A stepwise conflict analysis using the graph model

Mota, Raí dos Santos; Silva, Maisa Mendonca; Rêgo, Leandro Chaves Article A stepwise conflict analysis using the graph model Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Mota, Raí dos Santos; Silva, Maisa Mendonca; Rêgo, Leandro Chaves (2024) : A stepwise conflict analysis using the graph model, Games, ISSN 2073-4336, MDPI, Basel, Vol. 15, Iss. 6, pp. 1-16, https://doi.org/10.3390/g15060039 This Version is available at: https://hdl.handle.net/10419/330108 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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Games 2024,15, 39. https://doi.org/ 10.3390/g15060039 Academic Editors: Kjell Hausken and Ulrich Berge Received: 15 September 2024 Revised: 24 November 2024 Accepted: 26 November 2024 Published: 27 November 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). games Article A Stepwise Conflict Analysis Using the Graph Model Raí dos Santos Mota 1, Maisa Mendonça Silva 1,* and Leandro Chaves Rêgo 2 1Department of Management Engineering, Universidade Federal de Pernambuco, Recife 50670-901, Brazil; [email protected] 2Department of Statistics and Applied Mathematics, Universidade Federal do Ceara, Fortaleza 60020-181, Brazil; [email protected] *Correspondence: [email protected]g.br Abstract: Information about decision-makers’ preferences is essential for the efficient modeling of a conflict. However, obtaining this information becomes more challenging as the size of the conflict increases. To address this issue, this study proposes a new approach to the option prioritizing method within the graph model for conflict resolution. The approach aims to gather preferences from decisionmakers in a more consistent and practical manner. The proposed method involves partitioning the set of conflict options based on their importance, then applying the option prioritizing method and conflict stability analysis to subconflicts, where only the options in each partition set are considered. Additionally, only states that are equilibria in a given step are deemed feasible in subsequent steps. The main findings highlight a reduction in the cognitive effort required from decision-makers and the generation of more effective and consistent solutions that address the core needs of the conflict. By working with subsets of options incrementally, the method offers a more simplified and robust understanding of the problem. To demonstrate the proposed method, a real hydrological conflict was used as a case study. Keywords: group decisions and negotiations; graph model for conflict resolution; option prioritizing; preference elicitation 1. Introduction Strategic conflicts are increasingly prevalent in society’s daily life. They arise when multiple parties with decision-making authority have differing preferences regarding potential courses of action. These parties, referred to as decision-makers (DMs), can include individuals, companies, groups, or even countries. In scenarios where two or more DMs or groups of DMs hold conflicting decisionmaking powers and interests, a conflict situation arises. To address such situations, conflict modeling has been developed as a mathematical framework to evaluate and analyze the movements and strategies of the DMs involved in the conflict. In this sense, it is worth highlighting some formal methodologies used for modeling and analyzing conflicts, including game theory [ 1 ], metagame analysis [ 2 ], conflict analysis [3] and the graph model for conflict resolution (GMCR) [4]. Among the methodologies mentioned for conflict analysis and resolution, the GMCR stands out for its practicality and effectiveness in structuring and analyzing conflicts [ 5 ]. Moreover, this approach offers greater flexibility in eliciting DMs’ preferences, as it does not require cardinal utilities. As a result, the GMCR proves to be a highly valuable tool for analyzing conflicts and achieving consistent results in conflict resolution. For the development of this study, the option prioritizing technique will be used, as it is the most commonly applied method in practical problems involving the GMCR. This technique is effective in capturing DMs’ preferences in strategic conflicts, whether these preferences are unknown, uncertain, or crisp [6]. Games 2024,15, 39. https://doi.org/10.3390/g15060039 https://www.mdpi.com/journal/games Games 2024,15, 39 2 of 16 In this technique, each DM is asked to provide an ordered list of preference statements in descending order of priority, starting with the most important and ending with the least important. Each preference statement consists of a logical combination of the available options in the conflict. These options represent actions that the DMs can or cannot take during the conflict, and their combinations form the scenarios or states of the conflict. It is worth mentioning several works that have used the option prioritizing technique to elicit preference structures within the GMCR. Notable examples include: ref. [ 7 ], which explores two option prioritizing techniques for obtaining probabilistic preferences in the GMCR; ref. [ 