Independence, Additivity, Uncertainty

ISBN-13: 9783540416838 / Angielski / Twarda / 2002 / 277 str.

H. Holden; Karl Vind

Opis

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.

Recenzje

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)

Spis treści

1 Introduction.- 1.1 Economics.- 1.2 Statistics.- 1.3 Mathematic.- 1.4 Summary of results.- 1.5 Applications.- I Basic Mathematics.- 2 Totally preordered sets.- 2.1 Introduction.- 2.2 Order relations.- 2.2.1 Basic concepts.- 2.2.2 Completion.- 2.2.3 Representation.- 2.3 Topological concepts.- 2.4 The order topology.- 2.5 Representation.- 2.6 Notes.- 2.6.1 Basic concepts.- 2.6.2 Ordered sets.- 2.6.3 Topology and order topology.- 2.6.4 Ordered topological spaces.- 2.6.5 Lexicographic orders.- 2.6.6 Removing gaps.- 2.6.7 Further results.- 3 Preferences and preference functions.- 3.1 Introduction.- 3.2 Representations and representation theorems..- 3.3 Notes.- 4 Totally preordered product sets.- 4.1 Introduction.- 4.2 Independence assumptions.- 4.3 Order topologies on product sets.- 4.4 Existence of real continuous order homomorphisms.- 4.5 Note.- 5 A subset of a product set.- 5.1 Introduction.- 5.2 Independence.- 5.3 A total preorder on the set SA.- 5.4 The Thomsen and the Reidemeister conditions.- 5.5 Note.- 5.5.1 The Reidemeister and Thomsen conditions.- 6 Mean groupoids.- 6.1 Introduction.- 6.2 Definition of a commutative mean groupoid.- 6.3 Completion of commutative mean groupoid.- 6.4 The Aczél Fuchs theorem.- 6.5 Extension of a commutative mean groupoid.- 6.6 The bisymmetry equation.- 6.7 Notes.- 6.7.1 History and other results.- 6.7.2 Classifying commutative mean groupoids.- 6.7.3 Lexicographic “mean groupoids”.- 6.7.4 Totally ordered mixture spaces.- 6.7.5 Reducible.- 6.7.6 Products of mean groupoids.- 6.7.7 Completion.- 6.7.8 Measurement of magnitudes.- 6.7.9 The bisymmetry equation.- 6.7.10 Counter example (Andrew Gleason, Harvard).- 7 Products of two sets as a mean groupoid.- 7.1 Introduction.- 7.2 Thomsen’s and Reidemeister’s conditions.- 7.3 (S, % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafS4EIyMba0 % baaaa!37AC!$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } $$) = (X × Y/ ~) as a commutative mean groupoid.- 7.4 f (x, y) = f1 (x) + f2 (y).- 7.5 The functional equation F (x, y) = g?1 (f1 (x) +f2 (y)).- 7.6 Notes.- 7.6.1 History and further results.- II Relations on Function Spaces.- 8 Totally preordered function spaces.- 8.1 Introduction.- 8.2 Notation and definitions.- 8.3 Real order homomorphisms.- 8.4 The function space as a mean groupoid.- 8.5 Minimal independence assumptions.- 8.6 Existence of F : G — ? and f : G × A ? ?.- 8.7 X = {1, 2, ... , n}(?i?XYi, % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafS4EIyMba4 % baaaa!37AD!$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \succ } $$).- 8.8 Y = {0, 1}, (A, % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafS4EIyMba4 % baaaa!37AD!$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \succ } $$).- 8.9 Y a commutative mean groupoid.- 8.9.1 (H, % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafS4EIyMba4 % baaaa!37AD!$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \succ } $$, o).- 8.10 Y a commutative mean groupoid with zero.- 8.10.1 (H, % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafS4EIyMba4 % baaaa!37AD!$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \succ } $$ , ox, ?x).- 8.11 Related functional equations.- 8.12 Notes.