# MA3486 - Fixed Point Theorems and Economic Equilibria Dr. David R. Wilkins Internet Resources related to Correspondences, Hemicontinuity and Berge's Maximum Theorem.

## Lecture Notes

Lecture 6: Continuity of Correspondences and Berge's Maximum Theorem (Mathematics for Economics, Reihhard John, Universität Bonn)
Results are stated without proofs. There are a couple of examples at the end.
Lecture 5: Correspondences and Berge's Maximum Theorem (Math Camp, Department of Economics, Yale University)
This is the lecture from the lecture in the Math Camp of the graduate economics programme at Yale University that deals with correspondences. Note that these notes adopt a definition of upper semicontinuity that is equivalent to the definition of upper hemicontinuity adopted in MA3486 for compact-valued correspondences, but is not necessarily equivalent for correspondences that are not compact-valued.

## Videos

06-1 Continuity of Correspondences (Rajiv Sethi, YouTube)
This video is part of a lecture course entitled Mathematical Methods for Economists delivered by Rajiv Sethi, Ann Whitney Olin Professor of Economics in the Department of Economics of Barnard College, Columbia University. This video is the first of a series of videos of segments of a lecture course covering correspondences. The lectures in this course have been made available as a playlist on YouTube entitled Mathematical Methods for Economics, made available by courtesy of the Columbia University Economics Department. This video segment covers the definition of a correspondence, the definition of upper hemicontinuity and the definition of lower hemicontinuity, and gives a detailed example to illustrate these concepts.
Notes:
• In the definition of a correspondence, note that Prof. Sethi requires the values of a correspondence from a set X to a set Y to be non-empty subsets of Y. In TCD module MA3486, we do not require correspondences to be non-empty-valued. Conventions appear to vary between different textbooks and lecture courses worldwide with regard to whether the values of a correspondence are required to be non-empty-valued.
• Note that in the example discussed the regions of the graph between the black lines above the intervals x1 < x < x2 and x3 < x < x4 are shaded in grey, and are are part of the graph of the correspondence.
06-2 Continuity of Correspondences (Rajiv Sethi, YouTube)
This video is the sequel to that listed immediately above. It discusses, without proofs, the following two theorems:---
• if a correspondence has closed graph, and if the codomain is compact, then the correspondence is upper hemicontinuous;
• if a correspondence is upper hemicontinuous and closed-valued, then the correspondence has a closed graph.
• The lecture presents simple counter-examples that result when the hypotheses of the above theorems are relaxed. The lecture continues with a statement, without proof, of the Kakutani Fixed Point Theorem, and the lecturer explains how John Nash deduced the existence of equilibrium points in n-person games, quoting extensively from Nash's original paper.
07-1 The Theorem of the Maximum (Part I) (Rajiv Sethi, YouTube)
This video is the sequel to that listed immediately above. It states the Berge Maximum Theorem (The Theorem of the Maximum), and proves part of the theorem.
07-1 The Theorem of the Maximum (Part II) (Rajiv Sethi, YouTube)
This video is the sequel of the preceding and completes proof of Berge's Maximum Theorem

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