Matrix Representation of Solution Concepts in Multiple-Decision-Maker Graph Models

A matrix method is developed to apply four solution concepts to a graph model for conflict resolution in order to model human behavior in a multiple-decision-maker (DM) strategic conflict. A graph model represents the interaction of two or more DMs and can be analyzed to identify possible equilibrium states according to each solution concept. Previously, solution concepts were defined logically, in terms of the underlying graphs. However, procedures to identify stable states based on these definitions are difficult to code because of the nature of the logical representations. In this paper, a graph model and four graph model solution concepts are formulated explicitly using matrices. More specifically, matrix expressions are given for relative preferences, joint unilateral movements, and joint unilateral improvements in a multiple-DM model. Then, it is shown how to calculate stability under each of the four solution concepts using the matrix representation. Compared with the existing approach, matrix representation is more effective and convenient for calculating stabilities and predicting equilibria of a graph model. In particular, the proposed method is easy to code and extend to other contexts.