Fuzzy Preferences in the Graph Model for Conflict Resolution

Abstract

A Fuzzy Preference Framework for the Graph Model for Conflict Resolution (FGM) is developed so that real-world conflicts in which decision makers (DMs) have uncertain preferences can be modeled and analyzed mathematically in order to gain strategic insights. The graph model methodology constitutes both a formal representation of a multiple participant-multiple objective decision problem and a set of analysis procedures that provide insights into them. Because crisp or definite preference is a special case of fuzzy preference, the new framework of the graph model can include---and integrate into the analysis---both certain and uncertain information about DMs' preferences. In this sense, the FGM is an important generalization of the existing graph model for conflict resolution.

One key contribution of this study is to extend the four basic graph model stability definitions to models with fuzzy preferences. Together, fuzzy Nash stability, fuzzy general metarationality, fuzzy symmetric metarationality, and fuzzy sequential stability provide a realistic description of human behavior under conflict in the face of uncertainty. A state is fuzzy stable for a DM if a move to any other state is not sufficiently likely to yield an outcome the DM prefers, where sufficiency is measured according to a fuzzy satisficing threshold that is characteristic of the DM. A fuzzy equilibrium, an outcome that is fuzzy stable for all DMs, therefore represents a possible resolution of the conflict. To demonstrate their applicability, the fuzzy stability definitions are applied to a generic two-DM sustainable development conflict, in which a developer plans to build or operate a project inspected by an environmental agency. This application identifies stable outcomes, and thus clarifies the necessary conditions for sustainability. The methodology is then applied to an actual dispute with more than two DMs concerning groundwater contamination that took place in Elmira, Ontario, Canada, again uncovering valuable strategic insights.

To investigate how DMs with fuzzy preferences can cooperate in a strategic conflict, coalition fuzzy stability concepts are developed within FGM. In particular, coalition fuzzy Nash stability, coalition fuzzy general metarationality, coalition fuzzy symmetric metarationality, and coalition fuzzy sequential stability are defined, for both a coalition and a single DM. These concepts constitute a natural generalization of the corresponding non-cooperative fuzzy preference-based definitions for Nash stability, general metarationality, symmetric metarationality, and sequential stability, respectively. As a follow-up analysis of the non-cooperative fuzzy stability results and to demonstrate their applicability, the coalition fuzzy stability definitions are applied to the aforementioned Elmira groundwater contamination conflict. These new concepts can be conveniently utilized in the study of practical problems in order to gain strategic insights and to compare conclusions derived from both cooperative and non-cooperative stability notions.

A fuzzy option prioritization technique is developed within the FGM so that uncertain preferences of DMs in strategic conflicts can be efficiently modeled as fuzzy preferences by using the fuzzy truth values they assign to preference statements about feasible states. The preference statements of a DM express desirable combinations of options or courses of action, and are listed in order of importance. A fuzzy truth value is a truth degree, expressed as a number between 0 and 1, capturing uncertainty in the truth of a preference statement at a feasible state. It is established that the output of a fuzzy preference formula, developed based on the fuzzy truth values of preference statements, is always a fuzzy preference relation. The fuzzy option prioritization methodology can also be employed when the truth values of preference statements at feasible states are formally based on Boolean logic, thereby generating a crisp preference over feasible states that is the same as would be found using the existing crisp option prioritization approach. Therefore, crisp option prioritization is a special case of fuzzy option prioritization. To demonstrate how this methodology can be used to represent fuzzy preferences in real-world problems, the new fuzzy option prioritization technique is applied to the Elmira aquifer contamination conflict. It is observed that the fuzzy preferences obtained by employing this technique are very close to those found using the rather complicated and tedious pairwise comparison approach.