- 9 Relations on function spaces.- 9.1 Introduction.- 9.2 Existence of F : G ? ?, f : G × A ? ?.- 9.3 Existence of F : G × H ? ?, f : G × H × A ? ?.- 9.3.1 ((X, A), Y, G, P) Existence of F : G × G ? ? , f : G × G × A ? ?.- 9.4 X = {1, 2, ... , n} (?i?XYi, ?i?XZi, P).- 9.5 Y = Z = {0,1}, (X, A, P).- 9.6 Minimal independence assumptions.- 9.7 (Yx, Qx)x?X.- 9.7.1 (X, Y, G, Q, (Qx)x?X.- 9.7.2 ((G, % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafS4EIyMba4 % baaaa!37AD!$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \succ } $$ o), (Yx, Px)x?X) = ((G, % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafS4EIyMba4 % baaaa!37AD!$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \succ } $$, o), (Yx × Yx) / ~x, % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafS4EIyMba0 % badaWgaaWcbaGaamiEaaqabaaaaa!38D5!$${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } _x}$$, ox).- 9.7.3 (G, P),(Yx/ ~x, % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafS4EIyMba0 % badaWgaaWcbaGaamiEaaqabaaaaa!38D5!$${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } _x}$$, ox).- 9.7.4 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca % WGhbGaaiilaiaabccacaqGqbaacaGLOaGaayzkaaGaaiilaiaaygW7 % caqGGaWaaeWaaeaadaWcgaqaaiaadMfadaWgaaWcbaGaamiEaaqaba % aakeaacqWI8iIodaWgaaWcbaGaamiEaaqabaGccaGGSaGaaeiiaiqb % lUNiMzaaDaWaaSbaaSqaaiaadIhaaeqaaOGaaiilaiaabccacaqGVb % WaaSbaaSqaaiaadIhaaeqaaaaakiablgAjxnaaBaaaleaacaWG4baa % beaaaOGaayjkaiaawMcaaaaa!4DEF!$$\left( {G,{\text{ P}}} \right),{\text{ }}\left( {{{} \mathord{\left/ {\vphantom {{} {{ \sim _x},{\text{ }}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } }_x},{\text{ }}{{\text}_x}}}} \right. \kern-\nulldelimiterspace} {{ \sim _x},{\text{ }}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } }_x},{\text{ }}{{\text}_x}}}{\square _x}} \right)$$.- 9.7.5 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada % qadaqaaiaadEeacaGGSaGaamiuaaGaayjkaiaawMcaaiaacYcadaqa % daqaaiaadMfadaWgaaWcbaGaamiEaaqabaGccaGGSaGaamiuamaaBa % aaleaacaWG4baabeaaaOGaayjkaiaawMcaamaaBaaaleaacaWG4bGa % eyicI4SaamiwaaqabaaakiaawIcacaGLPaaacqGH9aqpdaqadaqaam % aalyaabaWaaeWaaeaacaWGhbGaey41aqRaam4raiaacYcacuWI7jIz % gaqhaiaacYcacaWGVbGaaiilaiablgAjxbGaayjkaiaawMcaaiaacY % cadaqadaqaaiaadMfacqGHxdaTcaWGzbaacaGLOaGaayzkaaaabaGa % eSipIOZaaSbaaSqaaiaadIhaaeqaaOGaaiilaiqblUNiMzaaDaWaaS % baaSqaaiaadIhaaeqaaOGaaiilaiaad+gadaWgaaWcbaGaamiEaaqa % baGccqWIHwYvdaWgaaWcbaGaamiEaaqabaaaaaGccaGLOaGaayzkaa % aaaa!65EF!$$\left( {\left( {G,P} \right),{{\left( {,} \right)}_{x \in X}}} \right) = \left( {{{\left( {G \times G,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } ,o,\square } \right),\left( {Y \times Y} \right)} \mathord{\left/ {\vphantom {{\left( {G \times G,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } ,o,\square } \right),\left( {Y \times Y} \right)} {{ \sim _x},{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } }_x},{\square _x}}}} \right. \kern-\nulldelimiterspace} {{ \sim _x},{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } }_x},{\square _x}}}} \right)$$.- 9.8 Notes.- III Relations on Measures.- 10 Relations on sets of probability measures.- 10.1 Introduction.- 10.2 Definitions and mathematics.- 10.3 Existence of a Bernoulli function.- 10.4 von Neumann Morgenstern preferences.- 10.4.1 The finite case.- 10.4.2 The general case.- 10.4.3 Special cases.- 10.5 Notes.- IV Integral Representations.- 11 A general integral representation by Birgit Grodal.