A solution concept (or stability definition) determines whether a state is stable for a decision-maker (DM) in a Graph Model for Conflict Resolution. An equilibrium according to that stability definition is a state stable for all DMs. Thus, stability definitions correspond to anticipated patterns of collective behavior in conflicts. This study focuses on a matrix method to identify stable states in a graph model with multiple DMs whose preferences are fuzzy relations, possibly incomplete. The four distinct approaches to incomplete fuzzy preference relations (IFPRs) are integrated with five classical stability definitions to produce matrix representations of twenty incomplete fuzzy stability definitions. Behavioral analysis and knowledge of DMs’ IFPRs can then be applied to identify which stability definitions are consistent with desired equilibrium states. Matrix representations and behavior analysis are illustrated using a model of a real-world conflict—a demolition dispute in Jiangsu Province, China.

Fair allocation of indivisible items among multiple individuals is a fundamental problem in collective choice. We consider the problem of allocating two of four indivisible items to each of two players, where the only known preference information is each player's strict ranking of the items. How is the rank correlation of preferences, as measured by Kendall Tau, related to properties that facilitate fair allocation such as the availability of envy-free, Pareto-optimal, maximin, and max BordaSum allocations? We also examine the relationship between the ranked correlation and features of Fallback Bargaining such as the depth of agreement and the probability of a (two-way) tie. We further categorize the players into two types, risk-averse and risk-acceptant, and analyze how player type affects various fair division properties. Our results suggest that increasing similarity of preferences tends to increase the number of Pareto-optimal and maximin allocations but to decrease the number of envy-free allocations. Higher rank correlation also makes Fallback Bargaining less compelling. Understanding how similarity of preference rankings influences the trade-offs among allocation properties gives new insight into the difficulties of fair allocation of indivisible items.

In strategic conflicts, the behavior of decision makers (DMs) is affected by their beliefs about how their opponents will respond, how they will counter respond, and more generally by the number of action-reaction steps they consider. In this article, we propose several notions of stability with variable horizon that vary in whether the focal DM and her opponents are allowed to make moves that are immediately disadvantageous, called unilateral disimprovements. Credible maximin stability reflects the assumption that opponents will attempt to deter a focal DM from a move by imposing the strongest possible sanction. We analyze the effects of varying the horizon on this form of stability, and on the relationships among the solution concepts that result. Requiring the focal DM to make only unilateral improvements does not influence the stability of states, but restricting opponents’ responses to unilateral improvements reduces the set of stable states. An application to a water-pricing conflict illustrates the analysis of a graph model using credible maximin stabilities.

A novel hierarchical graph model is developed to analyze conflicts interrelated on three levels. As an extension of the two-level hierarchical graph model, this new structure contains several smaller graph models, called local graphs, nested at three levels. Decision makers (DMs), states, moves, and preference relations in the three-level hierarchical graph model (3LHGM) are defined. The interrelationships between stabilities in local graph models and the overall graph model are investigated and utilized in developing algorithms to calculate stabilities in the hierarchical graph model. This novel methodology is then illustrated using a generic model of hierarchical climate change governance disputes. Stability calculations can uncover profitable courses of action. The 3LHGM aims to provide insightful resolutions for DMs with a broader vision of the hierarchical conflict they are participating in.

Hesitant fuzzy preference relations (HFPRs) are formally proposed to model the conflict situation in which each decision maker (DM) consists of multiple individuals and each individual has its own fuzzy preferences over the feasible states within the framework of the graph model for conflict resolution (GMCR). Based on HFPRs, new definitions for hesitant fuzzy Nash stability, hesitant fuzzy general metarationality, hesitant fuzzy symmetric metarationality, and hesitant fuzzy sequential stability permit stability analyses to be carried out. Moreover, a new option prioritization technique, called hesitant fuzzy option prioritization, is developed for modeling a DM’s HFPRs based on the DM’s priority sequence of preference statements, the DM’s fuzzy truth values and levels of confidence. The groundwater contamination conflict of Elmira, Ontario, Canada, is utilized as a case study to illustrate the usefulness and applicability of the hesitant fuzzy option prioritization technique and GMCR with HFPRs.

In many conflict situations, decision-makers (DMs) integrate multiple objectives rather than considering just one objective or dimension. A multi-objective graph model (MOGM) for two DMs is proposed to balance DMs' objectives in this paper. An intuitionistic fuzzy (IF) preference (IFP) framework is developed. By combining strength of preference and IF preference, integrating multiple objectives, an IFMOGM is developed for strategic conflicts. Four IF stability definitions: IF Nash stability, IF general metarationality (IFGMR), IF symmetric metarationality (IFSMR), and IF sequential stability (IFSEQ), are introduced and the dependence of these stability solutions on weights is shown. Finally, the IFMOGM is applied to greenhouse gas emission disputes between the US and China to demonstrate that the model and methodology.

The evolution of cooperation can be investigated using Prisoners' Dilemma game(PD) models on spatial networks. It has been shown that other-regarding preferences, such as inequality aversion, a taste for fairness, or reciprocal preference, can cause cooperative behavior to evolve. In this paper, we consider whether relationships among agents, alone or in combination with reciprocal preference, can drive cooperation. We study the emergence of cooperation in PD games on a two-dimensional spatial network where some individuals are reciprocators, altering their future behavior based on comparison with a randomly chosen neighbor. Simulation results show that, if the intensity of reciprocal preference is fixed, the frequency of cooperation increases with the fraction of reciprocators. When reciprocal preference intensity is high, a low level of cooperation can be sustained, even if there are few reciprocators -- though cooperation does not diffuse through the population. We also show that the particular linking rule matters, in that cooperators are more likely to survive under a Cooperate-Stay, Defect-Shift (CSDS) linking rule, as opposed to a Win-Stay, Lose-Shift (WSLS) rule. As the fraction of reciprocators increases, the CSDS rule provides a more favorable environment for the evolution of cooperation. In addition, there is a threshold fraction of reciprocators such that a large value of network evolution strength contributes to cooperation most the threshold is exceeded. On the other side of the threshold, stronger reciprocal preference intensity does not necessarily promote cooperation more powerfully. Our results provide insight into how relations between neighbors can be a potent force, in combination with reciprocal preference, in inducing cooperation.

