ISBN: 3540416838
TITLE: Independence, Additivity, Uncertainty
AUTHOR: Vind
TOC:

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
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 oder 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 an the set S_A 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 Aczl 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 Histor 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 groupoids 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) 65
7 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, \succeq) = (X x Y/ ~)as a commutative mean groupoid 73
7.4 f(x,y) =f_1(x)+f_2(y) 77
7.5 The functional equation F(x,y) = g^-1 (f_1 (x) + f_2 (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 : G -> R and f : G x A -> R 95
8.7 X={1,2,...,n}(Pi_i\inXY_i,\succsim) 97
8.8 Y={0,1},(A,\succsim) 98
8.9 Y a commutative mean groupoid 98
8.9.1 (H, \succsim=,o) 102
8.10 Y a commutative mean groupoid with zero 103
8.10.1 (H, \succsim,o_x, \Box_x) 106
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 : G -> R, f : G x A -> R 113
9.3 Existence of F : G x H -> R, f : G x H x A -> R 115
9.3.1 ((X, A), Y, G, P) Existence of F:GxG->R,f:GxGxA->R 116
9.4 x = {1,2,...,n} (Pi _i \inX Y_i, Pi\iinX Z_i, P) 117
9.5 Y=Z={0,1},(X,A,P) 118
9.6 Minimal independence assumptions 119
9.7 (Y_x, Q_x)_x\inX 120
9.7.1 (X, Y, G, Q, (Q_x)_x\inX) 121
9.7.2 (G \ succsi m, o), (Y_x,, P_x)_X\ inX) = ((G,\succsim,o),(Y_x xY_x)/~_x,\succeq_x,o_x) 123
9.7.3 (G,P), (Yx/ ~_x, \succ equ_x, o_x) 124
9.7.4 (G,P) , (Y_x/ ~_x, \succeq_x, o_x, \Box_x) 126
9.7.5 ((G, P), (Y_x, P_x)_x\inX) = ((G x G, \succsim o, \Box), (Y x Y) / ~_x, \succsim, o_x, \Box_x) 126
9.8 Notes 128
III Relations on Measures 131
10 Relations on sets of probability measures 133
10.1 Introduction 133
10.2 Definitions and mathematics 133
10.3 Existence of a Bernoulli function 135
10.4 von Neumann Morgenstern preferences 136
10.4.1 The finite case 137
10.4.2 The general case 138
10.4.3 Special cases 140
10.5 Notes 144
IV Integral Representations 147
11 A general integral representation by Birgit Grodal 149
11.l Introduction 149
11.2 Existence of u : X x Y -> R with f(g,A) = \_Au(x,g(x))dmu 152
11.3 Continuity and boundedness of u 159
11.4 Existence of u : X x Y -> R when G is a set of measurable selections 164
11.5 Notes 165
12 Special integral representations by Birgit Grodal 169
12.1 Introduction 169
12.2 f(g, A) = \int _A beta(x,u^-(g(x)))dmu 170
12.3 f(g, A) = \int_Au^-(g(x))alpha(x)dmu 174
12.4 f(g, A) = \int_Au^-(t)e^-deltatdlambda 177
12.5 Notes 183
V Decompositions and Uncertainty 187
13 Decompositions. Uncertainty 189
13.1 Introduction 189
13.2 von Neumann Morgenstern preferences 190
13.3 Function spaces 192
13.3.1 Y = Z = {0, 1}. Subjective probabilities and uncertainty 192
13.3.2 X = {1,2,...,n} (Pi _i \inX Y_i,P) 193
13.3.3 Y and X general 194
13.4 Historical notes 195
13.4.1 Knight 195
13.4.2 Keynes 195
13.4.3 von Neumann Morgenstern 196
13.4.4 Savage 197
13.4.5 Aumann 197
13.4.6 Friedman 198
13.4.7 Bewley 198
13.5 Conclusion 199
14 Uncertainty on products 201
14.1 Introduction 201
14.2 One level uncertainty on factors and products 203
14.2.1 Y=Z={0,1},X=X_1xX_2 203
14.2.2 Y = Z = {0, 1}, (X,A_i)_i\inI 206
14.2.3 Y and Z general 213
14.3 Two level uncertainty 215
14.4 Conclusions 220
14.5 Note 220
15 Conditional uncertainty 221
15.1 Introduction 221
15.2 Relations on function spaces 221
15.3 One probability-uncertainty measure 222
15.4 Several probability-uncertainty measures 223
15.4.1 Y = Z = {0,1} 223
15.4.2 Y and Z general 223
15.5 Two level uncertainty 223
15.6 Conclusion 223
VI Applications 225
16 Production, utility, preference 227
16.1 Introduction 227
16.2 Production functions 227
16.3 Additive preference functions 228
16.4 Additive utility functions 228
16.5 Notes 229
17 Preferences over time 231
17.1 Introduction 231
17.2 ((T, A), Y Z,G,H, P)
Existence of f : G x H x A -> R 232
17.2.1 Y general 235
17.3 Existence and decomposition of f:GxHxGxHxA->R 237
17.4 Notes 240
18 A foundation for statistics 243
18.1 Introduction and historical background 243
18.2 Basic concepts 244
18.3 Uncertainty about the parameter space 247
18.4 Robust Bayesian inference 247
18.5 Requirements for a foundation of statistics 247
18.6 A foundation of statistics 248
18.7 Notes 252
References
END
