Preference Modeling with Possibilistic Networks and Symbolic Weights: A Theoretical Study

The use of possibilistic networks for representing conditional preference statements on discrete variables has been proposed only recently. The approach uses non-instantiated possibility weights to define conditional preference tables. Moreover, additional information about the relative strengths of these symbolic weights can be taken into account. The fact that at best we have some information about the relative values of these weights acknowledges the qualitative nature of preference specification. These conditional preference tables give birth to vectors of symbolic weights that reflect the preferences that are satisfied and those that are violated in a considered situation. The comparison of such vectors may rely on different orderings: the ones induced by the product-based, or the minimum-based chain rule underlying the possibilistic network, the discrimin, or leximin refinements of the minimum-based ordering, as well as Pareto ordering, and the symmetric Pareto ordering that refines it. A thorough study of the relations between these orderings in presence of vector components that are symbolic rather numerical is presented. In particular, we establish that the product-based ordering and the symmetric Pareto ordering coincide in presence of constraints comparing pairs of symbolic weights. This ordering agrees in the Boolean case with the inclusion between the sets of preference statements that are violated. The symmetric Pareto ordering may be itself refined by the leximin ordering. The paper highlights the merits of product-based possibilistic networks for representing preferences and provides a comparative discussion with CP-nets and OCF-networks.