A new hybrid preference framework of the graph model for conflict resolution (GMCR) is proposed which allows decision makers (DMs) having both unknown preference and fuzzy preference to be taken into account in conflict modeling and analysis. The novel hybrid preference structure provides DMs with a more flexible technique to express preference. It is capable of covering the unknown preference of one feasible state over another, as well as fuzzy preference. Moreover, within the new hybrid preference structure, four extension forms of unknown preference are defined for different fuzzy stability definitions. These stability definitions under the new hybrid preference can be employed to thoroughly investigate complex conflicts existing in practical applications, and can offer enhanced strategic insights regarding the conflicts. A specific real-world water diversion conflict occurring in China, which includes multiple DMs and hybrid preference, is utilized to investigate how the new hybrid preference framework of the GMCR can be conveniently applied in practice.

Within the framework of the graph model for conflict resolution, the concept of coalition improvement is generalized, resulting in new definitions of Nash, general metarational, symmetric metarational, and sequential stability applicable to coalitions. To illustrate how these generalized stability definitions can be applied to a real conflict, they are used to analyze an offshore oil exploration dispute that occurred in the South China Sea in 2014. The novel approach reveals which coalitions can upset a noncooperative equilibrium or support new cooperative equilibria, and which decision makers can benefit from joining a coalition.

Using the observed or desired stability results of each decision maker (DM) and the equilibrium of a conflict, an algorithm is proposed that determines whether the Nash, general metarationality, symmetric metarationality or sequential stability definition best explains the observations or how to achieve the desired equilibrium. The matrix representation of the graph model for conflict resolution, normally used to determine stabilities that reflect different models of human interaction in conflict, is utilized here to keep track of available one-step moves for each DM, as well as the DMs' relative preferences among states. A key theorem involving matrix representation of graph models determines which stability types are consistent with the observations or achievement of the desired equilibrium, taking into account specified preferences of the DMs, for the case of two or more DMs. To illustrate the use of the algorithm in practice, behavioral analysis is applied to a softwood lumber dispute between Canada and the USA.

An extension of the Graph Model for Conflict Resolution (GMCR) is proposed to permit the investigation of a conflict model at the option level. The novel approach presented in this paper allows the computation of equilibrium strength and the calculation of option effectiveness in the GMCR model. The approach provides analysts with a better understanding of the options available to the decision makers (DMs) and permits the ranking of options in terms of their effectiveness. The information gained by utilizing this approach will help the DMs in ending disputes according to their preferred scenarios by selecting the options that are effective for them. As a means to evaluate this method, it is applied to a model of the conflict between China and the United States over global trade agreements.

The conflict over the Asian Infrastructure Investment Bank (AIIB) involving China, the United States of America (USA), and Japan is analyzed within the paradigm of the Graph Model for Conflict Resolution to gain strategic insights and to predict the possible compromise resolutions. The conflict starts after China proposed in October 2013 the establishment of the AIIB. The USA and Japan opposed the founding of AIIB because they were concerned that China would use the bank to set its own economic agenda and policies. The hypergame analysis of this dispute, which takes into account misperceptions, shows that the option of implementing the Trans-Pacific Partnership (TPP) agreement by the USA would not result in the expected resolution as predicted by it. The insights gained by analyzing the conflict include that the TPP might be beneficial for the Chinese which is contrary to what was expected by the USA and Japan.

A conflict over the fourth party logistics development is modeled, and a stability analysis is carried out based on the graph model for conflict resolution. In this conflict, corresponding DMs and their options are identified, infeasible states of this conflict are removed, and DMs' preferences over the feasible states are estimated based on option prioritization. Then, the directed graph for the fourth party logistics conflict is generated by using the decision support system entitled GMCRplus. Based on the calculation results, the interpretation on these calculated equilibria are provided, and the evolution of the conflict is analyzed.

A new hybrid preference framework of the Graph Model for Conflict Resolution (GMCR) is proposed to facilitate modeling and analysis of conflicts involving two decision makers (DMs) with fuzzy levels of preference. The new preference structure, a hybrid of models of three-level strength of preference and fuzzy preference, provides a more flexible technique to express DMs' relative preferences among states, and is more comprehensive and flexible than existing models. One key contribution of this paper is to extend four graph model stability definitions for fuzzy preferences, consisting of fuzzy Nash stability (FR), fuzzy general metarationality (FGMR), fuzzy symmetric metarationality (FSMR), and fuzzy sequential stability (FSEQ), to the new hybrid preference structure. This extension permits the analysis of the hybrid model, including both strong and weak stabilities. The resulting modeling and analysis system can be applied to complex two-decision-maker conflicts, thereby enhancing the ability of the graph model to provide strategic insights. A specific two decision-maker conflict is utilized to illustrate the practical application of the new hybrid preference framework for GMCR.

A two-level carbon emission conflict is investigated using Graph Model for Conflict Resolution. Chinese central government has conflict with local authorities at provincial levels regarding the implementation of new carbon emission reduction policy. As the conflict takes place in multiple regions, the governments at a higher level should form strategies to interact with governments at the lower level. The resolution of this conflict obtained by calculating the stabilities in the corresponding graph model can be used as guidance of actions for each decision maker to follow. The stabilities indicate that the central government should form distinct strategies to deal with provinces with different levels of economic development. Hierarchical graph model can provide decision makers a comprehensive understanding of the carbon emission conflict taking place at different locations.

