Continuous Representation of Preferences

The problem of constructing a representation of preferences appears when it is needed to assign utility functions to preferences in a well-defined way, since in general many utilities will represent the same preference. In models where the preferences of the agents vary, and the behaviour of the model depends on limiting properties of preferences, it is important that utilities be assigned to preferences so as to vary continuously as preferences change. In this paper we study a utility representation for preferences, and we prove its continuity, using a topology for preferences introduced by Chichilnisky (1977). Such utility representations were constructed under restrictive conditions first by Y. Kannai (1970), W. Hildenbrand (1970), W. Neuefeind (1972) and more recently by K. Mount and S. Reiter (1974, 1975). Our results are related more closely to those of Mount and Reiter (1974 and 1975). Starting from restrictive conditions, these works have attempted to enlarge the class of preferences that could be continuously represented. The class of preferences studied here is much larger than those considered earlier, and it therefore applies to a wider class of problems. Our preferences include ones which are not necessarily convex or monotone, and which may be locally satiated; furthermore, no completeness of the preferences is required. The assumption made in Mount and Reiter (1975) of the existence of an ez-threshold is also not required here. These results are possible due to the properties of the order topology introduced in Chichilnisky (1977). As we shall now discuss, this topology has quite desirable features for the study of preferences, and these make it a natural choice for the problem at hand. The order topology is liner than the Hausdorlf metric, and, as opposed to the Hausdorff metric, it is sensitive to the measure of the graphs of the preferences. This sensitivity is the main property that allows one to prove continuity of the representation, which is based on measure