8 ], which presents an option prioritizing technique for more efficiently capturing fuzzy preferences; ref. [ 9 ], which defines an option prioritizing method for eliciting three-level preferences; and ref. [ 10 ], which introduces the option prioritizing method for gray preferences. These methodologies share a common feature: they are adaptations of the classic option prioritizing method designed to handle different preference structures, such as fuzzy, gray, or probabilistic. However, they all share the limitation of being unsuitable for large conflicts, as they require all available options to be considered simultaneously. Moreover, according to [ 11 ], while these approaches offer the advantages of simplicity and usability, the evaluation results are highly susceptible to subjective factors such as the cognition, attitudes, and values of DMs, researchers, or experts involved in the conflict. The option prioritizing method requires DMs to provide an ordered list of preference statements that encompass all options in the conflict, which can be cognitively challenging in conflicts involving a large number of options. The aim of this work is to enhance the preference elicitation process for DMs, particularly in problems with numerous options. To address this, a new approach is proposed that divides both the elicitation of preferences and the stability analysis into multiple steps. The central idea is that large conflicts can be resolved incrementally, following a sequence based on priority. In each step, a sub-conflict is modeled and analyzed, and only the equilibrium states from one step remain feasible in the subsequent steps. By dividing the conflict analysis into steps, it becomes possible to manage conflicts involving a greater number of options, reducing the cognitive effort required from the DMs and thereby enabling more accurate results. The number of steps and the order in which options appear are determined through a pre-negotiation phase with the DMs involved in the conflict. In many complex conflict situations in real life, it is common for issues to be resolved incrementally. For instance, when voting on a major tax reform in a country, a parliament may choose to vote on individual items of the reform rather than the entire project at once. Similarly, in a war conflict, certain issues, such as humanitarian aid, may be addressed and agreed upon before other matters are resolved. In line with the proposed study, it is worth mentioning several existing works in the literature that are similar to the current one. The hierarchical graph model also proposes dividing a conflict into sub-conflicts to resolve them effectively. In this area, the study in [ 12 ] presents a hierarchical approach to a water distribution problem in China, where conflicts are analyzed separately using the GMCR. Ref. [ 13 ] introduces a hierarchical graph model represented by matrices, evaluating two sub-conflicts related to water resource distribution. Finally, ref. [ 14 ] demonstrates the use of the hierarchical graph model to analyze a conflict involving two locations in China, where a proposal to divert water from the south to the north is considered. Although hierarchical GMCR models also analyze conflicts separately, the goal in these studies is to assess multiple conflicts that occur simultaneously, where DMs and options may overlap. In contrast, our approach focuses on a single conflict that is analyzed in multiple steps, simplifying both the preference elicitation process and stability analysis. Generally, the main gap addressed by our proposed methodology is its design for situations where there are numerous issues (options) to be analyzed and resolved among the DMs. In such cases, the cognitive effort required to reach a resolution by considering all options simultaneously is either unfeasible or highly prone to assessment errors due to the Paradox of Choice [ 15 ]. The primary benefit of our proposal is that DMs can identify which Games 2024,15, 39 3 of 16 options should be prioritized for negotiation during a pre-negotiation phase. Naturally, this pre-negotiation phase itself may lead to conflicts of interest, and a conflict resolution method may be needed to generate a prioritized list of options to be negotiated. In this work, we do not address potential conflicts during the pre-negotiation phase; rather, we take the outcome of this phase as the input for our approach. Furthermore, our methodology does not require the conflict to be divided into multiple steps by prioritizing the options; if the DMs agree to analyze all options at once during the pre-negotiation phase, the classic version of the option prioritizing method can still be applied. The key advantage of our approach is that it provides flexibility, allowing DMs to resolve the conflict in parts, starting with the most urgent issues. To demonstrate the applicability of the method proposed in this study, we used a conflict presented in [ 16 ], which describes a water crisis in an irrigated perimeter located in the municipality of Limoeiro do Norte, in the eastern region of the state of Ceará, Brazil. This area