A methodology is developed to model a decision maker's (DM's) fuzzy and/or interval fuzzy preference over feasible scenarios or states within the framework of the graph model for conflict resolution. This technique uses the DM's fuzzy relative importance of its preference statements and their fuzzy truth values for the feasible states in a conflict under uncertain conditions. A preference statement of a DM is a preferable combination of DM's options or courses of action. The fuzzy importance for one preference statement over another, a value in the interval [0, 1], is interpreted as the degree to which the first preference statement is more important than the second to the DM. A fuzzy truth value of a preference statement at a feasible state is a number in the interval [0, 1] that represents the degree to which the statement is true at the state. When the DM is confident in its pairwise fuzzy importance degrees over the preference statements and their fuzzy truth values at the feasible states, the methodology provides a fuzzy preference over the states. When there is an ordinary or crisp importance ordering of preference statements and when the truth values of preference statements are classical or crisp at the feasible states, the technique generates a crisp preference over the states. The methodology is illustrated using a case study.

Three approaches to better understanding conflict resolution results are discussed and applied to a trans-boundary water conflict between the United States and Canada. A decision support system, called GMCR+, is utilized to carry out the analysis and provide strategic insights. These approaches will improve and simplify the analysis process especially for larger conflict models.

An expansion to the options structure is proposed to make the option-based graph model for conflict resolution more explicitly expressive of reality. The suggested improvement provides an opportunity to replace binary (on or off) options with multi-level options, wherein the decision maker (DM) selects one out of more than two distinct levels. The new approach simplifies the representation of conflicts and facilitates the understanding of their evolution by showing, in bargaining for example, outcomes that involve partial achievement of a DM's objectives. The preferences of decision makers can now be related to the possibility of tradeoffs between (or among) options.

Systems methodologies to model third-party intervention in international conflicts are developed within the framework of the graph model for conflict resolution (GMCR). An inverse GMCR is introduced to utilize the GMCR as a negotiation tool by altering the procedure of the original framework. The methodologies presented give a better understanding of how decision makers (DMs) can be motivated to undertake certain actions within the conflict. Moreover, the inverse GMCR tackles the problem of specifying which preferences for DMs lead to a particular resolution, thereby making it easier for a mediator or other third party to influence the course of the conflict. The methodologies are applied to a real-world dispute, a complex water conflict in the Middle East.

To capture uncertainty in preferences, definitions based on grey numbers are incorporated into the graph model for conflict resolution (GMCR), a realistic and flexible methodology to model and analyze strategic conflicts. A general grey number is a real number that may be a member of a discrete set of real numbers, or may fall within one or several intervals. It can represent uncertain preference of decision makers in a meaningful way. Here, a grey-based preference structure is developed and integrated with GMCR. Utilizing a number of grey-based ideas, solution concepts describing human behavior under conflict in the face of uncertain preference are defined for a conflict model. This grey-based GMCR is then applied to a generic sustainable development conflict with uncertain preferences in order to demonstrate how it can be conveniently utilized in practice.

Supply chain conflicts between Airbus and Boeing, along with potential suppliers are investigated using a hierarchical approach to obtain the strategic resolutions for all stakeholders. Two giant commercial aircraft makers, Airbus and Boeing, complete for better deals by selecting components for their newly designed aircraft, incurring conflicts due to their rivalry and the clash of interests with suppliers. To control the costs of their next generation aircraft and shorten the production period, both manufacturers demand better deals from potential suppliers with lower prices and more output. A conflict arises among the manufacturers and potential suppliers in the negotiation of the contract in each supply chain. To investigate these conflicts comprehensively, a hierarchical approach is adopted by constructing a Graph Model for Conflict Resolution (GMCR) with a hierarchical structure. The stakeholders may obtain strategic resolutions according to possible outcomes indicated by stability results. These findings in the hierarchical graph model are further compared to those obtained when each conflict is investigated separately.

An environmental conflict that occurred in Elmira, a small town in southwestern Ontario, Canada is re-examined using a modified version of the Graph Model for Conflict Resolution (GMCR). A newly developed decision support system (DSS) is utilized to carry out the analysis using the new “Inverse GMCR” approach. The new DSS features automatic graph drawing and in-depth conflict analysis, and provide new insights.

A conflict over water use and oil sands development in the Athabasca River Basin is modelled, and a stability analysis is carried out based on the Graph Model for Conflict Analysis. Oil sands require a great deal of water to extract bitumen, which may lead to insufficient water supply to the downstream area. Moreover, inappropriate waste water disposal can have serious effects on local fresh water resources. In order to balance industrial development and environmental protection, the conflict among the Alberta government, oil sands companies, and non-governmental organizations are analyzed, and insights and reasonable suggestions are provided.

Water diversion conflicts in China are studied using a hierarchical approach to obtain the strategic resolutions. To address the large population and uneven water distribution in China, the South-North Water Diversion Project (SNWDP) is being constructed to transfer water from the Yangtze River Basin to the North China Plain where there is a severe water shortage. As the eastern route of the project nears completion, conflicts have arisen along the central and western routes where the Chinese Central Government has to decide how to proceed. To obtain possible resolutions to these conflicts, a hierarchical approach is adopted by which conflicts along the two routes are analyzed separately using the Graph Model for Conflict Resolution (GMCR). The possible outcomes determined indicate that the central project can be fully initiated in spite of opposition, while the western route cannot be resumed without the agreement of the neighboring countries. These findings from the two separate conflicts are compared to those obtained when the two disputes are investigated as one larger conflict.

A historical water conflict along the Euphrates River in the Middle East is investigated and analyzed using the inverse approach to the Graph Model for Conflict Resolution (GMCR). The essence of this methodology is to inform the mediators of the possible approaches to reach a more desired resolution. The goal is to allow mediators to focus their efforts on influencing the involved parties efficiently in order to achieve an improved resolution.

A generic sustainable development conflict having uncertain preferences is investigated using a new methodology. In particular, definitions of grey numbers are introduced for incorporation into the Graph Model for Conflict Resolution (GMCR) in order to capture uncertainty in preferences. The GMCR constitutes a simple and flexible methodology model and analyze strategic conflicts. The goal of the study is to analyze the influence of uncertainty on the outcome of the conflict and to provide suggestion on guiding the conflict to move in a desired direction. General grey numbers are used to represent the uncertain preferences of decision makers. Following the explanation of four solution concepts based on grey preferences, a stability analysis is conducted and some new insights are provided in this research.

