Fixed Point Theorems with Applications to Economics and Game Theoryby Kim C. Border
Pub. Date: 07/28/1989
Publisher: Cambridge University Press
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… See more details below
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.
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Table of Contents
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.
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