Joint pseudo-utility representations (Q1206879)

Let \(X\) be a non-void set, \(P\) a binary relation on \(X\). The pseudo- utility representation (PUR) of \(P\) is a real-valued function on \(X^ 2\), the support of which equals the graph of \(P\). Let \(T\) and \(X\) be topological spaces, \(\mathcal K\) a family of subsets of \(X\), \(\Pi = \{\Pi_ t\mid t\in T\}\) an indexed family of relations \(\Pi_ t \subseteq X^ 2\). A joint PUR is a function \(\pi: T \times X^ 2 \to \mathbb{R}\) such that \(\pi_ t\) is a PUR of \(\Pi_ t\). The paper deals with the problem to find conditions for the existence of a continuous joint PUR such that for all \(t\in T\), \(x \in X\), and all real \(\alpha\), \(\{x'\mid \pi_ t(x,x') \geq \alpha\}\) belongs to \(\mathcal K\).