is responsible for producing various types of greens and vegetables. The conflict involves five options and four DMs: Government Organizations, Small Farmers, Agribusiness, and Civil Society. At the end of the application, it was found that the proposed approach yielded more consistent results compared to the conventional application of the option prioritizing technique described in [ 16 ], which suggested a different equilibrium from the one identified in this study. The remainder of this paper is organized as follows. Section 2contains the theoretical framework, including the concepts of the GMCR, the stabilities and the option prioritizing technique. Section 3details the approach proposed in this study, outlining the formal model for preference elicitation and conflict analysis. Section 4describes the application of the proposed methodology in a case study involving a water crisis conflict. Finally, Section 5 provides the conclusions of the work. 2. Materials and Methods In this section, we recall the main concepts necessary for understanding this work. 2.1. GMCR The GMCR consists of a set of directed graphs, where each graph corresponds to a DM involved in the conflict. Each graph has the same set of vertices, which will constitute the feasible states of the conflict [4]. The GMCR describes the conflict by describing the parties involved in the conflict, which are called DMs, denoted by the set N={ 1, 2, . . . , d} and the actions that each DM can or cannot take in the conflict, which are known as options, denoted by Ok , for k∈N . It is assumed that Ok=∅ and Ok∩ Oj=∅ , if k=j . Therefore, denote by O=∪k∈NOk the set of all options available to all DMs in the conflict. Since each option represents a course of action that may or may not be taken in the conflict, a conflict scenario or state is defined as a specification of which options are being chosen in that particular situation. Formally, a conflict state can be modeled as a function s:O → {Yes , No} , so that for o∈ Ok , if s(o) = Yes , then DM k takes option o at conflict state s , otherwise s(o) = No . In principle, if ||O|| =m, there are 2mpossible different states, but not every combination of options can be taken in a particular conflict, making some states unfeasible. The set of feasible states of the conflict is denoted by S={s1 , s2 , . . . , sq} . Once the set of feasible states is determined, the next step is to model how the DMs can change the conflict states by changing their own options. This is specified through a directed graph (S , Ak) , for k∈N , such that (sp , sl)∈Ak if DM k can in one step take the conflict from sp to sl by changing its options. We therefore have the restriction that sp(o) = sl(o),∀o/∈ Okif (sp,sl)∈Ak. To complete the modeling of the conflict, it is necessary to determine a preference relation for each DM over the set of feasible states. In this work, the preferences of a given DM k are represented by a binary preference relation, ≻k , over the set S [ 5 , 17 ]. This preference relation is assumed to be asymmetric, where sp≻ksq indicates that DM k strictly Games 2024,15, 39 4 of 16 prefers sp to sq . From the strict preference relation ≻k , the weak preference relation ≽k is derived, such that sp≽ksqif sq⊁ksp. 2.2. Stability Analysis After modeling the conflict, a stability analysis is performed to establish effective solutions for the analyzed situation. To conduct this stability analysis in the GMCR, it is essential to first understand that the concepts of stability reflect the strategies of the DMs regarding their view of the conflict and their perceptions of risk [18]. This stability analysis uses a DM k as a reference, known as the focal DM [ 19 ]. Intuitively, a state is stable for a DM k if it prefers to remain in that state based on the anticipated reactions of the other participants in the conflict. Since DMs may behave differently in a conflict situation, the possible reactions and counter-reactions lead to different stability concepts. Among the existing stability concepts are: Nash stability (R) [ 20 ], general meta-rationality (GMR) [ 2 ], symmetric meta-rationality (SMR) [ 21 ], sequential stability (SEQ) [21], and symmetric sequential stability (SSEQ) [22]. To formally recall the definition of these stability concepts, we need to introduce some more notation. First, consider the set of all states reachable by DM i from state s given by Ri(s) . This set consists of the states to where DM i can move unilaterally (in a single move) from state s. It is formally defined as follows: Ri(s) = {s1∈S:(s,s1)∈Ai}[17,23]. Furthermore, we need to identify which states from the set of reachable states by DM i are also preferable to DM i . These moves are known as unilateral improvement moves for DM i from state s [ 17 , 23 ]. The set of all unilateral improvement moves for DM i from state sis defined by: R+ i(s) = {s1∈S:(s,s1)∈Aiand s1≻is}. 2.2.1. Nash Stability According to [ 20 ], in this stability concept, for any state to which the focal DM can move from a state considered Nash stable for him/her, that state will not be preferable to the initial state. Definition 1. Let i ∈N; state s ∈S is Nash stable for DM i, if and only if R+ i(s) = ∅. 