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2003 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Independence, Additivity, Uncertainty

verfasst von: Professor Karl Vind

Verlag: Springer Berlin Heidelberg

Buchreihe : Studies in Economic Theory

Enthalten in: Professional Book Archive

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SUCHEN

Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Introduction

1. Introduction
Abstract
The most important contribution of this book to economic theory is to formalize a concept of uncertainty. Ideas about uncertainty and this concept’s importance for economics go back at least to Keynes (1921) [106] and possibly Knight (1921) [110]. The ideas have been rejected with the argument that the concept could not be made precise and therefore not included in serious economic theory. To the extend that what is meant by uncertainty is that agents can not compare any two alternatives, uncertainty can be represented by real numbers just like utility, probability and preference.
Karl Vind

Basic Mathematics

Frontmatter
2. Totally preordered sets
Abstract
The central topic in this chapter is totally preordered sets. Section 2.2 gives conditions for the existence of order homomorphisms between totally preordered sets and subsets of the real numbers. Some topological concepts are introduced in section 2.3 page 18. A total order gives rise to a topology and the main results from section 2.2 are in section 2.4 given in topological terms.
Karl Vind
3. Preferences and preference functions
Abstract
Chapter 2 treats totally ordered sets and gives representation theorems. Similar theorems for just relations — not assumed to be total — are trivial, but are convenient to have, because the main results in this book give conditions for particular additive representation.
Karl Vind
4. Totally preordered product sets
Abstract
The study of totally preordered product sets starts in this chapter. Section 4.3 contains definitions and some results concerning topological properties of totally preordered product sets. The main result is that a weak connectednes property (c.2)) combined with a strong continuity property (the product topology finer than the order topology) is equivalent to a strong connectedness property (c.3) combined with a weak continuity property (called continuity).
Karl Vind
5. A subset of a product set
Abstract
This chapter begins the study of the simplest possible structure on a product set.
Karl Vind
6. Mean groupoids
Abstract
The main problem in chapter 2 was the study of a set S with a total preorder \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \succ } \). This chapter studies a set S with an order relation ≻ and an algebraic operation o on S. Intuitively a o b can be thought of as the mean of or the midpoint between a and b, a o a = a, and if a ≻ b, then a ≻ a o b ≻ b. Later chapters will be concerned with totally preordered sets that have enough structure to define a mean operation on ((S/,~),≻).
Karl Vind
7. Products of two sets as a mean groupoid
Abstract
The problem in this chapter is to study a totally preordered product set (X x Y, \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \succ } \)) 1. In chapter 4 theorem 10 page 37 the existence of a real representation is proved.
Karl Vind

Relations on Function Spaces

Frontmatter
8. Totally preordered function spaces
Abstract
The totally preordered set in this chapter is a set g of functions g: X→ Y, where (X,A) is a measurable set and Y an arbitrary set. Section 8.2 gives definitions and notation.
Karl Vind
9. Relations on function spaces
Abstract
The equivalence theorem (theorem 11, chapter 5) can now be proved by combining the results from chapters 5 and 8. The results in chapter 8 combined with the equivalence theorem yields representation results for first an independent subset of a function space and then for a relation on a function space. (Sections 9.2 and 9.3). As in chapter 8 several special cases are of interest. Section 9.4 covers the case where X is a finite set. Y = {0, 1} gives subjective uncertainty in section 9.5. The minimal independence assumptions are given in section 9.6. And finally the consequences of special properties of the ranges of the functions for special representations are given in section 9.7.
Karl Vind

Relations on Measures

Frontmatter
10. Relations on sets of probability measures
Abstract
If it is assumed that the domain X of the functions g, H is not just a measurable space but a measure space, it becomes possible to assume that relations on g × H only depend on the distribution of (g, h). In this special case representation theorems can be obtained without using the machinery of the previous parts of the book. The conclusions of the theorems obtained are very similar to those from the earlier representation theorems, and some of the later integral representation theorems and decompositions can be used also on the representations from this chapter. These results are therefore included in this book even if the mathematics used is different — the Hahn-Banach theorem or just separating hyperplanes.
Karl Vind

Integral Representations

Frontmatter
11. A general integral representation by Birgit Grodal
Abstract
Chapter 8 studied a totally preordered set \((g,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\succ})\) of functions g : X → Y, where (X, A) is a measurable set and Y an arbitrary set. g was under an independence condition shown to be a commutative mean groupoid \((g,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \succ } ,o)\).
Karl Vind
12. Special integral representations by Birgit Grodal
Abstract
In chapter 11 we gave conditions on \( (g,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \succ } )\) which implied the existence of a measurable function.
Karl Vind

Decompositions and Uncertainty

Frontmatter
13. Decompositions. Uncertainty
Abstract
Agents in an economy, a game, or any other social system make decisions or choose between alternatives. Theories about such decisions or choices often include as an assumption that agents have preferences in the form of a relation on a set of alternatives. A very used special case is that the set of alternatives is a set of functions, that the preferences can be represented by real numbers, and can be decomposed into utility functions and measures, where the preference can be expressed as the expected value of the utility function with respect to the measure.
Karl Vind
14. Uncertainty on products
Abstract
For many applications the space of functions has not one but several preference relations expressing the knowledge and uncertainty. The set X may be a product space with separate uncertainty on each of the sets.
Karl Vind
15. Conditional uncertainty
Abstract
The fact that all representation theorems in this book contain measures and not more general non-additive set functions has the extremely convenient consequence that conditional relations have very natural representations via the conditional measures. Combining relations as in chapter 14 and taking conditional measures does however not commute. It is of course mathematically possible to perform the operations in any order and the results will be different. See example 13 and remark 56 page 211.
Karl Vind

Applications

Frontmatter
16. Production, utility, preference
Abstract
The results in this book have of course many applications. Whenever an integral or a sum is used, the use may be justified — via the independence assumption — as a representation of a subset, a total preorder, or just a relation.
Karl Vind
17. Preferences over time
Abstract
One of the most unsatisfactory assumptions in economic theory in general and in general equilibrium theory in particular is the assumption that consumers have total preorders as preferences for consumption in future periods. The results in this book allows for representation of preferences that are not total preorders. The direct interpretation of the results in this book gives the first set of results about preferences over time.
Karl Vind
18. A foundation for statistics
Abstract
Probability theory has had an almost unquestioned foundation since Kolmogorov (1933) [111]. Probability is a (normalized) measure on an algebra of events i.e. subsets of an arbitrary set. The probability of an event may be a result in a theory about any part of the real world. It may be an assumption that the beliefs or knowledge of agents can be expressed as total preorders on a system of subsets of events. Theorem 9 page 98 then gives probability as a representation of this relation. Under the assumptions of this representation theorem the two assumptions — a total preorder on an algebra or a probability measure on this algebra — are therefore equivalent.
Karl Vind
Backmatter
Metadaten
Titel
Independence, Additivity, Uncertainty
verfasst von
Professor Karl Vind
Copyright-Jahr
2003
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-540-24757-9
Print ISBN
978-3-540-41683-8
DOI
https://doi.org/10.1007/978-3-540-24757-9