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Equilibrium, fixed-points and computations

Teacher: Philippe Bich

E-mail: [email protected]
ECTS : 2.5
Evaluation:  written Exam or project. 
Previsional Place and time:  ROOM S2 of MSE, 15H30-17H, 12 sessions from January  to march. 

Prerequisites:  Logic and Set Theory. Analysis in finite dimensional spaces (compact subsets, open subsets, closed subsets, metric,  sequences, continuity, …),  convexity.
Aim of the course: During the last 30 years, fixed-point theory has entailed important progresses in Economic Theory, Finance, Game Theory, Decision Theory, Network theory… This course covers basic fixed-point methods that interact with these fields. We shall cover Three important questions: Existence, Uniqueness (or number of equilibria) and computation (of equilibria or of fixed-points), using several kind of methods. 
Syllabus: 
1)    Introduction; the fixed-point property.  Deformation retract, homeomorphism. 
2)    Topological degree (definition, properties, applications). 
3)    Brouwer fixed-point theorem; existence of zero for inward vector fields. 
4)    Sperner Lemma; proof of Brouwer; computation. 
5)    Multivalued functions,  Kakutani’s Theorem, existence of a maximal element. 
6)    Schauder’s theorem and the infinite dimensional case.
7)    Ordered fixed-point theorems. 
8)    Banach Fixed-point theorem and some applications. 
9)    Examples, applications (depending on time) 
 References: Lecture notes will be given. See also: “Fixed-point theory” Granas-Dugundji. 

Examen de l'année 2022: 
exam-point-fixe-2022.pdf
File Size: 78 kb
File Type: pdf
Download File

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  • Home
  • Courses
  • Timetable
  • Opportunities
  • Applying
  • Contact
  • Internship
  • Optimal transport
  • Algorithmic game theory
  • Neural network