Matrix Representation of Solution Concepts in the Graph Model for Conflict Resolution with Probabilistic Preferences and Multiple Decision Makers

In this paper, matrix methods are developed to determine stable states in the graph model for conflict resolution (GMCR) with probabilistic preferences with n decision makers. The matrix methods are used to determine more easily the stable states according to five stability definitions proposed for this model, namely: $$\alpha $$ α -Nash stability, ( $$\alpha $$ α , $$\beta $$ β )-metarationality, ( $$\alpha $$ α , $$\beta $$ β )-symmetric metarationality, ( $$\alpha $$ α , $$\beta $$ β , $$\gamma $$ γ )-sequential stability and ( $$\alpha $$ α , $$\beta $$ β , $$\gamma $$ γ )-symmetric sequential stability. With the help of such methods, we are able to analyze for which values of parameters $$\alpha $$ α , $$\beta $$ β and $$\gamma $$ γ the states satisfy each one of these stability notions. These parameters regions can be used to compare the equilibrium robustness of the states. As a byproduct of our method, we point out an existing problem in the literature regarding matrix representation of solution concepts in the GMCR.