Optima and Equilibria: An Introduction to Nonlinear Analysis

Front cover......Page 1
Series......Page 3
Books in the series......Page 4
Title page......Page 5
Date-line......Page 6
Dedication......Page 7
Foreword......Page 9
Table of Contents......Page 13
Introdction......Page 21
Part I Nonlinear Analysis: Theory......Page 27
1.2 Definitions......Page 29
1.3 Epigraph......Page 30
1.5 Lower Semi-continuous Functions......Page 31
1.6 Lower Semi-compact Functions......Page 33
1.7 Approximate Minimisation of Lower Semi-continuous Functions on a Complete Space......Page 35
1.8 Application to Fixed-point Theorems......Page 37
2.2 Definitions......Page 41
2.3 Examples of Convex Functions......Page 44
2.4 Continuous Convex Functions......Page 45
2.5 The Proximation Theorem......Page 47
2.6 Separation Theorems......Page 51
3.1 Introduction......Page 55
3.2 Characterisation of Convex Lower Semi-continuous Functions......Page 57
3.3 Fenchel\'s Theorem......Page 59
3.4 Properties of Conjugate Functions......Page 63
3.5 Support Functions......Page 68
3.6 The Cramer Transform......Page 72
4.1 Introduction......Page 77
4.2 Definitions......Page 81
4.3 Subdifferentiability of Convex Continuous Functions......Page 84
4.4 Subdifferentiability of Convex Lower Semi-continuous Functions......Page 86
4.5 Subdifferential Calculus......Page 87
4.6 Tangent and Normal Cones......Page 90
5.1 Introduction......Page 95
5.2 Fermat\'s Rule......Page 96
5.3 Minimisation Problems with Constraints......Page 100
5.4 Principle of Price Decentralisation......Page 102
5.5 Regularisation and Penalisation......Page 104
6.2 Definitions......Page 107
6.3 Elementary Properties......Page 111
6.4 Generalised Gradients......Page 115
6.5 Normal and Tangent Cones to a Subset......Page 117
6.6 Fermat\'s Rule for Minimisation Problems with Constraints......Page 119
7.1 Introduction......Page 121
7.2 Decision Rules and Consistent Pairs of Strategies......Page 122
7.3 Brouwer\'s Fixed-point Theorem (1910)......Page 124
7.4 The Need to Convexify: Mixed Strategies......Page 125
7.5 Games in Normal (Strategic) Form......Page 126
7.6 Pareto Optima......Page 128
7.7 Conservative Strategies......Page 130
7.8 Some Finite Games......Page 132
7.9 Cournot\'s Duopoly......Page 136
8.2 Value and Saddle Points of a Game......Page 145
8.3 Existence of Conservative Strategies......Page 150
8.4 Continuous Partitions of Unity......Page 155
8.5 Optimal Decision Rules......Page 157
9.1 Introduction......Page 163
9.2 Upper Hemi-continuous Set-valued Maps......Page 164
9.3 The Debreu-Gale-Nikaido Theorem......Page 168
9.4 The Tangential Condition......Page 169
9.5 The Fundamental Theorem for the Existence of Zeros of a Set-valued Map......Page 170
9.6 The Viability Theorem......Page 172
9.7 Fixed-point Theorems......Page 174
9.8 Equilibrium of a Dynamical Economy......Page 175
9.9 Variational Inequalities......Page 177
9.10 The Leray-Schauder Theorem......Page 179
9.11 Quasi-variational Inequalities......Page 180
9.12 Shapley\'s Generalisation of the Three-Poles Lemma......Page 182
10.1 Introduction......Page 187
10.2 Exchange Economies......Page 188
10.3 The Walrasian Mechanism......Page 189
10.4 Another Mechanism for Price Decentralisation......Page 193
10.5 Collective Budgetary Rule......Page 194
11.2 The Von Neumann Model......Page 199
11.3 The Perron-Frobenius Theorem......Page 204
11.4 Surjectivity of the M matrices......Page 207
12.2 Non-cooperative Behaviour......Page 209
12.3 $n$-person Games in Normal (Strategic) Form......Page 210
12.4 Non-cooperative Games with Constraints (Metagames)......Page 212
12.5 Pareto Optima......Page 213
12.6 Behaviour of Players in Coalitions......Page 216
12.7 Cooperative Games Without Side Payments......Page 217
12.8 Evolutionary Games......Page 225
