out through two hydraulic infrastructures: the Eixão das Águas and the Canal do Trabalhador. The Eixão das Águas extends for 255 km, with a maximum flow capacity of 22 m³/s (Estácio et al., 2022; CID, 2017). The Canal do Trabalhador, built in 1993, was the first hydraulic infrastructure in the state to transfer water through the Jaguaribe River, beginning its diversion in the municipality of Itaiçaba.
The main water source for the transfer is the Castanhão Reservoir. This reservoir accounts for approximately 90.5% of the storage capacity within the Middle Jaguaribe Hydrographic Region, with a total accumulation volume of 6.7 billion m³. Of this total, 2.2 billion m³ are allocated to flood control in the region (Cogerh, 2009; Almeida, 2015).
Short-term water allocation in this system is carried out through a negotiation mechanism known as negotiated water allocation. These meetings take place after the rainy season to establish water release volumes to meet the demands of multiple uses within the basin. They are usually held in July each year, with decision-making carried out by the members of the Jaguaribe River Basin Committees: the Upper Jaguaribe Sub-Basin Committee, the Middle Jaguaribe Sub-Basin Committee, the Lower Jaguaribe Sub-Basin Committee, the Salgado Sub-Basin Committee, and the Banabuiú Sub-Basin Committee. The MRF participates in these meetings; however, it does not hold decision-making power.
3. METHODOLOGY
The research was conducted in three main stages: (i) Identification and Classification of Real Conflicts; (ii) Selection of the Case Study Conflict; and (iii) Conflict Modeling and Analysis.
3.1 Identification and Classification of Real Conflicts
The literature indicates different approaches to identifying and classifying conflicts, depending on the context associated with water resources. Dessler (1994) employs the Actor–Intentional Model of Human Behavior, which uses a systematic classification of the roles that generate and sustain violent conflict: triggers, targets, channels, and catalysts. Through this qualitative typology, the author analyzes how environmental changes contribute to conflict.
In a geopolitical context, Gleick & Heberger (2014 apud Studart et al., 2021) developed a typology with three categories of global conflicts: trigger, weapon, and casualty. Homer-Dixon (1994), in turn, analyzed indicators of scarcity and water stress, considering scarcity as the origin of intergroup conflicts, arising from the interaction between environmental change, population growth, and unequal distribution of resources.
Studart et al. (2021), in contrast, based their typology on the Integrated Plan for the Exploitation of Water Resources of the Northeast (PLIRHINE) and the State Water Resources Plan of Ceará, also known as the “Plano Zero,” both developed in the 1980s. The typology derived from these studies is grounded in water conflicts related to governance.
Accordingly, the following elements are considered in the classification of conflicts: trigger, actors, scale, duration, arena, and location of occurrence. The trigger refers to the causes that may spark the conflict. In Studart et al. (2021), five types of triggers are identified: access to water, water quantity, water quality, water allocation, and water governance. The actors are the parties directly involved in the conflict. The scale corresponds to the geographic area affected by the conflict. Duration is defined as the period from the initial action to the final resolution of the conflict. Finally, the arena refers to the venue through which conflicts are mediated, being classified into two types: administrative and judicial.
3.2 Selection of the Case Study Conflict
In order to apply the typology proposed by Studart et al. (2021) to identify real conflicts, a documentary analysis was conducted using the minutes of meetings from the Middle Jaguaribe River Basin Committee. These documents are part of the public domain archive available at: http://www.csbhmj.com.br/.
By accessing the site, the minutes can be found under the quick access links, in the “Meeting Minutes” tab.
For this purpose, documents from 2004 to 2021 were selected. This time frame was defined because the main reservoir of the Middle Jaguaribe Basin, the Castanhão Dam, began operation in 2004. Its operation marked the onset of conflicts resulting from the inter-regional basin integration.
Subsequently, each recorded conflict in the documents was reviewed and classified. The data were then organized and separated according to the different water systems that make up the basin. This allowed for the identification of the types of conflicts and the monitoring of their profiles over time.
To provide greater clarity on the facts described in the minutes, semi-structured interviews were conducted between August and September 2022 with staff from the Ceará Water Resources Management Company (COGERH), Limoeiro do Norte regional office. These interviews were held remotely via the Microsoft Teams platform. The main objective of the interviews was to clarify information and obtain knowledge about actions that were not explicitly recorded in the CBHMJ meeting minutes.
