Combining preference relations: completeness and consistency

We introduce two criteria for judging "goodness" of the re- sult when combining preference relations in information sys- tems: completeness and consistency. Completeness requires that the result must be the union of all preference relations, while consistency requires that the result must be an acyclic relation. In other words, completeness requires that the re- sult contain all pairs appearing in the preference relations, and only those pairs; while consistency requires that for ev- ery pair (x, y) in the result, it must be able to decide which of x and y is preferred to the other. Obviously, when com- bining preference relations, there is little hope for the re- sult to satisfy both requirements. In this paper, we clas- sify the various methods for combining preference relations, based on the degree to which the result satisfies complete- ness and consistency. Our results hold independently of the nature of preference relations (quantitative or qualitative); and also independently of the preference elicitation method (i.e. whether the preference relations are obtained by the system using query-log analysis or whether the user states preferences explicitly). Moreover, we assume no constraints whatsoever on the preference relations themselves (such as transitivity, strict ordering and the like).