The fact that a conflict takes place in many rounds can obscure its nature. This research aims to answer the question of how to determine whether several rounds of a conflict happening at different points in time are connected. A conflict starts when a decision maker challenges a status quo. A decision maker can be involved in more than one conflict at a time, the features that differentiate conflicts are the objectives of the decision makers. The objectives are reflected in the decision makers' preferences over outcomes. The ultimate goal for a decision maker in a conflict is to obtain the most preferred achievable state.

A methodology is developed to use the fuzzy truth values of preference statements for feasible states in an option prioritization technique in order to rank states within the framework of the Graph Model for Conflict Resolution. Option prioritization ranks states based on the truth values of preference statements, which are compositions of the decision makers' courses of actions joined by logical connectives, ordered lexicographically. Fuzzy truth values are represented as truth degrees; so they include binary truth values, “true” and “false”, as well as other possible truth intensities that are reasonable according to the specific circumstances. Consequently, the assumption of fuzzy truth values of preference statements provides more realistic preference ordering of feasible states. The methodology is applied to the Elmira groundwater contamination dispute, which took place in Elmira, Ontario, Canada, for eliciting the preferences of decision makers, to demonstrate the applicability of this technique.

A water diversion conflict in the Danjiangkou area, China, is systematically studied in order to obtain an understanding of its strategic impacts. To address the large population and uneven distribution of water storage in China, the South-North Water Distribution Project (SNWDP) is being constructed to transfer water from the Yangtze River Basin to the North China Plain where there is a severe water shortage. The implementation of this project may not only affect water usage in the Danjiangkou area, which is the largest headwater region in the central route of SNWDP, but also have environmental impacts, resulting in conflict between local communities and the government which initiated this project. To formally investigate this conflict, the methodology of the Graph Model for Conflict Resolution (GMCR) is utilized. The equilibrium determined from a stability analysis indicates that the strategy used by the government benefits the residents, but neglects the environmental impacts. To understand these discrepancies, changes are made in the government's preferences to obtain an equilibrium which reflects the government's desire to achieve both economic equality and environmental sustainability.

Two approaches are introduced to model and analyze third party intervention, an important element in conflict resolution. These approaches are based on the GMCR framework. Although many studies discuss conflict resolution and conflict modeling, little work has been done on modeling third party interventions. The inverse approach to GMCR is introduced and investigated to find an optimal way to resolve a conflict. An historical conflict is analyzed using the introduced approach. Stability predictions based on different solution concepts are calculated, and the factors influencing third party intervention in a conflict are identified.

A new ranking approach based on Grey Systems Theory using PROMETHEE II (Preference Ranking Organization METHod for Enrichment Evaluation II) is applied to a Multiple Criteria Decision Making (MCDM) problem. The objective is to rank alternative suggestions to decision makers for maintaining source water quantity and quality . The method uses grey sets to represent uncertain information in the PROMETHEE II method. An inherent advantage of this approach is its ability to comprehensively consider quantitative and qualitative factors, making reliable decisions in an environment of uncertainty and subjective information. In this case study, involving the evaluation of source water strategies in the Waterloo Region, located in Ontario, Canada, the linguistic values of alternatives are expressed as grey sets, which are used to aggregate the opinions of experts and stakeholders on the impacts of the strategies according to selected criteria.

The negotiations between Brinco and Hydro-Québec, which led to the exploitation of the hydropower potential of the Upper Churchill Falls, is analyzed within the Graph Model for Conflict Resolution framework. Immediately after the agreement was signed, several rounds of conflicts arose between the Province of Newfoundland and Labrador and the Province of Québec. The equilibrium states of the negotiations are analyzed to understand the motivation and possible distinction for the evolved conflicts. The analysis shows strong motivation for the Province of Newfoundland and Labrador to renegotiate the terms of the contract to reach a better deal which may cause further conflicts. Analysis also suggests strategies and feasible circumstances in which the Province of Newfoundland and Labrador may be able to improve its situation.

A new framework for the graph model for conflict resolution is developed so that decision makers (DMs) with fuzzy preferences can be included in conflict models. A graph model is both a formal representation for multiple participant-multiple objective decision problems and a set of analysis procedures that add insights into them. Within the new framework, graph models can include-and integrate into the analysis-both certain and uncertain information about DMs' preferences. One key contribution of this study is to extend the four basic stability definitions for two or more DMs to models with fuzzy preferences. Together, fuzzy Nash stability, fuzzy general metarationality, fuzzy symmetric metarationality, and fuzzy sequential stability provide anuanced description of human behavior. A state is fuzzy stable for a DM if a move to any other state is not sufficiently likely to yield an outcome which the DM prefers, where sufficiency is measured according to a fuzzy satisficing threshold that is the characteristic of the DM. A fuzzy equilibrium, which is an outcome that is fuzzy stable for all DMs, therefore represents a possible resolution of the strategic conflict. The practical application and interpretation of these new stability definitions are illustrated with an example.

A Fuzzy Preference Framework for the Graph Model for Conflict Resolution is employed to analyze the sustainable development conflict in order to gain strategic insights. This framework is a novel methodology to identify outcomes favorable for all parties in a multiple-participant multiple-objective decision making setting with both certain and uncertain preference information. When applied to the sustainable development conflict, in which a developer plans to build or operate a project inspected by an environmental agency, the methodology identifies stable outcomes and thus clarifies the necessary conditions for sustainability.

The ongoing conflict among Shiites, Sunnis, and the Government of Bahrain is formally investigated using the graph model for conflict resolution in order to gain strategic insights into how this festering dispute can be better understood and resolved. At the modeling stage, the Bahrain conflict is meaningfully described in terms of the three main decision makers (DMs), as well as each DM's options or courses of action it controls and relative preferences among the possible states or scenarios that could arise. Subsequently, at the analysis stage, a stability analysis is carried out to determine the stability of a state from each DM's viewpoint, according to different descriptions of behavior under conflict, as well as the equilibria or compromise resolutions. The findings indicate that although the present strife could continue unless the Government steps down, a compromise solution in which the Government institutes reform is possible.

