Fixed Point Theorems with Applications to Economics and Game Theory

One of the problems in economics to which economists have devoted a considerable amount of attention in recent years has been to ensure consistency in the models they employ. Assuming markets to be generally in some state of equilibrium, it must be asked under what circumstances such an equilibrium is possible. The fundamental mathematical tools used to address this concern are fixed point theorems. These outline 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 cooperative and non-cooperative games. Special emphasis is given to economics and games in cases where the preferences of agents may not be transitive. In addition, the author presents new proofs of old results in order to further clarify the results. He also proposes new 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.

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.

Review :
'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