Matrix representation of stability definitions for the graph model for conflict resolution with reciprocal preference relations

Abstract

Within the framework of the graph model, a matrix formulation is developed to model and analyze conflicts in which decision makers (DMs) may have reciprocal preferences. Specifically, matrix expressions are employed to represent DMs' reciprocal preference relations, unilateral movements (UMs) and fuzzy unilateral improvements (FUIs), as well as joint UMs and joint FUIs for a coalition of two or more DMs. Furthermore, a matrix methodology is provided to calculate whether a state in a conflict model, or scenario, is stable for a particular DM under various solution concepts, or stability definitions, that reflect the diversity of possible behavioral patterns for a DM in a conflict when preferences can be reciprocal. Five solution concepts associated with reciprocal preferences, Fuzzy Nash Stability, Fuzzy Symmetric Metarationality, Fuzzy General Metarationality, Fuzzy Sequential Stability, and Fuzzy Symmetric Sequential Stability, are redefined for matrix representations of both two-DM and multiple-DM conflict models. To illustrate how the matrix representation can be conveniently employed in practice, it is applied to two real-world conflicts.