A Negotiation Support System (NSS) is developed based on the conceptual framework of the Evolutionary Model for Multilateral Negotiations (EMMN). The EMMN methodology is utilized to identify the most likely outcomes of asymmetric multilateral negotiations, in which each negotiator seeks to reach the most preferable outcome for which he or she can gain enough support from other negotiators. This NSS increases the speed of calculations, as well as avoids the errors typical of manual stability calculations and sensitivity analyses. All stable and feasible negotiation states are identified and illustrated in the output of this NSS, which is, therefore, a valuable aid in practical negotiations.

A bargaining procedure, or mechanism, is a set of rules for two bargainers to follow as they make offers in order to reach a mutually satisfactory agreement on, say, a price. The efficiency of a mechanism is the expected surplus it delivers to the bargainers, relative to the surplus that a social planner would deliver, or that the bargainers themselves might achieve if they truthfully revealed their reservation prices. A theoretical limit on this efficiency is known, as is a specific procedure that achieves this maximum. But this procedure induces players to make offers that do not truly reflect their reservation prices. This paper discusses three procedures that induce honest offers, although they necessarily fail to achieve maximum efficiency. Each procedure has its own characteristics and costs, and each may have some uses in particular circumstances.

A fuzzy preference framework is developed within the paradigm of the graph model for conflict resolution. This framework takes into account both certain and uncertain information about the preferences of decision makers (DMs) involved in a strategic conflict. The graph model is a solution methodology for conflict decision making that begins with a model of the problem and suggests possible resolutions through a number of stability definitions. Four basic fuzzy stability definitions are introduced for a two-DM graph model to analyze conflict behavior and identify possible resolutions even when preferences are fuzzy. Fuzzy stability definitions describe varied human behavior in a conflict model; a state is fuzzy stable for a DM according to a specific fuzzy stability definition if a move to any other state, evaluated according to that definition, does not meet the DM's fuzzy satisficing threshold. A state that is fuzzy stable for all DMs under a specific fuzzy stability definition constitutes a fuzzy equilibrium under that definition, and is interpreted as a possible resolution of the conflict. Fuzzy stability definitions include fuzzy Nash stability, fuzzy general metarationality, fuzzy symmetric metarationality, and fuzzy sequential stability.

A numerical technique for evaluating brownfields is proposed. The method is based on the representation of redevelopment strategies using fuzzy real options, which can be evaluated using an extended Least Squares Monte-Carlo (LSM) simulation. By employing a fuzzy boundary to represent the contaminated region of a brownfield, the parameters of various options for redevelopment can be estimated. Beginning with expert judgments on multiple factors at different locations, a decision support system can map a possibly polluted area using fuzzy real option parameters determined according to an ordered weighted averaging (OWA) technique. The results can be input directly into a decision process. A particular advantage of extended LSM lies in its ability to produce results whether or not an analytic solution exists, making fuzzy real options very widely applicable to brownfield development pricing.

A systematic study of recent developments in preferences based on fuzzy logic is carried out in order to identify an effective design for fuzzy preference models in conflict resolution. Fuzzy preference, defined via a fuzzy relation over the alternatives or states, attempts to represent a decision maker's preferences more realistically. It generalizes the usual preference structures in the sense that it provides a uniform description of both certain (crisp) and uncertain preferences. The potential applicability of fuzzy preferences to the modeling, analysis, and understanding of strategic conflicts is investigated and connected to a literature survey.

Formal analytical techniques are employed for better understanding the continuing conflict over Iran's nuclear program, including predicting various outcomes and recommending how to reduce the intensity of the conflict, or at least prevent its escalation. The Graph Model for Conflict Resolution is used to investigate the strategic interaction of Iran and its opponents. The analysis of our model shows that a military conflict is certainly possible but not the most likely outcome. We conclude with suggestions about how the decision makers could reach a peaceful resolution, demonstrating that the International Atomic Energy Agency should be as actively involved as possible.

A prototype decision support system is developed for private developers to evaluate brownfield redevelopment projects based on fuzzy real options. Redevelopment investments under high uncertainty can be better assessed using fuzzy real options than when employing the traditional Net Present Value (NPV) method. When using NPV, private developers tend to require high risk premiums and regard these projects as unprofitable. Hence, a better valuation method is needed to reveal the real value of a brownfield. The fuzzy real option model seems to be appropriate for this purpose. A Decision Support System (DSS) is proposed based on this model in order to evaluate a brownfield site and provide private developers with better investment advice. This DSS's architecture and user interface are introduced and discussed. Implementation techniques are compared and selected according to their advantages and disadvantages. An illustrative example is used to demonstrate this system's processing capability. It is found that the proposed DSS is flexible and user-friendly. It can generate not only a price, but also an optimized strategy for dealing with uncertainties, which has great potential in applications.

An algebraic approach is developed to calculate stabilities in two decision maker graph models with strength of preference. The original graph model uses ¿simple preference¿ to represent a decision maker's relative preference between two states. This preference structure includes only a relative preference relation and an indifference relation. Basic stability definitions, and algorithms to calculate them, assume simple preference. But difficulties in coding the algorithms, mainly because of their logical formulation, led to the development of matrix representations of preference and explicit matrix algorithms to calculate stability. Here, the algebraic approach is extended to representation of strength-of-preference graph models, which feature multiple levels of preference, and stability analysis for such models. Matrix representation of stability definitions facilitates the development of new stability concepts and algorithms to calculate them. The method is illustrated using a simple model of a conflict over sustainable development.

