An algebraic approach to calculating stabilities in the graph model with strength of preference

An algebraic approach is developed to calculate stabilities in two decision maker graph models with strength of preference. The original graph model uses ¿simple preference¿ to represent a decision maker's relative preference between two states. This preference structure includes only a relative preference relation and an indifference relation. Basic stability definitions, and algorithms to calculate them, assume simple preference. But difficulties in coding the algorithms, mainly because of their logical formulation, led to the development of matrix representations of preference and explicit matrix algorithms to calculate stability. Here, the algebraic approach is extended to representation of strength-of-preference graph models, which feature multiple levels of preference, and stability analysis for such models. Matrix representation of stability definitions facilitates the development of new stability concepts and algorithms to calculate them. The method is illustrated using a simple model of a conflict over sustainable development.