3.3 Conflict Modeling
The Graph Model for Conflict Resolution (GMCR), developed by Fang et al. (1993), is the original approach that employs game theory to model and analyze conflicts. It considers situations involving two or more decision makers, each with a set of possible actions (options) and different perceptions (preferences) regarding the potential outcomes of the dispute (states). Over time, several decision support systems have been developed to facilitate the application of GMCR to real conflicts, with GMCR + being the most advanced and up-to-date decision support system. GMCR + incorporates technical and computational improvements, offering a more robust interface for the practical application of the methodology in complex conflict analyses (Fang et al., 2003a; Fang et al., 2003b; Kilgour & Hipel, 2005; Kinsara, 2014). GMCR + is widely used to identify equilibrium solutions in conflicts, being particularly useful in scenarios that demand greater computational support (Zhao & Xu, 2019; Shahbaznezhadfard et al., 2020; Dowlatabadi et al., 2020; Yang et al., 2021; Rêgo et al., 2021).
The modeling of the real conflict was carried out using the GMCR + decision support system (Hipel et al., 2014). This system provides a user-friendly implementation of GMCR, which is a graph-based tool employed for the systematic study of conflicts. Based on game theory, GMCR was developed by Fang, Hipel, and Kilgour (1993), and its structure comprises two stages: modeling and analysis. For its operation, the analyst must input initial information regarding the players (or decision makers), their options within the conflict, the states, and the preferences of each player. After this stage, considered the modeling phase, the analysis phase begins, in which the equilibria of each state, the possible solutions to the conflict, and the stability analysis from the perspective of each player are examined. In this way, the negotiator (or analyst) is provided with a deeper understanding of the problem to support strategic decision making (Kilgour, 2007; Aires, 2018).
To structure the conflict in GMCR+, the following steps were carried out:
1.Collection of preliminary information;
2.Identification of decision makers;
3.Identification of each decision maker’s options;
4.Definition of feasible states;
5.Indication of decision makers’ preferences.
The collection of preliminary information was carried out through the review of CBH meeting minutes and interviews with the Regional Management of COGERH in Limoeiro do Norte. In this way, in addition to clarifying certain facts, it was possible to better identify the actors involved in the conflict, who, for the purposes of application in the software, are designated as decision makers (DMs). From this preliminary assessment, it was possible to establish the action options available to each decision maker.
The GMCR consists of a set of directed graphs and a set of preference relations, with one graph and one preference relation for each DM in the conflict. Let N = {1, 2, ... n} denote the set of DMs, and U= {1, 2, ...k} o
is used to model the conflict, identifying how DM
i can transition from one conflict state to another. Commonly, the accessibility of the DMs is described through the list of unilateral moves, such that
contains all states that DM 𝑖 can reach in a single step from state s. The preference relation,
, is an asymmetric binary relation on
U, such that
means that the DM
i prefers state
s to state
q. Finally, another binary relation can be derived, where
means that
. To aggregate accessibility and preference information, the list of unilateral improvements is used
, that is, the set of states preferred over state 𝑠 by DM 𝑖 which can be reached in a single step by DM 𝑖.
In conflicts involving more than two DMs, when one DM makes a move, it may be retaliated against by a sequence of moves from its opponents. In GMCR, it is assumed that this sequence must be legal, meaning that DMs may move more than once, but not consecutively within the sequence. Let
H be any non-empty subset of DMs, called a coalition. Then,
denotes the set of all states that can be reached through a legal sequence of unilateral moves by the DMs in
H, starting from state 𝑠. If all moves in the legal sequence are unilateral improvement moves for the DM making the move, then the set of states that can be reached by the DMs in 𝐻 from 𝑠 is denoted by
.
With the definition of the decision makers and their options, the states are generated. States are vectors formed from the total number of options of the DMs, where k = 2m, with “k” representing the number of states obtained and 𝑚 the number of decision makers’ options. Each state can be identified as an ordered sequence of length 𝑚 composed of “Y” or “N” where “Y” in the 𝑖-th position of the sequence means that option 𝑖 is selected, and “N” otherwise.
