Decision SupportMatrix representations of the inverse problem in the graph model for conflict resolution
Introduction
Conflicts occur whenever two or more decision makers (DMs) having differences in value systems, objectives or preferences, interact in the real world. In fact, each DM in a dispute strives to change the course of the conflict and reach a state of interest such as a more preferred state than the status quo. In order to better represent and analyze conflict, many available models to conflict resolution have been proposed within a broad field called game theory. The normal and extensive forms of the game, which are generally considered to be part of classical game theory, were developed by von Neumann and Morgenstern (1944). Classical game theory is considered to be quantitative in nature because it uses cardinal preferences often expressed as utility values. However, sometimes it is difficult for a DM to determine how much he prefers one state to another. Thus, Howard (1971) designed a fresh approach called metagame analysis which only assumes the availability of relative preference information in which a given DM either prefers one state over another or they are equally preferred. A methodology called conflict analysis put forward by Fraser and Hipel (1979, 1984) was an enhancement and expansion of metagame analysis. The graph model for conflict resolution (GMCR), which is more comprehensive than existing methodologies, was proposed by Kilgour, Hipel, and Fang (1987) and Fang, Hipel, and Kilgour (1993). The above three methodologies are regarded as qualitative techniques because only relative preference information between any two states is assumed. Because of the foregoing and other reasons, GMCR is widely employed by practitioners and researchers for investigating real world conflict in a highly flexible yet simple way (Madani, 2013).
According to the GMCR procedure, the elements used in this approach can be classified into three main parts which are input, analysis, and output (Fang et al., 2003, Kinsara et al., 2015b). The primary items in the input part are the DMs, feasible states in the dispute and DMs’ relative preferences over the states. Either an individual or a group, such as a company, can be a DM. A DM can control one or more options, each of which can be selected or not by the DM who controls it. A feasible state is formed as a specific selection of options by the DMs. The analysis part is employed to determine whether a given state is stable for a specified DM or not. The state is said to be stable for a DM if the DM cannot reach a more preferred state in the midst of moves and counter movements by other DMs. An equilibrium of a graph model is a state that is individually stable for all DMs under the same stability definition. A series of stability definitions have been proposed including Nash stability (Nash, 1950, Nash, 1951), general metarationality (GMR) (Howard, 1971), symmetric metarationality (SMR) (Howard, 1971), and sequential stability (SEQ) (Fraser and Hipel, 1979, Fraser and Hipel, 1984).
The DMs in conflicts may have different purposes when investigating a dispute with different known information. In most situations, one wishes to ascertain the output of an ongoing or a historical dispute by using the analysis engine to calculate various types of individual stability and equilibria after identifying the input part. This is called the forward perspective as portrayed at the top of Fig. 1. In Fig. 1, a check sign (√) means the associated information is known while a question mark (?) indicates an item to determine. Most of the extensions to enrich the theory and applicability of GMCR have been developed under the domain of the forward perspective (Bashar et al., 2012, Bristow et al., 2014, He et al., 2017, Xu et al., 2009, Xu et al., 2010). In some cases, the analyst may wish to determine the type of behavior needed to reach a state of interest. This is called the behavioral problem which is depicted as the middle diagram in Fig. 1 (Kinsara et al., 2015b) and for which a mathematical solution was recently provided (Wang, Hipel, Fang, Xu, & Kilgour, 2018).
In some conflict situations, one wishes to know the preferences required by DMs in order to reach an attractive resolution for all parties. In third party intervention, for example, a third party is invited to a negotiation in order to assist the disputants to reach a win/win resolution (see, for instance, Hipel, Sakamoto, and Hagihara (2015)). The third party facilitators may wish to ascertain which preferences are required by the parties in order to reach such an attractive outcome. In order to analyze the resolution of such conflicts in which the preferences for each DM are unknown or partially unknown, the inverse analysis in a graph model is proposed as displayed at the bottom of Fig. 1. As introduced by Kinsara, Kilgour, and Hipel (2015a), the main feature of the inverse analysis is that the preference information must be determined.
In summary, GMCR can be categorized into three perspectives based on the different given information and goals. As can be appreciated, each perspective solves a different kind of conflict problem. The differences among these three perspectives in a graph model are encapsulated as follows:
- (a)
The forward perspective determines the possible equilibria by carrying out the stability analysis based on the preferences of each DM contained in the input.
- (b)
The behavioral perspective ascertains the types of behavior which can produce the outcome of that dispute with the known preferences.
- (c)
The inverse perspective determines the unknown or partially unknown preference relationships for each DM which are required to make a state of interest be an equilibrium under a specific type of behavior.
Inverse analysis can provide all of the possible preferences for the DMs to reach a desired resolution. For instance, a third party in a conflict (Hipel et al., 2015), who may be a mediator or analyst, can use the results of the inverse approach to determine how to persuade each DM to select the options resulting in the desired equilibrium according to the needed preferences. In other words, a third-party intervenor can employ inverse analysis to design his mediation strategy based on the required preference relationships to reach a more desired outcome. On the other hand, a particular DM involved in the dispute can take advantage of inverse analysis to change his own preferences and attempt to influence a competitor such that an equilibrium of interest can be reached (Kinsara et al., 2015a). In fact, within engineering and science, inverse analysis is referred to as inverse engineering and constitutes a crucial field of study when addressing physical systems problems (Gladwell, 2005). The topic of this paper is inverse engineering within societal systems in the presence of conflict.
