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This cutting-edge book presents the theory and practice of the Graph Model for Conflict Resolution (GMCR), which is used for strategically investigating disputes in any field to enable informed decision making. It clearly explains how GMCR can determine what is the best a particular decision maker (DM) can independently achieve in dynamic interaction with others. Moves and counter-moves follow various stability definitions reflecting human behavior under conflict. The book defines a wide range of preference structures to represent a DM’s comparisons of states or scenarios: equally preferred, more or less preferred; unknown; degrees of strength of preference; and hybrid. It vividly describes how GMCR can ascertain whether a DM can fare even better by cooperating with others in a coalition. The book portrays how a conflict can evolve from the status quo to a desirable resolution, and provides a universal design for a decision support system to implement the innovative decision technologies using the matrix formulation of GMCR. Further, it illustrates the key ideas using real-world conflicts and supplies problems at the end of each chapter. As such, this highly instructive book benefits teachers, mentors, students and practitioners in any area where conflict arises.



Chapter 1. Conflict Resolution in Practice

An encompassing methodology, the Graph Model for Conflict Resolution (GMCR), is applied to a controversial groundwater contamination dispute to demonstrate how to obtain valuable strategic insights that can lead to informed decisions. Because of GMCR’s inherently flexible systems design, both researchers and practitioners can utilize it to investigate conflict in any field. Appreciating the ability of GMCR to rigorously model and analyze actual conflict makes it easier to follow the mathematical developments in later chapters. This book contains many specific contributions, such as structures for representing preference and solution concepts describing human interactions under conflict. It describes both independent and cooperative behavior within GMCR, in both the logical and matrix formulations. A road map will help the reader navigate through these ideas and procedures. A new decision support system captures these recent GMCR developments to realize fully the capability of GMCR to address the broader scope of human strategic conflict. This chapter is essential reading as preparation for each of the other chapters in this book.

Haiyan Xu, Keith W. Hipel, D. Marc Kilgour, Liping Fang

Chapter 2. Decision-Making in Perspective

The crucial role of the Graph Model for Conflict Resolution (GMCR) in making wise decisions to resolve complex real-world problems is put into perspective in this important chapter. First, the relationship of GMCR to approaches in game theory is explained, emphasizing how GMCR has been purposely designed to model and analyze real-world conflict efficiently. Second, GMCR is a powerful decision technology for tackling multiple participant-multiple objective decision problems, complementing other formal decision tools within Operations Research and Systems Engineering. Third, when utilized within a system of systems viewpoint and an integrative and adaptive approach to responsible governance, the many benefits of GMCR enable it to address many of the complex problems challenging society. To appreciate this chapter fully, it should be read in combination with Chaps. 1 and 10. As an introduction to Chaps. 3 to 9, it reminds the reader of the big picture of decision-making, which complements the technical aspects of GMCR.

Haiyan Xu, Keith W. Hipel, D. Marc Kilgour, Liping Fang

Chapter 3. Conflict Models in Graph Form

The basic design of the Graph Model for Conflict Resolution (GMCR) is presented in this chapter, which includes both the logical and matrix formulations of this simple and flexible approach to tackling challenging disputes. The chapter starts by defining the normal and option forms of a game so the reader will fully appreciate their exact connections to GMCR. The Graph Model is then defined in terms of decision makers (DMs), the feasible states or scenarios of the conflict model under study, the movements among states under the control of each particular DM, and the DM’s relative preferences over the states. The transitions among states controlled by each DM are represented by a directed graph in which the vertices represent the states and the arcs the movements between them, explaining some of the GMCR terminology. Because movements in a graph can be recorded using a matrix, the logical ways to express moves, and the relative preferences of states, have equivalent matrix representations. A generic sustainable development conflict and a groundwater contamination dispute are utilized to illustrate the concepts presented in this foundational chapter. The contents of this chapter provide background definitions for additional definitions and theorems developed in Chaps. 4–9.