13.2 Coalitions, Fuzzy Coalitions and Generalised Coalitions of $n$ Players......Page 231
13.3 Action Games and Equilibrium Coalitions......Page 236
13.4 Games with Side Payments......Page 238
13.5 Core and Shapley Value of Standard Games......Page 246
Part II Nonlinear Analysis: Exercises and Problems......Page 255
14.1 Exercises for Chapter 1 - Minimisation Problems: General Theorems......Page 257
14.2 Exercises for Chapter 2 - Convex Functions and Proximation, Projection and Separation Theorems......Page 262
14.3 Exercises for Chapter 3 - Conjugate Functions and Convex Minimisation Problems......Page 267
14.4 Exercises for Chapter 4 - Subdifferentials of Convex Functions......Page 276
14.5 Exercises for Chapter 5 - Marginal Properties of Solutions of Convex Minimisation Problems......Page 283
14.6 Exercises for Chapter 6 - Generalised Gradients of Locally Lipschitz Functions......Page 290
14.7 Exercises for Chapter 8 - Two-person Zero-sum Games: Theorems of Von Neumann and Ky Fan......Page 297
14.8 Exercises for Chapter 9 - Solution of Nonlinear Equations and Inclusions......Page 302
14.9 Exercises for Chapter 10 - Introduction to the Theory of Economic Equilibrium......Page 307
14.11 Exercises for Chapter 12 - $n$-person Games......Page 312
14.12 Exercises for Chapter 13 - Cooperative Games and Fuzzy Games......Page 319
15.2 Problem 2 - Upper Semi-continuous Set-valued Maps......Page 323
15.4 Problem 4 - Inverse Image of a Set-valued Map......Page 324
15.6 Problem 6 - Marginal Functions......Page 325
15.8 Problem 8 - Approximate Selection of an Upper Semi-continuous Set-valued Map......Page 326
15.10 Problem 10 - Interior of the Image of a Convex Closed Cone......Page 327
15.11 Problem 11 - Discrete Dynamical Systems......Page 330
15.12 Problem 12 - Fixed Points of Contractive Set-valued Maps......Page 332
15.14 Problem 14 - Open Image Theorem......Page 333
15.15 Problem 15 - Asymptotic Centres......Page 335
15.16 Problem 16 - Fixed Points of Non-expansive Mappings......Page 336
15.17 Problem 17 - Orthogonal Projectors onto Convex Closed Cones......Page 337
15.18 Problem 18 - Gamma-convex functions......Page 338
15.19 Problem 19 - Proper Mappings......Page 339
15.20 Problem 20 - Fenchel\'s Theorem for the Functions $L(x,Ax)$......Page 341
15.21 Problem 21 - Conjugate Functions of $x \\to L(x, Ax)$......Page 342
15.22 Problem 22 - Hamiltonians and Partial Conjugates......Page 343
15.23 Problem 23 - Lack of Convexity and Fenchel\'s Theorem for Pareto Optima......Page 344
15.24 Problem 24 - Duality in Linear Programming......Page 345
15.25 Problem 25 - Lagrangian of a Convex Minimisation Problem......Page 346
15.26 Problem 26 - Variational Principles for Convex Lagrangians......Page 347
15.27 Problem 27 - Variational Principles for Convex Hamiltonians......Page 348
15.29 Problem 29 - Transposes of Convex Processes......Page 349
15.31 Problem 31 - Regularity of Tangent Cones......Page 351
15.32 Problem 32 - Tangent Cones to an Intersection......Page 352
15.33 Problem 33 - Derivatives of Set-valued Maps with Convex Graphs......Page 353
15.34 Problem 34 - Epiderivatives of Convex Functions......Page 354
15.36 Problem 36 - Values of a Game Associated with a Covering......Page 355
15.37 Problem 37 - Minimax Theorems with Weak Compactness Assumptions......Page 356
15.38 Problem 38 - Minimax Theorems for Finite Topologies......Page 357
15.39 Problem 39 - Ky Fan\'s Inequality......Page 358
15.40 Problem 40 - Ky Fan\'s Inequality for Monotone Functions......Page 359
15.41 Problem 41 - Generalisation of the Gale-Nikaido-Debreu Theorem......Page 360
15.43 Problem 43 - Eigenvectors of Set-valued Maps......Page 361
15.44 Problem 44 - Positive Eigenvectors of Positive Set-valued Maps......Page 362
15.46 Problem 46 - Generalised Variational Inequalities......Page 363