2.2.2. GMR Stability According to [ 2 ], in GMR stability, DM i evaluates his/her moves conservatively, believing that by unilaterally moving the conflict, his/her opponent DM j will react in a way that leads to a non-preferable state for DM icompared to the initial state s. Definition 2. Let i∈N ; state s∈S is GMR stable for DM i , if and only if ∀s1∈R+ i(s) , there exists at least one state s2∈Rj(s1)such that s ≽is2. 2.2.3. SMR Stability Based on study [ 21 ], in SMR stability, the focal DM evaluates which move to perform, considers the reactions of their opponent DM j in response to the initial move, and then assesses his/her own counter-reaction to the opponent’s response. However, there will be no state that can be reached by DM i and is preferred to the starting state s , if s is SMR stable for DM i. Definition 3. Let i∈N ; state s∈S is SMR stable for DM i if and only if ∀s1∈R+ i(s) , there exists at least one state s2∈Rj(s1), such that s ≽is2and s ≽is3,∀s3∈Ri(s2). 2.2.4. SEQ Stability As described by [ 21 ], in SEQ stability, DM i assumes that DM j will not only attempt to sanction i ’s unilateral improvements, but will also aim to achieve his/her own improvements. In other words, DM j unilaterally moves the conflict to a state that is preferable for him/her, but not preferable for DM icompared to the initial state. Games 2024,15, 39 5 of 16 Definition 4. Let i∈N ; state s∈S is SEQ stable for DM i if and only if ∀s1∈R+ i(s) , there exists at least one state s2∈R+ j(s1), such that s ≽is2. 2.2.5. SSEQ Stability As proposed by [ 22 ], SSEQ stability incorporates aspects of both SEQ and SMR stability. In this case, DM i assumes that DM j is not solely focused on sanctioning DM i ’s unilateral improvements, but also seeks his/her own improvements. Furthermore, DM i cannot escape the sanction imposed by DM j’s move. Definition 5. Let i∈N ; state s∈S is SSEQ stable for DM i if and only if ∀s1∈R+ i(s) , there exists at least one state s2∈R+ j(s1), such that s ≽is2and s ≽is3,∀s3∈Ri(s2). 2.3. Option Prioritizing One of the main challenges in conflict modeling is obtaining preferences. To address this, one of the most widely used techniques in GMCR analysis is the option prioritizing technique [ 24 ]. This method was developed based on the preference tree concept introduced in [ 25 , 26 ]. The option prioritizing technique involves eliciting, for each DM, an ordered list of preference statements, ranked from highest to lowest priority. Each preference statement is a logical formula that involves Boolean combinations of the conflict options. Consequently, each state may or may not satisfy a given preference statement. As previously described, the options in a conflict are actions that DMs may choose to take or not take during the dispute. The set of conflict options available to DM k is denoted by Ok={ok 1,ok 2,...,ok mk}. Moreover, preference statements are denoted by ψ(O) , taking a corresponding truth value, either “True” (T) or “False” (F), based on whether the associated options are taken or not in a given state. According to [ 24 ], preference statements can be categorized into three types: non-conditional, conditional, or biconditional. Thus, an unconditional preference statement refers to a combination of available options and logical connectives, such as: negation (“not” or -), conjunction (“and” or “&”) and disjunction (“or” or “ | ”). Parentheses (“(” and “)”) are also used to control the priority of the connectives in a preference statement. On the other hand, a conditional (or biconditional) preference statement involves two preference statements connected by the logical operator “IF” (or “IFF”, respectively). According to the option prioritizing technique, the states that satisfy the first statements of the ordered list are preferable to those that do not. Formally, if Ψk(O) = (ψ1 k(O) , ψ2 k(O) , . . . , ψlk k(O)) is the ordered list of DM k ’s preference statements, state s is preferred to s′ if there exists 1 ≤t∗≤lk such that ψt k(O)(s) = ψt k(O)(s′) , for all t<t∗ , ψt∗ k(O)(s) = T and ψt∗ k(O)(s′) = F . In other words, the above definition establishes that state s is preferable to state s′ by DM k if, according to the ordered sequence of preference statements, state sis the first that uniquely satisfies a preference statement. Among the three approaches for obtaining DMs’ preferences in a conflict (option weighting, option prioritizing, and direct ranking), option prioritizing is considered the most flexible. It is regarded as the most suitable for most models, as it only requires DMs to provide preference statements about specific options being selected or not, in descending order of priority [6]. Several studies have explored the application of the option prioritizing method within the GMCR framework. For instance, ref. [ 27 ] proposed a methodology incorporating fuzzy truth values into the option prioritizing technique to achieve a more realistic preference ordering of feasible states. Similarly, ref. [ 28 ] introduced enhancements to the option prioritizing technique in GMCR