Independence, Additivity, Uncertainty

275 Pages·2003·12.06 MB·English

by  Professor Karl Vind (auth.)

Studies in Economic Theory
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Charalambos D. Aliprantis
Purdue University
Department of Economics
West Lafayette, IN 47907-1310
USA
Nicholas C. Yannelis
University of Illinois
Department of Economics
Champaign, IL 61820
USA
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Karl Vind
Independence,
Additivity,
Uncertainty
With Contributions
by Birgit Grodal
With 10 Figures
, Springer
Professor Karl Vind
University of Copenhagen
Institute of Economics
Studiestraede 6
DK 1455 Copenhagen K
Denmark
ISBN 978-3-540-41683-8 ISBN 978-3-540-24757-9 (eBook)
DOI 10.1007/978-3-540-24757-9
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Preface
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.
The realization at MSRl, Berkeley 1985-86 that the same math
ematics could be used to get results for relations which were not.
total or transitive rp,sulted in the papers [180, 179]. The results in
these papers are included in and extended in this book. They were
presented January 1987 in Oberwolfach, where I also heard Be
wley's ideas about Knightian uncertainty. The importance of the
results about not totally ordered function spaces for formalizing
uncertainty became clear in conversations with 'fruman F. Bewley
in Bonn in the summer of 1990.
The Notes gives references, open problems and a few theorems.
The References contains the references. No attempts have been
made to make a complete bibliography of any of the fields touched
VI Preface
upon in the bookI, but some attempts have been made to include
references to papers and books which may be relevant to or ex
tending the results in this book
The work on this book started in 1965, and parts of the re
sults have since been used in courses and presented at seminars
in mathematical economics at the University of Copenhagen, the
University of California, Berkeley, Stanford University, etc. I have
had very useful discussions with and comments from a large num
ber of economists, mathematicians, and statisticians. I should like
to mention in particular Gerard Debreu, Werner Fenchel, Birgit
Grodal, and S0ren Johansen.
Mansoor Hussain did a good job proofreading, the remaining
mistakes were probably added after he finished.
My research has been supported by grants from the Social Sci
ence Research Council, Denmark, the Carlsberg Foundation, by
grant from the Ford Foundation to the Graduate School of Business
Administration, University of California, Berkeley, and by NSF
grant 8120790. Many of the new results in the book have been
obtained at the University of California, Berkeley, as a visitor to
the Department of Economics (1962-63, 1964-66, 1970-71, 1981-
82, 1983, 1990-91) or to Mathematical Sciences Research Institute
(1985-86).
August 2002 Karl Vind
1 For most of the fields see Peter Wakker's useful references [182].
Contents
1 Introduction 1
1.1 Economics 1
1.2 Statistics. 1
1.3 Mathematics. 2
1.4 Summary of results 4
1.5 Applications.... 6
I Basic Mathematics 7
2 Totally preordered sets 9
2.1 Introduction ..... . 9
2.2 Order relations ... . 9
2.2.1 Basic concepts. 9
2.2.2 Completion 14
2.2.3 Representation 16
2.3 Topological concepts 18
2.4 The order topology 20
2.5 Representation .. . 23
2.6 Notes.. . .... . 23
2.6.1 Basic concepts. 23
2.6.2 Ordered sets. . 24
2.6.3 Topology and order topology . 24
2.6.4 Ordered topological spaces. 24
2.6.5 Lexicographic orders 25
2.6.6 Removing gaps 25
2.6.7 Further results ... 25
3 Preferences and preference functions 27
3.1 Introduction.............. .., 27
3.2 Representations and representation theorems. 27
3.3 Notes......................· 29
viii Contents
4 Totally preordered product sets 31
4.1 Introduction........... 31
4.2 Independence assumptions . . . 31
4.3 Order topologies on product sets 33
4.4 Existence of real continuous order homomorphisms 37
4.5 Note.......................... 38
5 A subset of a product set 39
5.1 Introduction....... 39
5.2 Independence . . . . . . 40
5.3 A total preorder on the set SA 40
5.4 The Thomsen and the Reidemeister conditions . 43
5.5 Note........................ 45
5.5.1 The Reidemeister and Thomsen conditions 45
6 Mean groupoids 49
6.1 Introduction.................. 49
6.2 Definition of a commutative mean groupoid 49
6.3 Completion of commutative mean groupoid. 52
6.4 The Aczel Fuchs theorem. . . . . . . . . . . 54
6.5 Extension of a commutative mean groupoid 57
6.6 The bisymmetry equation .. . . 59
6.7 Notes...................... 60
6.7.1 History and other results. . . . . . . 60
6.7.2 Classifying commutative mean groupoids 61
6.7.3 Lexicographic "mean groupoids" 61
6.7.4 Totally ordered mixture spaces 61
6.7.5 Reducible . . . . . . . . . . 62
6.7.6 Products of mean gToupoids 62
6.7.7 Completion . . . . . . . . . 62
6.7.8 Measurement of magnitudes 63
6.7.9 The bisymmetry equation . 63
6.7.10 Counter example (Andrew Gleason, Harvard)l 65
1 Letter of May, 1983 to Paul Samuelson, copy to me summer 2000.
Contents ix
1 Products of two sets as a mean groupoid 69
7.1 Introduction...................... 69
7.2 Thomsen's and Reidemeister's conditions . . . . .. 70
7.3 (S, t:) = (X x Y/ rv) as a commutative mean gToupoid 73
7.4 f(x,Y)=fdx)+h(Y) ............... 77
7.5 The functional equation F (x, y) = g-1 (It (x) + h (y)) 77
7.6 Notes................. 78
7.6.1 History and further results. . . . . . . . .. 78
II Relations on Function Spaces 81
8 Totally preordered function spaces 83
8.1 Introduction.. ...... 83
8.2 Notation and definitions . . . . . . 85
8.3 Real order homomorphisms .. 87
8.4 The function space as a mean groupoid 88
8.5 Minimal independence assumptions . . 91
8.6 Existence of F : 9 lR and f : 9 x A lR . 95
---t ---t
8.7 X={1,2, ... ,n}(ITiEx}i,;:).. 97
8.8 Y={O,l},(A,;:)......... 98
8.9 Y a commutative mean groupoid 98
8.9.1 (1t,;:, 0) . . . . . . . . . . 102
8.10 Y a commutative mean groupoid with zero . 103
8.10.1 (1t,;:, Ox) . . 106
Ox,
8.11 Related functional equations 107
8.12 Notes. . . . . . . . .. .. 108
9 Relations on function spaces 113
9.1 Introduction.. .................. 113
9.2 Existence of F : 9 lR, f : 9 x A lR . . . . . 113
---t ---t
9.3 Existence of F : 9 x 1t lR,J : 9 x 1t x A lR 115
---t ---t
9.3.1 ((X, A), Y, g, P) Existence of
F : 9 x 9 lR, f : 9 x 9 x A lR 116
---t ---t
9.4 X = {1, 2, ... ,n} (ITiEX }i, ITiEX Zi, p) 117
9.5 Y = Z = {O, I}, (X,A, P) ..... 118
9.6 Minimal independence a,.<;snmptions 119
9.7 (YX,QX)XEX ... . . . . . . 120
9.7.1 (X, Y, g, Q, (QX)XEX) . . . . 121