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Fixed Point Theorems with Applications to Economics and Game Theory

Fixed Point Theorems with Applications to Economics and Game Theory

$67.00 (P)

  • Date Published: July 1989
  • availability: Available
  • format: Paperback
  • isbn: 9780521388085

$ 67.00 (P)
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About the Authors
  • 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.

    Reviews & endorsements

    "The book cleanly separates the mathematical theory from the economic applications. Mathematicians, as well as economists, will value this book as a basic, helpful handbook on finite-dimensional fixed-point theorems." Joel Franklin, SIAM Review

    "Perhaps the book's greatest strength is that the level of mathematical sophistication it requires is uniform throughout." Timothy J. Kehoe, Journal of Economic Literature

    "The organization of the material is novel and interesting and should make the book a useful reference work and perhaps a textbook for a rigorous class on static equilibrium theory of finite competitive economics." Erwin Klein, Mathematical Reviews

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    Product details

    • Date Published: July 1989
    • format: Paperback
    • isbn: 9780521388085
    • length: 140 pages
    • dimensions: 225 x 152 x 9 mm
    • weight: 0.22kg
    • availability: Available
  • 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.

  • Author

    Kim C. Border, California Institute of Technology

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