Haiyan Xu, Keith W. Hipel, D. Marc Kilgour, Liping Fang

Chapter 4. Stability Definitions: Simple Preference

Stability definitions describing a range of human behavior under conflict are presented for decision makers (DMs) having simple preferences in order to establish the concepts of stable states and equilibria in both the logical and matrix formulations of the Graph Model for Conflict Resolution (GMCR). For a DM with simple preference, a state or scenario is either more, less or equally preferred to another state in a conflict model. A state is stable for a DM if all of the DM’s available unilateral improvements can be blocked by specified patterns of countermoves by the other (sanctioning) DMs. The logical interpretation of GMCR, and the idea of interactive moves, are the basis of the four stability definitions presented in this chapter: Nash stability, general metarationality (GMR), symmetric metarationality (SMR) and sequential stability (SEQ). The logical definitions can be converted to equivalent matrix representations, which improve the computational performance of the engine of the decision support system described in Chap. 10. The contents of Chap. 3 provide background modeling for the definitions for this chapter, which in turn constitutes the launching pad for the preference structures built up in Chaps. 5, 6 and 7.

Haiyan Xu, Keith W. Hipel, D. Marc Kilgour, Liping Fang

Chapter 5. Stability Definitions: Unknown Preference

To account for uncertainty in the preferences of a decision maker (DM), the simple preference framework put forward in Chap. 4 is expanded to handle unknown preference, in which a DM simply does not know the preference relationship of one or more pairs of states. The strategic impacts of unknown preferences can be ascertained by employing variations of the earlier stability definitions. As presented in this chapter, Nash, general metarational, symmetric metarational and sequential stability can determine stable states for each DM, as well as the equilibria or potential resolutions, even when some preferences are unknown. They are defined for conflicts with two, or more than two, DMs, for both the logical and matrix representations of the Graph Model for Conflict Resolution (GMCR). A sustainable development dispute and a conflict over the bulk export of water illustrate how the versatile techniques put forward in this chapter can be used to ascertain rigorously the strategic effects of unknown preference. Readers interested in unknown preferences should also refer to the foundational material on simple preference in Chap. 4, as well as Chaps. 6 and 7, which describe approaches to address other preference structures.

Haiyan Xu, Keith W. Hipel, D. Marc Kilgour, Liping Fang

Chapter 6. Stability Definitions: Degrees of Preference

Two nations sharing an oil field spanning their common border may greatly prefer a peaceful division of the benefits to warfare, but may be at an impasse because each is uncertain how seriously the other takes the issue. The objective of this chapter is to expand the concept of simple preference to address situations in which the strength, level or degree of preference is important. To account for the strategic consequences of degree of preference, the definitions of Nash stability, general metarationality (GMR), symmetric metarationality (SMR) and sequential stability (SEQ) must be extended. Logical representations of Nash, GMR, SMR and SEQ are determined for conflicts in which two decision makers (DMs), or more than two, may have any degree of preference. Matrix representations of these four stability definitions are formulated for up to three degrees of preference for any number of DMs. An international environmental conflict is utilized to demonstrate how four degrees of preference can affect strategy choices and outcomes of a dispute. Foundational material on simple preference is furnished in Chap. 4, while other preference structures are described in Chaps. 5 and 7.

Haiyan Xu, Keith W. Hipel, D. Marc Kilgour, Liping Fang

Chapter 7. Stability Definitions: Hybrid Preference

In order to handle simultaneously simple preference, unknown preference and degrees of preference strength within the paradigm of the Graph Model for Conflict Resolution (GMCR), this chapter combines the structures described in the previous three chapters into what is called hybrid preference. Logical and matrix representations of Nash stability, general metarationality (GMR), symmetric metarationality (SMR) and sequential stability (SEQ) are developed for conflicts with hybrid preference. Detailed definitions are provided for three degrees of preference and any number of decision makers (DMs), which is sufficient to handle most conflicts arising in practice. The analysis of a conflict over proposed bulk exports of freshwater from the province of Newfoundland and Labrador, Canada, demonstrates how to determine stability results for graph models including unknown preferences as well as three degrees of preference. Readers of this chapter may wish to refer to Chaps. 4–6 for basic definitions of simple preference, unknown preference, and degrees of preference, respectively.