Next, situations that cannot occur in practice (infeasible states) and the preferences of each decision maker regarding each feasible state are defined. At this point, the modeling stage is concluded, and the analysis stage begins.
In the analysis stage, the individual stability of each state is verified according to a solution concept and from the perspective of each DM, referred to as the focal DM. When a given state is stable for all decision makers under the same stability concept, it is said to be an equilibrium under that stability notion (Hipel et al., 2020).
Thus, once a state is found to be stable for all DMs, it can be considered a possible resolution to the conflict. Table 1 presents some definitions of stability according to Fang et al. (1989), Kassab et al. (2006)d go & Vieira (2017).
Table 1
Definitions of Stability within GMCR
Stability Concept | Definition |
|---|
Nash Stability | |
General Metarationality (GMR) | |
Symmetric Metarationality (SMR) | |
Sequential Stability (SEQ) | |
Symmetric Sequential Stability (SSEQ) | |
4. RESULTS
4.1Identification and Classification of Real Conflicts
The conflict typology was applied according to the methodology of Studart et al. (2021). It was used to categorize and assess the existing conflicts in the Middle Jaguaribe River Basin (BHMJ). A total of 90 minutes from Committee meetings recorded between 2004 and 2021 were analyzed.
In total, 111 conflicts were identified in the donor basin. These conflicts were evaluated considering the following elements: trigger, actors, scale, duration, arena, and location of occurrence.
In the category of water allocation, 19 conflicts were identified involving the Castanhão, Riacho de Sangue, Figueiredo, Feiticeiro, and Nova Floresta reservoirs, in addition to disputes related to discussions within the scope of the CSBH MJ. Among these conflicts, 7 were of long duration, 7 of moderate duration, and 5 of short duration. The predominant arena was the administrative one, representing 90.47% of the cases.
The allocation conflicts revealed tensions between different uses and inter-uses of water, highlighting situations such as: irrigation vs. irrigation, human supply vs. human supply, human supply vs. irrigation, and supply vs. shrimp farming. In particular, disputes emerged between irrigators in the Jaguaribe Valley and the Federation of Associations of the Jaguaribe-Apodi Irrigation District (FAPIJA), related to water availability and the lack of reduction in irrigated areas. Conflicts were also identified between human water supply in the Middle Jaguaribe and in the Metropolitan Region of Fortaleza (MRF), the latter being an inter-regional conflict already mentioned by Pinheiro (2002).
Regarding disputes over water among irrigation users, interviewees from the COGERH regional office in Limoeiro do Norte highlighted three major irrigation districts supplied by the perennialization of the Castanhão Reservoir: the Mandacaru Project in Jaguaribara; the Tabuleiro de Russas Irrigation District (DISTAR) in Tabuleiro de Russas; and FAPIJA in Limoeiro do Norte. In addition to these districts, there are users along the 159 km perennialized stretch who withdraw water for irrigation.
With respect to different categories of users, the main protests generally involved human water supply, with demands to reduce withdrawals by irrigators and shrimp farmers. This conflict was confirmed during interviews, which highlighted that discussions surrounding shrimp producers and irrigators intensified from 2014 onwards, becoming one of the main points of contention in committee meetings.
Another prominent conflict related to human supply concerns the population residing in municipalities around the Castanhão Reservoir and along the perennialized stretch, as well as the population supplied directly by this reservoir, such as Fortaleza and the Metropolitan Region of Fortaleza (MRF). This conflict had already been mentioned in 1993. During the interview with COGERH in Limoeiro do Norte, it was reported that, in 2016, the Orós Reservoir water system was diverted to support the supply of the MRF, with withdrawals in the municipality of Itaiçaba, complemented by the Eixão das Águas. Only from 2020 onwards did the Pacajus, Pacoti, Riachão, and Gavião reservoirs begin supplying Fortaleza and the cities of the MRF.
Beyond the conflicts documented in the minutes, interviews with the COGERH team also revealed other conflicts not officially recorded. One example was the difficulty in understanding the technical data presented during allocation processes, especially at the beginning of the formation of the Middle Jaguaribe River Basin Committee (CSBHMJ). This situation generated mistrust among committee members. However, with the inclusion of specialists, such as professors from IFCE, university professors, representatives of social groups, and Caritas, discussions improved, which contributed to enriching the debate and, consequently, to pacifying conflicts within the committee.
