# Applications/generalizations of a theorem of Debreu

I would like to know how the last theorem in Debreu's paper "Neighboring economic agents" (La Decision 171 (1969): 85-90; reprinted in G. Debreu, Mathematical Economics: Twenty Papers of Gerard Debreu (1986), pp. 173-178) has been used:

Theorem. For a topological space $M$ and a metric space $H$, let $\varphi$ be a set-valued mapping from $M$ to $H$ that is compact-valued (i.e. $\varphi(e)$ is compact for every $e \in M$) and continuous. Further, for each $e \in M$ let $\lesssim_e$ be a total preorder on $\varphi(e)$ such that the set $\{(e, x, y) \in M \times H \times H : x \lesssim_e y\}$ is closed. Then the set-valued mapping $\varphi^0$ from $M$ to $H$ where

$\varphi^0(e) = \{z \in \varphi(e) : x \lesssim_e z \ \ \mbox{for all} \ x \in \varphi(e)\}, \quad e \in M,$

is compact-valued and upper hemi-continuous.

Note that the theorem looks similar to the well-known Berge Maximum Theorem. Prior to the statement of the theorem, Debreu writes that special cases of it "have been repeatedly used in the theory of economic equilibrium and in game theory", but doesn't give any references; in the paper itself, it's used to prove the upper hemi-continuity of the demand correspondence for an agent in an exchange economy.

I'm especially interested in whether there have been any recent uses or generalizations of this theorem, e.g. to mappings that are not compact-valued.

Questions: What are some good examples of and/or references for applications of the above theorem? Has it been generalized to mappings that are not compact-valued?

This result is indeed a version of Berge's maximum theorem. If there is a continuous function $u:M\times H\to\mathbb{R}$ such that $x\preceq_e z$ if and only if $u(e,x)\leq u(e,z)$, one can derive the result directly from Berge's maximum theorem. If $H$ is locally compact, as it is the case if $H=\mathbb{R}^n$, then such a function can always be found, this follows from Theorem 1 in Mas-Colell's On the Continuous Representation of Preorders (at least if $M$ is metrizable, I'm not sure on that point). More on such "jointly continuous utility functions" can be found in chapter 8 of Representations of preference orderings, 1995, by Bridges & Mehta.