MA3486 - Fixed Point Theorems and Economic Equilibria
Dr. David R. Wilkins
Internet Resources related to Correspondences,
Hemicontinuity and Berge's Maximum Theorem.
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.
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
x_{1} < x < x_{2}
and
x_{3} < x < x_{4}
are shaded in grey, and are are part of the graph of
the correspondence.
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.
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.