4.2Selection of the Conflict for Analysis
Among the identified water allocation conflicts, the dispute over the allocation of water from the Castanhão Reservoir stands out. This conflict is triggered by the inter-basin transfer of water from the Middle Jaguaribe Hydrographic Region to the Metropolitan Region of Fortaleza, involving users from the irrigation and water supply sectors, as well as the water resources management authority in Ceará, COGERH. According to documents from the committee, the conflict began in 2015, when, due to the prolonged drought affecting the state since 2012, the reservoir exhibited low water storage levels and restrictions were implemented to ensure human supply and promote water savings in irrigation. These measures included reducing cultivated and irrigated areas along the perennialized stretch of the Jaguaribe Valley.
4.3 Conflict Modeling and Analysis
The conflict typology was useful in identifying the decision makers and their interests. These were grouped into three categories: Irrigation Users, COGERH, and Human Water Supply Users (Table 2).
Table 2
Characteristics and/or Interests of Each Decision Maker
DM | Characteristics and/or interests |
|---|
Irrigation Users (DM1) | Users, either individually or through irrigation districts or associations, who use water from the basin for agricultural production through irrigation. |
COGERH (DM2) | The Water Resources Management Company (Companhia de Gestão dos Recursos Hídricos) is a mixed-capital company created by State Law No. 12.217 of November 18, 1993, with the mission of managing water resources in surface and groundwater bodies under the jurisdiction of the State of Ceará and the Union, by delegation, in an integrated, participatory, and decentralized manner, promoting their rational, social, and sustainable use. This decision maker seeks effective water resources management during the prolonged period of water scarcity in order to ensure the supply of multiple uses in the Metropolitan Region of Fortaleza (an important consumer region of the state). |
Human Water Supply Users (DM3) | Users who, through the Autonomous Water and Sewage Supply Systems or the Ceará Water and Sewage Company (CAGECE), demand water from the Castanhão Reservoir for human supply purposes. |
Based on the available options (Table 3) for the three decision makers (DMs), it was possible to generate the set of possible states of the conflict. Considering the 8 binary options (activated or not) assigned to the DMs, a total of 256 possible states were obtained, resulting from the combination 28 (i.e., 2m with m = 8).
Table 3 presents the status quo of the conflict, that is, the state corresponding to the situation observed in 2015, the initial year of the analyzed period. In this table, the letters "Y" and "N" respectively indicate whether a given option was in effect ("Y") or not ("N") in the status quo state. The table also details the actions attributed to each of the decision makers, providing an overview of the system’s initial configuration.
Table 3
Decision Makers and Their Respective Options
Decision Makers | Option | Description | Status quo |
|---|
DM1 | 1 | Pressure to increase the flow allocated for their use | Y |
2 | Invest in obtaining water from alternative sources | Y |
3 | Illegally extract water for use | Y |
4 | Voluntarily comply with the limit established by the management authority | N |
DM2 | 5 | Carry out water transfer to the MRF | Y |
6 | Restrict water use for flood irrigation and/or drilling wells within 100 m of the perennialized riverbed | Y |
7 | Intensify inspection and monitoring of water use | Y |
DM3 | 8 | Request that their water demand be given priority | Y |
After combining all states, the infeasible states were defined—those alternatives that could not occur in practice. In the case of the conflict under study, an infeasible state could be, for example, simultaneously pressuring to increase the flow allocated for irrigation use (option 1) and voluntarily complying with the limits established by the management authority (option 4). Thus, the infeasible states defined for this application were the following option combinations: 1 and 4; “–6 and 3”; 3 and 4; “4 and − 6.” Once the infeasible states were established, 144 states were removed, leaving a total of 112 feasible states for modeling.
The definition of the order of preference of actions for each DM was carried out based on documentary analysis of the meetings’ minutes and on interviews conducted with representatives of the COGERH regional unit. This step is fundamental for the parameterization of the GMCR + model, which requires the structured establishment of individual preferences.
In GMCR+, preference modeling is expressed through logical connectives that allow for the representation of different forms of strategic prioritization. Among these connectives, the following stand out: the symbol “–” to indicate rejection of a given option; the conditional operator if to express conditional preferences; the biconditional operator iff to express biconditional preferences; and the operator “&” to indicate preference for multiple actions together (for example, A & B).
