The Computation of Fixed Points and Applications

1. Front Matter

   Pages I-VII

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2. Brouwer’s Theorem

   • Michael J. Todd

   Pages 1-16

3. Some Applications of Brouwer’s Theorem

   • Michael J. Todd

   Pages 17-23

4. Triangulations

   • Michael J. Todd

   Pages 24-39

5. Algorithms to Find Completely-Labelled Simplices

   • Michael J. Todd

   Pages 40-53

6. Extensions of Brouwer’s Theorem

   • Michael J. Todd

   Pages 54-63

7. Applications of Kakutani’s Theorem and its Extensions

   • Michael J. Todd

   Pages 64-72

8. Eaves’ First Algorithm

   • Michael J. Todd

   Pages 73-81

9. Merrill’s Algorithm

   • Michael J. Todd

   Pages 82-93

10. Homotopy Algorithms

    • Michael J. Todd

    Pages 94-100

11. Triangulations with Continuous Refinement of Grid Size

    • Michael J. Todd

    Pages 101-115

12. Measures of Efficiency for Triangulations

    • Michael J. Todd

    Pages 116-125

13. Back Matter

    Pages 126-132

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Fixed-point algorithms have diverse applications in economics, optimization, game theory and the numerical solution of boundary-value problems. Since Scarf's pioneering work [56,57] on obtaining approximate fixed points of continuous mappings, a great deal of research has been done in extending the applicability and improving the efficiency of fixed-point methods. Much of this work is available only in research papers, although Scarf's book [58] gives a remarkably clear exposition of the power of fixed-point methods. However, the algorithms described by Scarf have been super~eded by the more sophisticated restart and homotopy techniques of Merrill [~8,~9] and Eaves and Saigal [1~,16]. To understand the more efficient algorithms one must become familiar with the notions of triangulation and simplicial approxi- tion, whereas Scarf stresses the concept of primitive set. These notes are intended to introduce to a wider audience the most recent fixed-point methods and their applications. Our approach is therefore via triangu- tions. For this reason, Scarf is cited less in this manuscript than his contri- tions would otherwise warrant. We have also confined our treatment of applications to the computation of economic equilibria and the solution of optimization problems. Hansen and Koopmans [28] apply fixed-point methods to the computation of an invariant optimal capital stock in an economic growth model. Applications to game theory are discussed in Scarf [56,58], Shapley [59], and Garcia, Lemke and Luethi [24]. Allgower [1] and Jeppson [31] use fixed-point algorithms to find many solutions to boundary-value problems.

• Computation
• Invariant
• economic growth
• economics
• game theory

• School of Operations Research and Industrial Engineering, Cornell University, Ithaca, USA

  Michael J. Todd

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