by proposing a score function based on a confidence level function and utilizing Games 2024,15, 39 6 of 16 a preference tree. These improvements aim to quantitatively express differences in DMs’ preferences across various states. Ref. [ 29 ] introduced a dynamic conflict model that incorporates the evolving attitudes of DMs using the option prioritizing technique within the GMCR framework. This model was applied to analyze the planning of an urban transport system project in Pakistan. Additionally, ref. [ 30 ] proposed an alternative approach that involves third-party interventions within the GMCR framework. Their method aims to achieve satisfactory agreements by promoting minimal adjustments in the prioritization of preference statements. Thus, the option prioritizing method stands out for its applicability, flexibility, and the advantages it brings to both the structuring and analysis of conflicts. It contributes to improved understanding and ensures more consistent information. However, when dealing with conflicts involving a large number of options, it becomes cognitively challenging to generate an ordered sequence of preference statements that accurately reflects the DMs’ preferences. To address this issue, this work introduces a stepwise approach to option prioritizing. 3. Stepwise Option Prioritizing This new approach aims to facilitate the process of eliciting DMs’ preferences in conflicts involving a large number of options. The strategy for managing these conflicts involves dividing the available options based on their importance and resolving the conflicts incrementally. Once resolutions are determined for the most important options, the feasible outcomes in subsequent steps must align with the equilibria established in earlier steps. The overall conflict resolution is achieved in the final steps, where all available options are considered. In general, this approach aims to adapt option prioritization by dividing the elicitation process into multiple steps, rather than requiring preference statements for the entire set of options at once. At each step, DMs provide preference statements for a subset of the available options, prioritizing the most important options first. Below, we formally describe the stepwise option prioritization method. Let O∗={α1 , α2 , . . . , αn} , where αi⊆ O is the subset of options to be negotiated at the i-th conflict step, αi⊊αi+1, for i=1, 2, . . . , n−1 and αn=O. In this approach, the preference elicitation process for DMs is divided into multiple steps. In the first step, the preference statements involve only the options within the set α1 . In the i -th step, new preference statements are added to the list, involving only the options within the set αi . The number of steps and the order in which the options appear at each step result from a pre-negotiation phase. This phase must consider the number of options in the conflict, the DMs’ priorities, and the interdependencies between the options. In this work, we assume that this pre-negotiation phase has already been completed, and the sets αi’s are provided as input data for our problem. The idea is that, at each step, new preference statements are added to the end of the list for each DM. These new statements involve the additional options introduced in that step, capturing the DMs’ preferences for those options. Since not all options are present in every step, conflict states are described partially by the corresponding partial states at each step. Formally, a partial state at the i -th step is a function s:αi→ {Y , N} . The set of all feasible partial states at the i -th step is denoted by Sαi . These states involve only the options in αi that are being negotiated, subject to the restriction that the agreements made in previous steps must be upheld. In this context, during the first step, the options initially prioritized by the DMs are described in α1 , resulting in a set of states Sα1 . Thus, for αi , αj∈ O∗ , with j>i , we have αi⊊αj , so Sαj is considered more refined than Sαi , in that the partial states in the set Sαj reflect all the options described in the partial states in Sαi and more. Formally this is denoted by Sαj≥Sαi. As the name suggests, partial states provide partial descriptions of conflict states, such that these states are interconnected. We will utilize the concept of projection, as proposed Games 2024,15, 39 7 of 16 in [ 31 ], to model this relationship. For αi⊆αj and a state s∈Sαj , we define the projection of this state into Sαito be rαj αi(s)∈Sαiso that rαj αi(s)(o) = s(o), for all o∈αi. Therefore, the same options in αi are taken in s∈Sαj and in rαj αi(s)∈Sαi . Furthermore, we assume that Sαi={rαn αi(s):s∈S} , i.e., the set of feasible states at the i -th step is the set of projections of feasible conflict states considering the set of options available in the n-th step. In this way, let Ni⊆N represent the set of DMs participating in the i -conflict step. In this i -th step of the conflict analysis, we will obtain, by the options prioritization method, the preferences ≻i k , for k∈Ni , on the set of partial states Sαi . However, when carrying