LAUSR

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.