Applications/generalizations of a theorem of Debreu

Question

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?

Answer 1

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.

Now Debreu did not have such a result available, so he worked with preference relations and essentially reproved Berge's maximum theorem (the generalization is mathematically straightforward). Why did he do so? To understand that, one needs to understand the point of Debreu's paper, which is finding a topology on preference relations that has nioce properties and makes economic behavior continuous. The need for such a result comes from the literature on economies with a continuum of agents.

What does it mean that a continuum of agents economy is the limit of a sequence of finite eonomies? One answer is that the distribution on characteristics of agents converges to the distribution of characteristics in the continuum economy, so the notion of convergence is convergence in distribution. To make this idea operational, one needs to topologize the characteristics of agents. Now an agent is characterized by her endowment and by her preferences (and in more general models by her consumption set). There is a natural topology on endowments, the Euclidean topology, but it is less straightforward to topologize preferences, and that is what Debreu did in his paper. An exposition of this distributional approach can be found in Hildenbrand 1974, Core and equilibria of a large economy.

Now, there are cases where one would like to apply Berge's theorem for non-compact sets of choices. This can be important when studying economies with infinite dimensional commodity spaces, in which being closed and bounded does not imply compactness. One way to deal with this problem is to find a compact set so that the correspondence is compact-valued and nonempty-valued when restricted to this set. There is a large, very technical, literature on "generalized games" or "abstract economies" (basically normalform games in which strategy spaces depend on the actions of others), and they implicitely often contain non-compact generalizations of Berge's theorem. If you can get your hands on the book, check chapter 4 of Xian-Zhi Yuan 1999, KKM Theory and Applications in Nonlinear Analysis. My impression, however, is that these results proved to be not that useful in economic applications. To prove existence of Walrasian equilibria in models with infinite dimensional commodity spaces, one usually uses different methods.