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Independence, Additivity, Uncertainty

  • Book
  • © 2003

Overview

Authors:
  1. Karl Vind

    1. Institute of Economics, University of Copenhagen, Copenhagen K, Denmark

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  • New insights into:
  • The mathematics of additive utility and preference functions
  • A formalization of uncertainty
  • Includes supplementary material: sn.pub/extras

Part of the book series: Studies in Economic Theory (ECON.THEORY, volume 14)

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Table of contents (18 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Introduction

    1. Introduction

      • Karl Vind
      Pages 1-6

  3. Basic Mathematics

    1. Front Matter

      Pages 7-7
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    2. Totally preordered sets

      • Karl Vind
      Pages 9-26

    3. Preferences and preference functions

      • Karl Vind
      Pages 27-30

    4. Totally preordered product sets

      • Karl Vind
      Pages 31-38

    5. A subset of a product set

      • Karl Vind
      Pages 39-47

    6. Mean groupoids

      • Karl Vind
      Pages 49-67

    7. Products of two sets as a mean groupoid

      • Karl Vind
      Pages 69-80

  4. Relations on Function Spaces

    1. Front Matter

      Pages 81-81
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    2. Totally preordered function spaces

      • Karl Vind
      Pages 83-111

    3. Relations on function spaces

      • Karl Vind
      Pages 113-129

  5. Relations on Measures

    1. Front Matter

      Pages 131-131
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    2. Relations on sets of probability measures

      • Karl Vind
      Pages 133-146

  6. Integral Representations

    1. Front Matter

      Pages 147-147
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    2. A general integral representation by Birgit Grodal

      • Karl Vind
      Pages 149-167

    3. Special integral representations by Birgit Grodal

      • Karl Vind
      Pages 169-185

  7. Decompositions and Uncertainty

    1. Front Matter

      Pages 187-187
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    2. Decompositions. Uncertainty

      • Karl Vind
      Pages 189-199

    3. Uncertainty on products

      • Karl Vind
      Pages 201-220
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Keywords

About this book

The work on this book started many years ago as an attempt to simplify and unify some results usually taught in courses in mathematical economics. The economic interpretation of the re­ sults were representations of preferences as sums or integrals and the decomposition of preferences into utilities and probabilities. It later turned out that t.he approach taken in the earlier versions were also the proper approach in generalizing from preferences which were total preorders to preferences which were not total or tran­ sitive. The same mathematics would even in that situation give representations which were additive. It would also give decomposi­ tions where concepts of uncertainty appeared. Early versions of some of the results appeared as Working Pa­ pers No. 135, 140, 150, and 176 from The Center for Research in Management Science, Berkeley. A first version of chapters 2, 4, 6, 7, and 8 appeared 1969 with the title" Mean Groupoids" [177]. They are essentially unchanged -except for some notes especially in chapter 6. Another version appeared 1990 as [178]. Chapter 10 contains results from the same versions and from [181]. Chapter 11 by Birgit Grodal is based on [91] by Grodal and Jp,an-Francois Mertens. Chapters 11 and 12 - also by Birgit Gro­ dal -contains the results from the earlier versions, but have been extended (by Karl Vind) to take into account the new corollaries of the results in the other chapters.

Reviews

From the reviews of the first edition:

"The most important contribution of this book is to formalize, simplify and unify concepts of uncertainity that occur in mathematical economics. … This well-written book provides an elegant and serious introduction to the basic concepts and results of representations of preferences as sums and integrals. It discusses in detail the decomposition of preferences into utilities and probabilities. This lively presentation … is impressive. The book will serve as an inspiration for further research and can be recommended … ." (P.R. Parthasarathy, Zentralblatt MATH, Vol. 1080, 2006)

Authors and Affiliations

  • Institute of Economics, University of Copenhagen, Copenhagen K, Denmark

    Karl Vind

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