Searching for the dimension of valued preference relations

A new concept of dimension is proposed, taking into account inconsistencies in source of information, and a general representation theorem for arbitrary crisp binary relations is obtained, showing the difference in representation between incomparability-related to the intersection operator-and other inconsistencies to the union operator.

Abstract

The more information a preference structure gives, the more sophisticated representation techniques are necessary, so decision makers can have a global view of data and therefore a comprehensive understanding of the problem they are faced with. In this paper we propose to explore valued preference relations by means of a search for the number of underlying criteria allowing its representation in real space. A general representation theorem for arbitrary crisp binary relations is obtained, showing the difference in representation between incomparability––related to the intersection operator––and other inconsistencies––related to the union operator. A new concept of dimension is therefore proposed, taking into account inconsistencies in source of information. Such a result is then applied to each α-cut of valued preference relations.