Based on the information gathered, it was possible to construct the preference statements of each DM, as presented in Table 4. This structuring made it possible to capture the priority strategies of the actors involved in the water allocation conflict, considering both desirable actions and those that decision makers seek to avoid.
Table 4
Preference Statements of Each Decision Maker
DM1 | DM2 | DM3 |
|---|
-5 -7 -6 1 if 5 2 (3 iff − 7) if 5 -4 | 5 -3 2 6 4 7 if 3 | -5 8 if 5 2 -3 7 |
The conflict analysis was conducted through a game in which the DMs acted unilaterally (individual players). Among the 112 feasible states generated in the GMCR + model, only one stood out as an equilibrium according to all the criteria discussed in this study: Nash Stability, General Metarationality (GMR), Symmetric Metarationality (SMR), Sequential Stability (SEQ), and Symmetric Sequential Stability (SSEQ). This state — identified as 106 — constitutes a potential solution in which none of the DMs would have individual incentives to unilaterally change their strategies (Table 5).
Table 5
Equilibrium Analyses through the Application of Individual Stability Concepts
Options | Scenarios |
|---|
Status quo | 106 |
|---|
1 | Y | Y |
2 | Y | Y |
3 | Y | N |
4 | N | N |
5 | Y | Y |
6 | N | Y |
7 | Y | Y |
8 | Y | Y |
The state encompasses a wide range of common decisions, reflecting the complexity of interactions among the three DMs. However, an analysis focused specifically on the actions of DM1 — the irrigation users — reveals a fundamental strategic difference: whether or not to adopt irregular water withdrawal (option 3). Table 6 summarizes the decisions of DM1 in the identified state:
Table 6
Analysis of State 106 for DM1
Option | Action | State 106 | Evaluation for DM1 |
|---|
1 | Pressure for increased flow | Yes | Represents an active stance of demand. It indicates that, despite the restrictions imposed by the scarcity context, irrigators attempt to maintain their water security through political/institutional pressure. |
2 | Invest in alternative sources | Yes | Demonstrates strategic rationality: irrigators recognize the limits of the current system and seek autonomous solutions. This action reduces the risk of exclusive dependence on the centralized allocation system. |
3 | Irregular water withdrawal | No | The explicit refusal of illegality strengthens the legitimacy of the irrigators’ position, suggesting commitment and avoiding penalties and institutional wear. |
4 | Voluntary compliance with restrictions | No | Indicates passive resistance, as irrigators do not accept the limits on their own initiative. This stance creates an impasse but avoids direct rupture with the management institution. |
This analysis shows that State 106 represents an institutionally stable and politically defensible solution for DM1. By rejecting irregular water withdrawal, irrigation users maintain their conduct within the allocation agreement, even if they resist the use restrictions imposed by water management. This stance reinforces their legitimacy as stakeholders in the negotiation process and prevents punitive reactions from the management authority, such as intensified monitoring (option 7).
State 106 is robust from both a strategic and normative perspective, particularly in contexts of prolonged water scarcity, in which compliance with legality and the possibility of institutional negotiation are central elements for the sustainable management of conflicts.
In this regard, the results of this study are consistent with the findings of Rêgo et al. (2021), who applied the GMCR + model to analyze a water conflict involving the Jaguaribe–Apodi Irrigation District. In that case, marked by years of drought and low storage levels in the Castanhão Reservoir, the most stable solution identified combined reduced consumption by irrigators and the adoption of alternative water supply sources—precisely the same strategies observed in State 106 of this study. The convergence of results in distinct contexts, yet subject to the same framework of structural scarcity, confirms the methodological consistency and analytical robustness of the applied model.
Overall, the findings of this work demonstrate that, even in a scenario of intense disputes between productive uses and human consumption, it is possible to identify resolution pathways based on the minimum compatibility among stakeholders’ interests and on strengthening legality and adaptive governance. State 106 therefore emerges as a preferential equilibrium state, both for the strategic stability identified and for the degree of acceptance among decision makers—representing a viable alternative for resolving inter-regional water allocation conflicts in the semi-arid context of Northeastern Brazil.
