On the jointly continuous multi-utility representation problem

One of the main problems in Decision Theory is to represent preference relations (or preorders) by means of continuous utility functions. We are interested in the existence of the multi-utility representations, jointly continuous in consumption sets and preferences, of a family P of non-total (partial) preorders defined on closed subsets of a topological space X. Non-total preorders are usually considered in the context of decision-making under uncertainty and risk. The continuous multi-utility representation of a non-total preorder is more functional than the classical Richter-Peleg utility representation as it allows a complete characterization of the preorder. The research in this paper can be described as a jointly continuous multi-utility representation problem. Problems of this type are important in Mathematical Economics, where the closed subsets of X are the consumption sets and P can be considered as a family of preference relations related to a set of economics agents.