A benchmark-based method is proposed for multiple-criteria ranking, and a case study is presented to demonstrate that the procedure can be efficient and effective in practice. Multiple-criteria ranking aims to help a decision maker (DM) assess a finite set of alternatives according to several criteria, usually conflicting, in order to rank the full set. The relation of benchmarks to multiple-criteria decision analysis is investigated systematically, and then, an approach based on distance from a benchmark is designed to incorporate information about a DM's judgements so as to produce a full ranking. The procedure is applied to rank 81 U.S. brownfield redevelopment projects based on available data and an accepted benchmark.

Perceptual graph model systems are designed to be employed when there are discrepancies in decision makers' (DMs) perceptions of a conflict, which may be caused, for instance, by the presence of negative emotion or asymmetric information among DMs. In this case, conventional stability analysis cannot be used; perceptual stability analysis is proposed as a new theoretical procedure that extends existing stability algorithms to situations when DMs have independent perceptions or awarenesses of a conflict. The overriding objective of perceptual stability analysis is to predict possible resolutions, and unveil the dependence of these predictions on variability in a DM's awareness. Perceptual stability analysis takes a two-phase approach. In Phase 1, individual stability analysis is applied to each DM's graph model (a perceptual graph model) from the point of view of the owner of the model, for each DM in the model, using standard or perceptual solution concepts, depending on the owner's awareness of others' perceptions. Then, in Phase 2, metastability analysis is employed to consolidate the stability assessments of a state in all perceptual graph models and across all variants of awareness-i.e., in all possible graph model systems. The distinctive modes of equilibria thus defined reflect the incompatibilities in DMs' perceptions and viewpoints, but nonetheless provide important insights into possible resolutions of the conflict. To demonstrate the practical application of these new developments, a model of the conflict in Chechnya is presented and analyzed.

A matrix method is developed to apply four solution concepts to a graph model for conflict resolution in order to model human behavior in a multiple-decision-maker (DM) strategic conflict. A graph model represents the interaction of two or more DMs and can be analyzed to identify possible equilibrium states according to each solution concept. Previously, solution concepts were defined logically, in terms of the underlying graphs. However, procedures to identify stable states based on these definitions are difficult to code because of the nature of the logical representations. In this paper, a graph model and four graph model solution concepts are formulated explicitly using matrices. More specifically, matrix expressions are given for relative preferences, joint unilateral movements, and joint unilateral improvements in a multiple-DM model. Then, it is shown how to calculate stability under each of the four solution concepts using the matrix representation. Compared with the existing approach, matrix representation is more effective and convenient for calculating stabilities and predicting equilibria of a graph model. In particular, the proposed method is easy to code and extend to other contexts.

A system that aggregates case-based linguistic decision rules using a hybrid of the dominance-based rough set approach (DRSA) and the Dempster-Shafer (DS) theory of evidence is proposed for multiple criterion-multiple participant sorting. First, DRSA is employed to estimate decision makers' (DMs) preferences by deducing linguistic decision rules from representative case sets evaluated by different DMs. Next, DS theory is used to evaluate each alternative by aggregating the decision rules triggered by the DMs, thereby generating an overall decision recommendation for the alternative. The proposed method is demonstrated using a numerical example.

New definitions and methodology are proposed in this paper to predict the most likely outcomes of multilateral negotiations. The negotiation problems modeled and analyzed here are distinctive because the set of possible agreements is discrete and specified in advance. Each decision maker has two concerns: first, achieving an alternative that is as preferable as possible; second, building support among the other decision makers for this alternative. The general objective of this research is to study movements from one state to another in multilateral negotiations, to predict stable agreements, and to study their properties. In addition, we propose some procedures that measure the relative likelihood of moves from an unstable state to stable agreements. An example is used to clarify how this methodology can be applied to predict outcomes of negotiations.

This paper uses the graph model for conflict resolution (GMCR) to model and analyze strategic conflict in Brownfield redevelopment, which is a major issue in Canada and around the world. As a case study, a graph model of the conflict surrounding the ERASE program, implemented in Hamilton, Ontario, Canada, in 2001, is analyzed using the recently-developed matrix representation. Final equilibria (stable states) include no redevelopment, redevelopment by the private sector, and redevelopment by the public sector. These three equilibria seem to correspond to the outcome of brownfields conflicts in various cities in Ontario, including the ERASE program in Hamilton, showing the potential of GMCR to predict impacts of policy modification and to improve policy-making in Brownfield redevelopment.

A new preference structure that includes preference strength and preference uncertainty is developed within the paradigm of the graph model for conflict resolution with two decision makers. This structure allows a decision maker to strongly or mildly prefer one state or scenario over another, to have equal preference for the two states, or to be uncertain about the preference between them, thereby creating a more general preference structure. Four solution concepts, Nash Stability, General Metarationality, Symmetric Metarationality, and Sequential Stability, are defined for the graph model with this extended preference structure, and relationships among stability types are determined. A specific graph model with two decision makers is used to illustrate how the new solution concepts can be applied in practice.

A strategic decision support procedure is developed for brownfield redevelopment planning and demonstrated with a practical case study. The procedure contains both higher and lower levels of strategic decisions. The higher-level strategic decision is designed to assess the feasibility of a specific brownfield redevelopment. A well-known business strategic planning tool, SWOT analysis, is adapted to help decision makers (DMs) to identify the strengths, weaknesses, opportunities, and threats involved in implementation of a brownfield redevelopment. The lower-level strategic decision is meant to find the best brownfleld redevelopment solution. Several multiple criteria decision analysis tools are employed to assist DMs to evaluate brownfield redevelopment designs using economic, social and environmental criteria. Finally, a practical case study is carried out to demonstrate the proposed procedure.

The general objective of this research is developing methodologies to predict stable agreements in multilateral negotiations and study their properties. To model and analyze negotiations, a wide range of integrative or cooperative approaches can be used to identify appropriate tradeoffs for the negotiators. The negotiation problems considered here are distinctive, however, because the set of possible agreements is discrete, and specified in advance. Each decision maker has two concerns: first, achieving an alternative that is as preferable as possible; second, building support for this alternative among the other decision makers. New definitions and methodologies are proposed in this paper to model and analyze multilateral negotiations within a context of discrete choices.