15.47 Problem 47 - Monotone Set-valued Maps......Page 365
15.48 Problem 48 - Walrasian Equilibrium for Set-valued Demand Maps......Page 366
16.2 Problem 2 - Solution. Upper Semi-continuous set-valued Maps......Page 369
16.4 Problem 4 - Solution. Inverse Image of a Set-valued Map......Page 370
16.6 Problem 6 - Solution. Marginal Functions......Page 372
16.8 Problem 8 - Solution. Approximate Selection of an Upper Semi-continuous Set-valued Map......Page 373
16.10 Problem 10 - Solution. Interior of the Image of a Convex Closed Cone......Page 374
16.11 Problem 11 - Solution. Discrete Dynamical Systems......Page 378
16.12 Problem 12 - Solution. Fixed Points of Contractive Set-valued Maps......Page 380
16.13 Problem 13 - Solution. Approximate Variational Principle......Page 381
16.14 Problem 14 - Solution. Open Image Theorem......Page 382
16.15 Problem 15 - Solution. Asymptotic Centres......Page 384
16.16 Problem 16 - Solution. Fixed Points of Non-expansive Mappings......Page 385
16.17 Problem 17 - Solution. Orthogonal Projectors onto Convex Closed Cones......Page 387
16.18 Problem 18 - Solution. Gamma-convex Functions......Page 388
16.19 Problem 19 - Solution. Proper Mappings......Page 389
16.20 Problem 20 - Solution. Fenchel\'s Theorem for the Functions L(x,Ax)......Page 390
16.22 Problem 22 - Solution. Hamiltonians and Partial Conjugates......Page 391
16.23 Problem 23 - Solution. Lack of Convexity and Fenchel\'s Theorem for Pareto Optima......Page 392
16.24 Problem 24 - Solution. Duality in Linear Programming......Page 394
16.25 Problem 25 - Solution. Lagrangian of a Convex Minimisation Problem......Page 395
16.27 Problem 27 - Solution. Variational Principles for Convex Hamiltonians......Page 396
16.28 Problem 28 - Solution. Approximation to Fermat\'s Rule......Page 397
16.29 Problem 29 - Solution. Transposes of Convex Processes......Page 398
16.30 Problem 30 - Solution. Cones with a Compact Base......Page 399
16.31 Problem 31 - Solution. Regularity of Tangent Cones......Page 400
16.32 Problem 32 - Solution. Tangent Cones to an Intersection......Page 401
16.33 Problem 33 - Solution. Derivatives of Set-valued Maps with Convex Graphs......Page 403
16.34 Problem 34 - Solution. Epiderivatives of Convex Functions......Page 404
16.35 Problem 35 - Solution. Subdifferentials of Marginal Functions......Page 405
16.36 Problem 36 - Solution. Values of a Game Associated with a Covering......Page 406
16.37 Problem 37 - Solution. Minimax Theorems with Weak Compactness Assumptions......Page 407
16.38 Problem 38 - Solution. Minimax Theorems for Finite Topologies......Page 408
16.39 Problem 39 - Solution. Ky Fan\'s Inequality......Page 409
16.40 Problem 40 - Solution. Ky Fan\'s Inequality for Monotone Functions......Page 410
16.41 Problem 41 - Solution. Generalisations of the Gale-Nikaido-Debreu Theorem......Page 411
16.42 Problem 42 - Solution. Equilibrium of Coercive Set-valued Maps......Page 412
16.45 Problem 45 - Solution. Some Variational Principles......Page 413
16.46 Problem 46 - Solution. Generalised Variational Inequalities......Page 415
16.47 Problem 47 - Solution. Monotone Set-valued Maps......Page 417
16.48 Problem 48 - Solution. Walrasian Equilibrium for Set-valued Demand Maps......Page 419
Appendix......Page 421
17.1 Nontrivial, Convex, Lower Semi-continuous Functions......Page 423
17.2 Convex Functions......Page 425
17.3 Conjugate Functions......Page 426
17.4 Separation Theorems and Support Functions......Page 427
17.5 Subdifferentiability......Page 430
17.6 Tangent and Normal Cones......Page 431
17.7 Optimisation......Page 433
17.8 Two-Person Games......Page 435
17.9 Set-valued Maps and the Existence of Zeros and Fixed Points......Page 437
References......Page 443
Index......Page 449
Graduate Texts in Mathematics (continued)......Page 455
Back cover......Page 459