Haiyan Xu, Keith W. Hipel, D. Marc Kilgour, Liping Fang

Chapter 8. Coalitional Stabilities

The Graph Model for Conflict Resolution (GMCR) paradigm includes a comprehensive approach to coalition analysis that enables the analyst to ascertain whether a decision maker (DM) can fare better by cooperating with other DMs in a coalition, as opposed to acting independently. Both logical and matrix representations of coalitional stabilities are developed for Nash stability, general metarationality, symmetric metarationality and sequential stability under different preference structures: simple (Chap. 4), unknown (Chap. 5), three degrees (Chap. 6), and hybrid (combination of unknown and three degree preference, Chap. 7). For a given preference structure and stability definition, a state is general coalitional stable for a particular coalition if all of the coalition’s improvements can be sanctioned by coalitions of other DMs. The state is universally coalition stable if it is stable for every possible coalition. An application of coalition analysis to the controversy over the proposed bulk export of water from Lake Gisborne in the province of Newfoundland and Labrador, Canada, demonstrates the strategic insights that can be garnered using coalition analysis. There are good reasons to include coalition analysis in any conflict study, and therefore to embed it within a decision support system for GMCR (Chap. 10).

Haiyan Xu, Keith W. Hipel, D. Marc Kilgour, Liping Fang

Chapter 9. Follow-Up Analysis: Conflict Evolution

This chapter is devoted to algorithms that trace the paths a conflict could follow as it evolves from a selected starting state, often the status quo, to a state of interest, such as an attractive resolution. For instance, if a group of decision makers (DMs) could make a sequence of individual unilateral improvements to achieve an outcome they all prefer, it would be a win/win resolution for all of them, and therefore in their interests to do so. Path-following algorithms are developed within logical and two matrix representations of the Graph Model for Conflict Resolution (GMCR) under four different preference structures: simple (Chap. 4), unknown (Chap. 5), three degree (Chap. 6) and hybrid (unknown combined with three degree, Chap. 7). The value of the logical approach is its ability to explain how the evolution can occur, while the matrix approach facilitates the calculations. To illustrate the algorithms, they are applied to graph models of three real-world case studies: a groundwater contamination dispute, a conflict arising over the proposed bulk export of fresh water, and an international conflict over the development of a large-scale irrigation system.

Haiyan Xu, Keith W. Hipel, D. Marc Kilgour, Liping Fang

Chapter 10. Design of a Decision Support System for Conflict Resolution

In this chapter, a universal design for a Decision Support System (DSS) for the Graph Model for Conflict Resolution (GMCR) is proposed. Its objective is to permit researchers and practitioners to take full advantage of the rich range of capabilities of GMCR developed in this book and elsewhere. Using highly informative graphs, the overall design of the DSS is described, along with its three main subsystems: input, analysis engine, and output. The modular design allows the initial basic system to be extended over time as new definitions are developed and users’ needs for additional capabilities are established. For instance, an initial DSS implementation based on a matrix formulation of simple preference (Chap. 4) will permit the analysis engine to compute noncooperative (Chap. 4) and cooperative (Chap. 8) behavior efficiently. Like a lego structure, the DSS will allow for the later incorporation of additional components, such as unknown preference (Chap. 5) and conflict evolution (Chap. 9). Many of the exciting new frontiers for GMCR referred to in Sect. 10.3, such as inverse engineering to identify the preferences required to reach and stabilize a desirable resolution, can be operationalized within this proposed DSS.

Haiyan Xu, Keith W. Hipel, D. Marc Kilgour, Liping Fang


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