out the stability analysis of the i -th step, not all states in Sαi will be considered, since the states that do not satisfy the notion of equilibrium in the previous step will be disregarded in the next step. Accessibility sets at each step should also take into account which option changes are allowed in the original conflict. Next, we formally describe each step of conflict modeling and analysis: Step 1. In this first step, we perform the usual stability analysis considering the GMCR (Sα1 , {A1 k}k∈N1 , {≻1 k}k∈N1) , where (s1 p , s1 l)∈A1 k if and only if (sp , sl)∈Ak , s1 p=rαn α1(sp) and s1 l=rαn α1(sl) . Moreover, if Ψk(α1) = (ψ1 k(α1) , ψ2 k(α1) , . . . , ψl1 k k(α1)) is the ordered list of preference statements for DM k at Step 1, then s1 p≻1 ks1 l if there exists 1 ≤t∗≤l1 k , such that ψt k(α1)(s1 p) = ψt k(α1)(s1 l) for t<t∗ , ψt∗ k(α1)(s1 p) = T , and ψt∗ k(α1)(s1 l) = F . For STAB ∈ {Nash , GMR , SMR , SEQ and SSEQ} , let SSTAB 1 be the subset of Sα1 consisting of all states that satisfy the equilibrium notion STAB at Step 1. Step i . For Step i , the set of states considered in the analysis is given by the subset of Sαi whose states have projection in Sαi−1 within the set SSTAB i−1 . Formally, ΩSTAB i= {s∈Sαi:rαi αi−1(s)∈SSTAB i−1} . The accessibility set for the DM k in the i -th step, Ai k⊆ΩSTAB i×ΩSTAB i will be given by (si p , si l)∈Ai k if and only if (sp , sl)∈Ak , where si p=rαn αi(sp)and si l=rαn αi(sl). Furthermore, if Ψk(αi) = Ψk(αi−1)◦(ϕ1 k(αi),ϕ2 k(αi), . . . , ϕni k k(αi)) = (ψ1 k(αi),ψ2 k(αi), . . . , ψli k k(αi)), where ◦ is the concatenation of two lists of preference statements, li k=li−1 k+ni k and Ψk(αi) is the ordered list of preference statements DM k at Step i , then si p≻i ksi l if there exists 1 ≤t∗≤li k such that ψt k(αi)(si p) = ψt k(αi)(si l) for t<t∗ , ψt∗ k(αi)(si p) = T , and ψt∗ k(αi)(si l) = F . Finally, a usual stability analysis is made considering the GMCR (ΩSTAB i , {Ai k}k∈Ni , {≻i k}k∈Ni) . For STAB ∈ {Nash , GMR , SMR , SEQ and SSEQ} , define by SSTAB i the subset of ΩSTAB i consisting of the states satisfying the equilibrium notion STAB at Step i. Definition 6. State s∈S is a stepwise equilibrium according to the stability notion STAB if s∈SSTAB n. 4. Application 4.1. Conflict Description The conflict used to demonstrate the applicability of the method proposed in this work is presented in [ 16 ]. This is a conflict related to the water scarcity of an irrigated perimeter located in the municipality of Limoeiro do Norte, eastern region of the state of Ceará, Brazil, which is responsible for the production of various types of vegetables. Table 1presents the DMs involved in the conflict. Games 2024,15, 39 8 of 16 Table 1. DMs in the conflict. Source: [16]. DMs Description Governmental Organizations (DM1) COGERH, DNOCS, SOHIDRA. Small Farmers (DM2) Members of local families owners of small farms. Agribusiness (DM3) Members of big agribusiness companies settler in the irrigated perimeter. Civil Society (DM4) Members of community entities, technicalscientific associations (UFC, IFCE) and professional associations. The DMs, described in Table 1can make decisions based on the choice of options that best suit their preferences. These options are shown in Table 2. Table 2. Conflict options. Source: [16]. Options Description Governmental Organizations (DM1) o1 Demand the implementation of a water reuse system by large companies. o2Increase water pumping tariffs for producers. Small Farmers (DM2) o3Drill deep water wells. Agribusiness (DM3) o4Grow crops that consume less water. Civil Society (DM4) o5 Provide political training and social mobilization in support of small farmers. In line with study [ 16 ], Government Agencies appear to have two potential options to address the water crisis in the Jaguaribe-Apodi irrigated perimeter of Chapada do Apodi. These options, aimed at reducing water consumption, are outlined in Table 2. However, they are not favored by other decision-makers (DMs), such as Agribusiness, Small Farmers, and Civil Society, as implementing these measures would result in increased costs for them. In addition, since there is groundwater in the region of the irrigated perimeter, farmers can drill water wells to increase their access to water resources. However, this option involves higher costs. From the perspective of agribusiness, there is the possibility of adopting crops that require less water, but this is not desirable due to the accompanying changes in market dynamics. Regarding civil society, promoting political training and social mobilization to support small farmers is a viable strategy. Therefore, as shown in Table 3, the conflict described in [ 16 ] comprises 32 possible states, as there are no infeasible states in this scenario. Additionally, each state includes options that decision-makers (DMs) can either adopt or reject, represented by Yes (Y) and No (N), respectively. Games 2024,15, 39 15 of 16 Data Availability Statement: The authors confirm that all data generated or analyzed during this study are included in this published article. 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