New matrix representations are developed for characterizing potential human interactions under conflict within the structure of the Graph Model for Conflict Resolution for the case of a dispute having two decision makers. The matrix representations are devised for calculating individual stability and equilibria for strategic interactions defined according to the solution concepts consisting of Nash stability, general metarationality, symmetric metarationality, and sequential stability. A potential military confrontation is utilized to illustrate the procedures for applying the matrix representation of solution concepts to a practical problem. Compared with the existing systems, matrix representation of solution concepts for graph models is more effective and convenient for calculating stabilities and predicting equilibria, especially in complex conflict models with many feasible states. Moreover, the method is easy to code and can be extended to other contexts such as those that allow preference uncertainty.

A case-based distance model to solve sorting problems in multiple-criteria decision analysis (MCDA) is developed, and its application in water resources management is presented. The sorting problem in MCDA is to arrange a set of alternatives into ordered groups. MCDA is introduced as consequence-based preference aggregation, whereby consequence and preference expressions (values and weights) are defined and combined in a sequence of steps. Then, sorting problems are defined, and some properties are explained. Based on weighted Euclidean distance, two case-based distance models are developed for sorting using weights and group thresholds obtained by assessment of a case set provided by a decision maker (DM). This case-based method can elicit the DM's preferences more expeditiously and accurately than direct inquiry. Case-based sorting model I is designed for cardinal criteria, while its extension, i.e., case-based sorting model II, can handle both cardinal and ordinal criteria. Optimization programs are employed to find the most descriptive weights and group thresholds. A case study in which Canadian municipalities are sorted according to water usage is presented.

A policy equilibrium is defined, and its properties investigated, for conflicts with more than two decision makers (DMs). A fundamental construction is the metarational tree, which expresses DMs' interactions as sequences of rounds, each consisting of an initial move by the focal DM followed by countermoves by the opponents. Using the metarational tree, the stability definitions of the graph model for conflict resolution can be adapted to apply to policies. These generalized metarational stabilities are shown to generalize Nash, general metarational, and symmetric metarational stabilities. Relationships among generalized metarationalities are derived, as are their connections with policy equilibria. Finally, the refinement that allows only credible moves (moves that are in the immediate interest of the mover) produces a new family of credible generalized metarational stabilities that generalizes the concept of sequential stability in the graph model.

Over the past twenty years, many decision analysis tools have been developed to apply multiple-criteria techniques to sorting. In both of the main approaches, direct judgment and case-based reasoning, alternatives are arranged into a predefined number of groups. In practical applications, however, a decision maker may prefer a sorting method that facilitates effective organization of the alternatives based on interactive adjustment of the group count. This paper proposes an efficient sorting method with this flexibility. A case study in inventory management is carried out to demonstrate the approach.

Sorting is a process for multiple criteria decision analysis that arranges a set of alternatives into ordinally defined groups. Based on weighted Euclidean distance, a case-based distance model is proposed for sorting using weights and group thresholds obtained by assessment of a case set. A quadratic optimization program is employed to find optimal weights and group thresholds. Instead of direct inquiry, this case-based method can elicit the decision maker's preferences more expeditiously and accurately. A case study on Canadian municipal water usage analysis is presented.

A procedure is put forward for using the graph model for conflict resolution to account for the role of emotion in strategic conflict. The concept of possibility provides a framework for modeling decision makers' subjective perceptions of outcomes in a conflict, which can conceal logically valid states or expose states that are otherwise invisible. The implicit assumption of the graph model, that all decision makers' directed graphs have the same set of states, and hence that all decision makers share the same conceptualization of the conflict, is no longer appropriate. The proposed analysis technique for asymmetric state sets is suitable for use when emotions have caused discrepancies in the apprehension of a situation, and it permits exploration of the behavioral consequences of these perceptual differences. The 1990s confrontation in Chechnya between Russia and Chechen Rebels, an emotionally-leaden struggle with inconsistent perceptions, is used to illustrate the new procedure

A new preference structure is introduced into the graph model for conflict resolution. This structure can handle a decision-maker's (DM) strict preference for one state or scenario over another, equal preference for states, and uncertain or unknown preference in the comparison of two states. Built upon this preference structure, four types of solution definitions modeling human behavior under conflict are extended to accommodate uncertainty in preferences. Four distinct ways to consider uncertain preference information are identified, producing sixteen extended stability definitions. Interrelationships of these definitions within and across the four definition sets are investigated. Illustrative examples of two-DM and multi-DM conflict models are presented to show how the new solution concepts can be applied in practice.

A new approach is developed to conduct status quo analysis within the paradigm of the graph model for conflict resolution. An algorithm is formulated to generate a diagram starting from the status quo state to characterize possible moves and countermoves by the decision-makers as they move the conflict towards a desirable state. To present the result to users more clearly, another algorithm is developed to generate a table based on the diagram. An illustrative environmental conflict model is employed to demonstrate how these algorithms can be applied in practice and what significant insights can be garnered from the status quo analysis.

A conceptual framework is proposed to understand the role of emotions in conflict. Taking a multidisciplinary approach, we suggest why the evaluation of emotions is crucial to the improvement of current conflict analysis and resolution techniques. We begin by outlining the critical features of a conflict. The use of appraisal theory for emotion activation and differentiation shows why conflict is often laden with emotion, and why emotion is relevant to conflict analysis. The confrontation between the United States and North Korea over nuclear weapons proliferation illustrates the emotional states associated with decision makers' positions. Our goal is to draw attention to the centrality of emotion in conflict and to the need for research on the incorporation of emotions into conflict analysis and resolution methods.

The concept of a policy is developed within the framework of the graph model for conflict resolution. More specifically, a policy is defined as an action plan from every possible state in a conflict for each player. A state forms what is called a policy stable state (PSS) if every player in a game does not prefer to move to another state given the other players' policies. The policies that cause a PSS constitute a policy equilibrium. Moreover, a refinement to PSS is suggested in order to take into account credible responses by players. Relationships with existing stability concepts in the graph model for conflict resolution are explored.