4.4 Analysis of the Strategic Pathway
Based on the GMCR + model, it is possible to evaluate not only equilibrium states but also the strategic pathways through which the system may evolve from the status quo. This type of analysis is particularly relevant in the context of water allocation in the Brazilian semi-arid region, where unilateral decisions often determine the evolution of conflicts.
In the present study, the status quo (Table 5) represents a state of latent tension, marked by illegal actions and the absence of effective regulatory responses. The transition to equilibrium state 106—considered more stable and institutionally acceptable—can occur through a sequence of unilateral improvement moves, following the logic of Game Theory as applied in GMCR+:
Step 1: Move by DM1
Change: Ceases irregular water withdrawal (Y → N).
Rationale: Pressured by intensified monitoring (option 7 = Y), DM1 may seek to maintain legitimacy and avoid legal sanctions.
Impact: Reduces illegality and opens space for institutional solutions.
Step 2: Move by DM2 (COGERH)
Change: Formally enforces the restriction on water use for irrigation (option 6 = N → Y).
Rationale: By observing that DM1 has ceased acting illegally, the management authority gains legitimacy to enforce rules with greater social adherence.
Impact: Consolidates governance through clear normative measures.
With these two changes (option 3 of DM1 and option 6 of DM2), the system reaches state 106, since the other actions remain unchanged from the status quo. For urban users, no change is necessary, as the prioritization of their demand is guaranteed in all states considered.
This trajectory illustrates that conflict resolution does not always require a simultaneous agreement among decision makers but may instead emerge through gradual and rational adjustments, consistent with the incentives and constraints of each decision maker.
4.5 Sensitivity Analysis
In this section, we investigate how sensitive the stability of state 106 is to changes in the DMs’ preferences. For example, suppose that the original preference statements of the DMs presented in Table 4 is altered and replaced by the configuration shown in Table 7.
Table 7
Preference Statements of Options for Each Decision Maker
DM1 | DM2 | DM3 |
|---|
-5 -7 1 if 5 (3 iff − 7) if 5 -6 2 -4 | 5 -3 2 7 if 3 4 6 | -5 8 if 5 -3 2 7 |
By performing the stability analysis again using GMCR+, it was found that scenario 106 remained the only one to satisfy all the stability criteria presented in this study. Thus, it can be observed that the equilibrium of this state is robust to small variations in the DMs’ preferences.
5. Conclusions
This study identified existing conflicts in the area of the Middle Jaguaribe River Basin, in the eastern portion of the state of Ceará, Brazil, through a documentary review of minutes from the River Basin Committee between 2004 and 2021. In total, 111 conflicts were identified and classified, divided into five categories of triggers and their subtypes: access to water, water quantity, water quality, water allocation, and governance.
Subsequently, in the application of the Graph Model for Conflict Resolution (GMCR+), a water allocation conflict was selected, specifically the transfer of water from the Castanhão Reservoir to the MRF. From this, the actors were identified: water users, irrigators, and the water resources management authority in Ceará, COGERH; along with their available options and preferences. A total of 256 possible states were generated, of which 144 were defined as infeasible states, leaving 112 feasible options. For the conflict analysis, individual games among the different DMs were employed. By applying the stability criteria of Nash, GMR, and SEQ, equilibrium state 106 was identified.
When evaluating each of the two identified scenarios and the preference order of each scenario from the perspective of each DM, it was found that, for the decision makers, state 106 would be the most suitable to be considered as a potential resolution to the conflict. In this scenario, among other options, water use restrictions for irrigation are enforced, monitoring actions are intensified along the perennialized stretch of the Castanhão Reservoir, irrigators refrain from irregular withdrawals, and alternative sources are sought to ensure water supply during this period.
The trajectory analysis reinforces the value of GMCR + not only as a tool for identifying equilibria but also as an instrument for outlining viable pathways for the incremental resolution of complex conflicts in the context of water allocation in semi-arid regions.
Credit authorship contribution statement
Renata Rochelly: Writing – original draft; Writing – review & editing; Formal analysis. Leandro Chaves Rêgo: Supervision; Review; Validation. Samiria Maria Oliveira da Silva: Supervision; Conceptualization; Methodology; Investigation; Data curation; Visualization; Project administration.