Fixed Point Theorems with Applications to Economics and Game Theory

This book explores fixed point theorems and its uses in economics, co-operative and noncooperative games.

Kim C. Border (Author)

9780521388085, Cambridge University Press

Paperback, published 28 July 1989

140 pages
22.5 x 15.2 x 0.9 cm, 0.22 kg

'This book provides a clear exposition of the most important results and techniques of fixed point theory with applications to many areas of current interest in analysis … It is a well written book … the material is completed by well collected exercises at the end of every chapter. We have to emphasize also, the accuracy of applications to differential mathematicians interested in functional analysis, operator theory, differential equations and also to students for the first view on the fixed point theory.' Acta Scientiarum Mathematicarum

One of the problems in economics that economists have devoted a considerable amount of attention in prevalent years has been to ensure consistency in the models they employ. Assuming markets to be generally in some state of equilibrium, it is asked under what circumstances such equilibrium is possible. The fundamental mathematical tools used to address this concern are fixed point theorems: the conditions under which sets of assumptions have a solution. This book gives the reader access to the mathematical techniques involved and goes on to apply fixed point theorems to proving the existence of equilibria for economics and for co-operative and noncooperative games. Special emphasis is given to economics and games in cases where the preferences of agents may not be transitive. The author presents topical proofs of old results in order to further clarify the results. He also proposes fresh results, notably in the last chapter, that refer to the core of a game without transitivity. This book will be useful as a text or reference work for mathematical economists and graduate and advanced undergraduate students.

Preface
1. Introduction: models and mathematics
2. Convexity
3. Simplexes
4. Sperner's lemma
5. The Knaster-Kuratowski-Mazurkiewicz lemma
6. Brouwer's fixed point theorem
7. Maximization of binary relations
8. Variational inequalities, price equilibrium, and complementarity
9. Some interconnections
10. What good is a completely labelled subsimplex?
11. Continuity of correspondences
12. The maximum theorem
13. Approximation of correspondence
14. Selection theorems for correspondences
15. Fixed point theorems for correspondences
16. Sets with convex sections and a minimax theorem
17. The Fan-Browder theorem
18. Equilibrium of excess demand correspondences
19. Nash equilibrium of games and abstract economies
20. Walrasian equilibrium of an economy
21. More interconnections
22. The Knaster-Kuratowski-Mazurkiewicz-Shapley lemma
23. Cooperative equilibria of games
References
Index.

Subject Areas: Economic theory & philosophy [KCA]