In the field of conflict resolution, techniques for tackling the inverse problem possess some drawbacks. More specifically, the inverse model studied by Kinsara et al. (2015a) assumes the employment of ordinal preferences which mean the preferences are transitive and hence intransitive preferences cannot be handled. Moreover, Sakakibara, Okada, and Nakase (2002) and Kinsara et al. (2015a) investigated the inverse problem using the logical representation of GMCR and an enumeration method which encounters the issue of computational complexity. In order to address the aforementioned two shortcomings, the preferences of each DM are defined by using pairwise comparisons of every pair of states in conflict as given later in this paper in Definition 1, which means the findings in this paper are valid for both transitive and intransitive preferences. Preference matrices are defined later in Definition 2 to mathematically represent the binary relation between each two states. Preference matrices are employed to derive the mathematical representations of an inverse problem in a graph model having two or more DMs in this research. The preferences for each DM required to make a state of interest be an equilibrium with the given solution concept can be obtained using the inequalities provided in Section 4. As shown in Section 5.4, the computational complexity of the inverse analysis in a graph model can be enormously reduced by using the matrix representations of the inverse GMCR approach proposed in this paper instead of the enumeration method given by Kinsara et al. (2015a).
The remainder of this paper is organized as follows. In Section 2, two potential general applications of this inverse engineering approach to GMCR are described. Within Section 3, matrix representations of preferences, unilateral movements and improvements, and joint movements and improvements are given. Matrix formulations of the inverse analyses for Nash stability, GMR, SMR and SEQ are presented and proven in Section 4. Section 5 consists of a case study of a controversial fracking dispute among the Elsipogtog First Nation, New Brunswick Provincial Government and Southwestern Energy (SWN) Resources in Eastern Canada, which is used to demonstrate how the proposed matrix representations of the inverse problem can be conveniently employed in practice. Finally, some conclusions and ideas for future work are presented in Section 6.
Section snippets
Application approaches
In some real life conflicts, the output and part of the input preference information to a graph model are known, but the required preferences of one or more DMs to generate the output are unknown, as portrayed at the bottom of Fig. 1. More specifically, an analyst may wish to ascertain the possible preferences of DMs which satisfy the given final stability results in a historical conflict. For instance, in the past two DMs have reached an equilibrium in a conflict after tough negotiations.
Modeling of a graph model with matrix representations
GMCR was originally formulated using what is called the “logical” form since solution concepts defining possible human behavior are explained in terms of moves and counter moves. For instance, if all possible unilateral improvements from a given state by a particular DM can be blocked by counter moves by other DMs, the DM is better off not to move and the state under consideration is deemed to be stable. Xu et al., 2009, Xu et al., 2010 provided a clever matrix or algebraic formulation of GMCR
Inverse analysis using matrix representations
In the inverse problematique of a graph model, one may wish to study the preferences of the DMs given the known behavior of the DMs and a specified equilibrium. The DMs in the conflict may want to ascertain what preferences are needed to cause a state of interest to be stable according to the specified behavior. Four definitions of stabilities consisting of Nash stability (Nash, 1950, Nash, 1951), general metarationality (GMR) (Howard, 1971), symmetric metarationality (SMR) (Howard, 1971), and
Application of inverse analysis to the Elsipogtog First Nation fracking dispute
A New Brunswick fracking dispute in Canada between the provincial Government of Premier Alward and Elsipogtog First Nation is employed to illustrate how the proposed approach for determining the required preferences of each DM for a given state of interest to be an equilibrium works in practice. Because this conflict was previously formulated and analyzed by O'Brien and Hipel (2016), the possible equilibria to this dispute under the preferences provided in their paper have already been
Conclusions
In this paper, the matrix representations of the inverse problem for a graph model are formulated to ascertain the required preferences of each DM for reaching a given equilibrium or outcome of interest. Inverse analysis constitutes a powerful extension of the forward and behavioral GMCR methodologies for determining all of the possible preferences causing a state of interest to be an equilibrium as depicted at the bottom of Fig. 1. Four matrix expressions, which are Eq. (2), Inequality (4),
Acknowledgments
The authors would like to express their sincere appreciation to the anonymous reviewers and Editor for their constructive comments which improved the quality of the paper. The authors are grateful for the financial support supplied by the National Natural Science Foundation of China (71371098, 71071077, 71301060 and 71471087), Funding of Jiangsu Innovation Program for Graduate Education (KYZZ15_0093), Funding for Outstanding Doctoral Dissertation at the Nanjing University of Aeronautics and
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2023, European Journal of Operational ResearchCitation Excerpt :There are many stability definitions, including Nash Stability (Nash) (Nash, 1950), General Metarationality (GMR) (Howard, 1971), Symmetric Metarationality (SMR) (Howard, 1971), Sequential Stability (SEQ) (Fraser & Hipel, 1979), and Symmetric Sequential Stability (SSEQ) (Rêgo & Vieira, 2017). Different definitions may be appropriate for different DMs (He, Kilgour, & Hipel, 2017; Wang, Hipel, Fang, & Dang, 2018; Wang, Hipel, Fang, Xu, & Kilgour, 2019; Zhao, Xu, Hipel, & Fang, 2019). Stability analysis is used to determine how well each DM can fare when acting independently.
Solving the inverse graph model for conflict resolution using a hybrid metaheuristic algorithm
2023, European Journal of Operational ResearchCitation Excerpt :Given the foregoing issues, Wang, Hipel, Fang, & Dang (2018) proposed the matrix representation of inverse GMCR, establishing matrix inequalities for inverse analysis to reduce the computational complexity, but did not provide computational algorithms to find unknown preferences. Then, Han et al. (2022) and Han, Xu, Hipel, & Fang (2021) developed an integer programming approach for solving the inverse GMCR formulation presented by Wang et al. (2018) for the cases of two DMs and multiple DMs, respectively. In the second direction, Wu et al. (2019) made a first attempt to apply a 0–1 nonlinear programming method for computing minimal priority adjustments of preference statements for a two-DM model under the Nash stability, where only the order of the preference statements can be adjusted.