Players are assumed to rank each other as coalition partners. Two processes of coalition formation are defined and illustrated:•Fallback (FB): Players seek coalition partners by descending lower and lower in their preference rankings until some majority coalition, all of whose members consider each other mutually acceptable, forms.•Build-up (BU): Same descent as FB, except only majorities whose members rank each other highest form coalitions.BU coalitions are stable in the sense that no member would prefer to be in another coalition, whereas FB coalitions, whose members need not rank each other highest, may not be stable. BU coalitions are bimodally distributed in a random society, with peaks around simple majority and unanimity; the distributions of majorities in the US Supreme Court and in the US House of Representatives follow this pattern. Other examples of real-life coalition processes are discussed.

A comprehensive decision support system, GMCR II, is developed for the systematic study of real-world interactive decision problems. Model formulation is presented in Part I, and analysis and output interpretation in Part II. GMCR II is based upon existing and new research developments of the graph model for conflict resolution. In Part I, specially designed data structures and corresponding algorithms are implemented for generating and representing possible states, removing infeasible states, coalescing indistinguishable states, and specifying and storing allowable state transitions. Algorithms implementing different approaches to the elicitation of preferences over states enable GMCR II to construct and manage an efficient, flexible, and complete graph model of a strategic conflict.

For pt.I see ibid., p.42-55 (2003). The development of a comprehensive decision support system, GMCR II, for the systematic study of real-world interactive decision problems is presented. The companion paper (Part I), discusses how GMCR II elicits, stores, and manages conflict models; here (Part II), the focus is on GMCR IIs analysis and output interpretation subsystems. Specifically, this paper describes the powerful and efficient analysis engine contained in GMCR II, its informative output presentation and interpretation facilities, and a number of follow-up analyses. Furthermore, an illustrative case study is used to demonstrate how GMCR II can be conveniently applied in practice.

Four types of solution concepts for modeling human behavior under conflict are defined to accommodate uncertainty in preferences. For a given decision-maker participating in a conflict the preference structure includes components to handle strict preference of one state over another, equal preference between states, and uncertainty or unknown preference when comparing two states or scenarios. Four distinct ways to incorporate uncertain preference information into the four types of solution concepts are considered and the "sustainable development game" is used to illustrate how the new solution concepts can be applied in practice and strategic insights that can be garnered.

Conflicts associated with fisheries are common and frequently attract significant public attention given the economic and social importance of this sector to coastal communities. The introduction of species-at-risk legislation, clarification of First Nations (aboriginal) fishing rights, globalization, and changing public expectations with respect to conservation and sustainability have significantly complicated the problem of conflict resolution. A number of examples of fisheries conflicts are described, and approaches to resolving them are discussed. Then the recent salmon aquaculture conflict in British Columbia, Canada is examined within the framework of the graph model for conflict resolution. The graph model permits the consideration of how parties in the conflict may strategically interact with one another as a consequence of each party selecting from the options under its control. The participants adopt relative preferences among the possible states or scenarios a priori and in response to the actions taken by other players in the conflict. The relative stability of different solutions provides useful insights into the behavior of the participants and the likelihood of future conflicts.

A novel graphical input system for conflict modeling is designed for use as an interface to the analysis engine of the decision support system GMCR II, which implements the graph model for conflict resolution. The main advantage of the new input system is that it enables the user to draw the conflict model as a graph on the screen, i.e., modeling while drawing. Not only is the graphical input simpler, but it also helps the user to envision the conflict in proper perspective, getting a "big picture" uncluttered by details. Visio 2000 is selected as the graph drawing tool and the modeling is implemented through a Visio 2000 application program. Graphical input should be particularly useful for modeling real-world disputes, especially in early stages when the analyst only has a general idea of the strategic issues. When completed, the full system, Graphical GMCR, is expected to facilitate brainstorming and other techniques to generate creative solutions.

Game theory deals with decision-making processes involving two or more parties, also known as players, with partly or completely conflicting interests. Decision-makers in a conflict must often make their decisions under risk and under unclear or fuzzy information. In this paper, two distinct fuzzy approaches are employed to investigate an extensively studied 2/spl times/2 game model-the game of Chicken. The first approach uses a fuzzy multicriteria decision analysis method to obtain optimal strategies for the players. It incorporates subjective factors into the decision-makers' objectives and aggregates objectives using a weight vector. The second approach applies the theory of fuzzy moves (TFM) to the game of Chicken. The theory of moves (TOM) is designed to bring a dynamic dimension to the classical theory of games by allowing decision-makers to look ahead for one or several steps so that they can make a better decision. TOM is the crisp counterpart of TFM, the approach we implement here to deal with games that include fuzzy and uncertain information. The application of fuzzy approaches to the game of Chicken demonstrates their effectiveness in manipulating subjective, uncertain, and fuzzy information and provides valuable insights into the strategic aspects of Chicken.

The graph model for conflict resolution is put forward as a valuable service tool for providing strategic advice in multiple participant-multiple objective decision making situations. An international trading conflict over the sale of Canadian softwood lumber to customers in the United States is employed for demonstrating how the decision support system GMCR II can be utilized by managers and consultants for better understanding a conflict and its strategic implications. The objective is to furnish clients with insightful advice on how to make sage decisions within a competitive economic environment.

More efficient enforcement of environmental regulations would enhance environmental conditions around the globe by improving compliance to existing regulations and by allowing regulatory agencies to broaden their scope without increasing their budgets. This research analyzes comply-evade decisions by regulated firms in the context of their relationships with environmental agencies, and determines which enforcement policies effectively encourage voluntary compliance. More specifically, the enforcement conflict is modeled as a 2/spl times/2 non-cooperative game called the Enforcement Dilemma. This game is examined as a simultaneous game, and also as a more realistic sequential game with asymmetric information. Then a supergame based on the Enforcement Dilemma is developed and analyzed. An effective long-term policy called the review strategy is proposed to overcome the dilemma in this context.>