Prioritising Preference Relations

We describe some ideas and results about the following problem: Given a set, a family of \preference relations" on the set, and a \priority" among those preference relations, which elements of the set are best? That is, which elements are most preferred by a consensus of the preference relations which takes account of their relative priority? The problem is posed in a deliberately general way, to capture a wide variety of examples. Our main result gives suucient conditions for the existence of`best' elements for an important instance of the problem: preference relations are pre-orders, the priority among them is a partial order, and the definition of best elements uses a generalisation of lexicographic ordering.