- 11.1 Introduction.- 11.2 Existence of u : X × Y ? ? with % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm % aabaGaam4zaiaacYcacaWGbbaacaGLOaGaayzkaaGaeyypa0Zaa8qe % aeaacaWG1bWaaeWaaeaacaWG4bGaaiilaiaadEgadaqadaqaaiaadI % haaiaawIcacaGLPaaaaiaawIcacaGLPaaaaSqaaiaadgeaaeqaniab % gUIiYdGccaWGKbGaamyDaaaa!482F!$$f\left( {g,A} \right) = \int_A {u\left( {x,g\left( x \right)} \right)} d\mu $$.- 11.3 Continuity and boundedness of u.- 11.4 Existence of u: X × Y ? ? when G is a set of measurable selections..- 11.5 Notes.- 12 Special integral representations by Birgit Grodal.- 12.1 Introduction.- 12.2 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm % aabaGaam4zaiaacYcacaWGbbaacaGLOaGaayzkaaGaeyypa0Zaa8qe % aeaacqaHYoGydaqadaqaaiaadIhacaGGSaGabmyDayaaraWaaeWaae % aacaWGNbWaaeWaaeaacaWG4baacaGLOaGaayzkaaaacaGLOaGaayzk % aaaacaGLOaGaayzkaaaaleaacaWGbbaabeqdcqGHRiI8aOGaamizai % abeY7aTbaa!4C2D!$$f\left( {g,A} \right) = \int_A {\beta \left( {x,\bar u\left( {g\left( x \right)} \right)} \right)} d\mu $$.- 12.3 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm % aabaGaam4zaiaacYcacaWGbbaacaGLOaGaayzkaaGaeyypa0Zaa8qe % aeaaceWG1bGbaebaaSqaaiaadgeaaeqaniabgUIiYdGcdaqadaqaai % aadEgadaqadaqaaiaadIhaaiaawIcacaGLPaaaaiaawIcacaGLPaaa % cqaHXoqydaqadaqaaiaadIhaaiaawIcacaGLPaaacaWGKbGaeqiVd0 % gaaa!4B7B!$$f\left( {g,A} \right) = \int_A {\bar u} \left( {g\left( x \right)} \right)\alpha \left( x \right)d\mu $$.- 12.4 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm % aabaGaam4zaiaacYcacaWGbbaacaGLOaGaayzkaaGaeyypa0Zaa8qe % aeaaceWG1bGbaebadaqadaqaaiaadshaaiaawIcacaGLPaaacaWGLb % WaaWbaaSqabeaacqGHsislcqaH0oazcaWG0baaaaqaaiaadgeaaeqa % niabgUIiYdGccaWGKbGaeq4UdWgaaa!4972!$$f\left( {g,A} \right) = \int_A {\bar u\left( t \right){e^{ - \delta t}}} d\lambda $$.- 12.5 Notes.- V Decompositions and Uncertainty.- 13 Decompositions. Uncertainty.- 13.1 Introduction.- 13.2 von Neumann Morgenstern preferences.- 13.3 Function spaces.- 13.3.1 Y = Z = {0, 1}. Subjective probabilities and uncertainty.- 13.3.2 X = {1, 2, ... , n}(?i?XYi, P).- 13.3.3 Y and X general.- 13.4 Historical notes.- 13.4.1 Knight.- 13.4.2 Keynes.- 13.4.3 von Neumann Morgenstern.- 13.4.4 Savage.- 13.4.5 Aumann.- 13.4.6 Friedman.- 13.4.7 Bewley.- 13.5 Conclusion.- 14 Uncertainty on products.- 14.1 Introduction.- 14.2 One level uncertainty on factors and products.- 14.2.1 Y = Z = {0, 1}, X = X1 × X2.- 14.2.2 Y = Z = {0, 1}, (X, Ai)i?I.- 14.2.3 Y and Z general.- 14.3 Two level uncertainty.- 14.4 Conclusions.- 14.5 Note.- 15 Conditional uncertainty.- 15.1 Introduction.- 15.2 Relations on function spaces.- 15.3 One probability-uncertainty measure.- 15.4 Several probability-uncertainty measures.- 15.4.1 Y = Z = {0, 1}.- 15.4.2 Y and Z general.- 15.5 Two level uncertainty.- 15.6 Conclusion.- VI Applications.- 16 Production, utility, preference.- 16.1 Introduction.- 16.2 Production functions.- 16.3 Additive preference functions.- 16.4 Additive utility functions.- 16.5 Notes.- 17 Preferences over time.- 17.1 Introduction.- 17.2 ((T, A), Y, Z, G, H, P) Existence of f : G × H × A ? ?.- 17.2.1 Y general.- 17.3 Existence and decomposition of f : G × H × G × H × A ? ?.- 17.4 Notes.- 18 A foundation for statistics.- 18.1 Introduction and historical background.- 18.2 Basic concepts.- 18.3 Uncertainty about the parameter space..- 18.4 Robust Bayesian inference.- 18.5 Requirements for a foundation of statistics..- 18.6 A foundation of statistics.- 18.7 Notes.- References.