Fixed Points and Economic Equilibria

Preface......Page 8
2 This Book and Current Stream of Economic Theory......Page 9
3 For General Readers......Page 12
4 Relation to Other Works......Page 13
5 Acknowledgments......Page 14
Contents......Page 16
1.1 Mathematics is Language......Page 20
1.2.1 Convexity......Page 21
1.2.2 Duality......Page 24
1.2.3 Algebraic methods......Page 26
1.2.4 Axiomatic set theory......Page 30
1.3 Basic Mathematical Concepts and Definitions......Page 31
Sets......Page 32
Cartesian products and relations......Page 33
Axiom of choice and products of a family of sets......Page 34
Continuity......Page 35
Subnet and cluster point......Page 36
Quotient topology......Page 37
1.3.2 Compact sets, open coverings, and partition of unity......Page 38
1.3.3 Vector space duality and hyperplane......Page 40
Notes on References......Page 44
2.1.1 Classical fixed-point arguments......Page 46
2.1.2 Extensions of classic results......Page 50
2.2 Convexity and Duality for General Spaces......Page 58
2.2.1 Convex structure and convex combination......Page 59
2.2.2 System of duality......Page 61
2.2.3 Topological condition for dual-system structure......Page 62
2.2.4 Differentiability and dual-space representation......Page 63
2.3 Extension of Classical Results to General Spaces......Page 65
Notes for Section 2.1......Page 73
Notes for Section 2.3......Page 74
3.1 Multi-Agent Product Settings for Games......Page 76
3.2 Nash Equilibrium......Page 86
3.3 Abstract Economy......Page 94
Notes for Section 3.1......Page 102
Notes for Section 3.3......Page 103
4.1 Gale–Nikaido–Debreu’s Theorem......Page 106
4.2 Market Equilibria in General Vector Spaces......Page 110
4.3 Demand-Supply Coincidence in General Spaces......Page 118
Bibliographic Notes......Page 119
5.1 General Preferences and Basic Existence Theorems......Page 120
5.2 Pareto Optimal Allocations......Page 128
5.3 Existence of General Equilibrium......Page 130
5.3.1 Non-compact consumption and production sets......Page 131
5.3.2 For cases with general dual-system operators......Page 133
5.3.3 Price domains having no compact bases......Page 135
Bibliographic Notes......Page 141
6.1.1 Structures and categories......Page 144
6.1.2 Inverse and direct limits......Page 145
6.1.3 Simplex......Page 146
6.1.4 Homology theory......Page 147
6.1.5 Star refinement of covering......Page 150
6.1.6  Cech homology......Page 152
6.1.7 Vietoris homology......Page 154
6.1.8 Vietoris and Cech cycles......Page 156
6.2.1 Vietoris–Begle mapping......Page 161
6.2.2 Locally connected spaces......Page 164
6.3.1 Vietoris–Begle barycentric subdivision......Page 169
6.3.2 Vietoris–Begle mapping theorem......Page 176
6.3.3 Analogue of Sperner’s lemma......Page 178
6.4 Eilenberg–Montgomery’s Theorem......Page 183
Bibliographic Notes......Page 184
7.1 Lefschetz’s Fixed-Point Theorem and Its Extensions......Page 186
7.1.1 Convex structures and mappings of Browder type (class B)......Page 187
7.1.2 Convex structures and mappings of Kakutani type (class K)......Page 189
7.1.3 Acyclic valued directional structures and mappings of class D......Page 191
7.2 Cohomology Theory for General Spaces......Page 194
7.4 Linear Approximation for Isolated Fixed Points......Page 196
7.5 Indices for Compact Set of Fixed Points......Page 199
Bibliographic Notes......Page 200
8.1 KKM, KKMS, and Core Existence......Page 202
8.2 Eaves’Theorem......Page 206
8.3 Fan–Browder’s Coincidence Theorem......Page 208
8.4 L-majorizedMappings......Page 209
8.5 Variational Inequality Problem......Page 210
8.6 Equilibrium with Cooperative Concepts......Page 211
8.6.1 Social systems with coordination......Page 212
8.6.2 Social coalitional equilibrium......Page 213
8.7 System of Inequalities and Affine Transformations......Page 214
9.1 Basic Concepts in Axiomatic Set Theory......Page 220
9.1.1 Formal theories, models, and theory of sets......Page 221
List of syntactical rules......Page 222
List of axioms......Page 224
Axioms for set theory......Page 228
9.2 Individuals and Rationality......Page 237
9.2.1 Views of the world......Page 240
9.2.2 Rationality......Page 243
9.2.3 Rational acceptability......Page 245
9.3 Society and Values......Page 251
9.3.1 Society......Page 252
9.3.2 Social validity and mathematical truth......Page 254
Bibliographic Notes......Page 260
10.1.1 Equilibrium concepts in economics......Page 262
10.1.2 Fixed points and social validity......Page 264
10.1.3 Rationality and economic equilibria......Page 265
10.1.4 Fixed points and view of the world......Page 267
10.2 Rationality and Fixed-Point Views of the World......Page 269
10.2.3 Possible worldviews and behaviors......Page 271
10.2.4 Justification and refutability......Page 272
10.2.5 Equilibrium under social recognition......Page 274
P-adic Expansion of a Real Number......Page 276
Urysohn’s Lemma......Page 279
Partition of Unity Theorem......Page 280
Vector-Space Topology......Page 282
Hahn–Banach Theorem......Page 286
Separation Theorem......Page 289
Ordinal......Page 292
Cardinal......Page 293
Continuum Hypothesis......Page 294
References......Page 296
Index......Page 304