Mathematical Methods of Game and Economic Theory Revised Edition

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ISBN: 9780486462653 | 048646265X
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Copyright: 11/2/2007

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Non-linear functional analysis is the focus of this text, which offers both formal definitions and proofs. Its central application is the fundamental economic problem of allocating scarce resources among competing agents, which leads to considerations of the interrelated applications in game theory and the theory of optimization. 1982 edition.

Preface to the Dover Edition	p. iii
Preface (1982)	p. vii
Summary of Results: A Guideline for the Reader	p. xxi
Contents of Other Possible Courses	p. xxvii
Notations	p. xxix
Optimization and Convex Analysis	p. 1
Minimization Problems and Convexity	p. 3
Strategy sets and loss functions	p. 4
Optimization problem	p. 4
Allocation of available commodities	p. 5
Resource and service operators	p. 6
Extension of loss functions	p. 8
Sections and epigraphs	p. 10
Decomposition principle	p. 11
Product of a loss function by a linear operator	p. 11
Example: Inf-convolution of functions	p. 12
Decomposition principle	p. 13
Another decomposition principle	p. 15
Mixed strategies and convexity	p. 17
Motivation: extension of strategy sets and loss functions	p. 18
Mixed strategies and linearized loss functions	p. 19
Interpretation of mixed strategies	p. 21
Case of finite strategy sets	p. 21
Representation by infinite sequences of pure strategies	p. 22
Linearized extension of maps and the barycentric operator	p. 24
Interpretation of convex functions in terms of risk aversion	p. 25
Elementary properties of convex subsets and functions	p. 25
Indicators, support functions and gauges	p. 27
Indicators and support functions	p. 28
Reformulation of the Hahn-Banach theorem	p. 31
The bipolar theorem	p. 32
Recession cones and barrier cones	p. 34
Interpretation: production sets and profit functions	p. 35
Gauges	p. 38
Existence, Uniqueness and Stability of Optimal Solutions	p. 42
Existence and uniqueness of an optimal solution	p. 43
Structure of the optimal set	p. 43
Existence of an optimal solution	p. 45
Continuity versus compactness	p. 45
Lower semi-continuity of convex functions in infinite dimensional spaces	p. 45
Fundamental property of lower semi-continuous and compact functions	p. 46
Uniqueness of an optimal solution	p. 47
Non-satiation property	p. 48
Minimization of quadratic functionals on convex sets	p. 48
Hilbert spaces	p. 49
Existence and uniqueness of the minimal solution	p. 49
Characterization of the minimal solution	p. 50
Projectors of best approximation	p. 51
The duality map from an Hilbert space onto its dual	p. 52
Minimization of quadratic functionals on subspaces	p. 54
The fundamental formula	p. 54
Orthogonal right inverse	p. 56
Orthogonal left inverse	p. 57
Another decomposition property	p. 58
Interpretation	p. 59
Perturbation by linear forms: conjugate functions	p. 60
Conjugate functions	p. 60
Characterization of lower semi-continuous convex functions	p. 61
Examples of conjugate functions	p. 62
Elementary properties of conjugate functions	p. 64
Interpretation: cost and profit functions	p. 65
Stability properties: an introduction to correspondences	p. 66
Upper semi-continuous correspondences	p. 66
Lower semi-continuous correspondences	p. 68
Closed correspondences	p. 70
Construction of upper semi-continuous correspondences	p. 73
Compactness and Continuity Properties	p. 75
Lower semi-compact functions	p. 76
Coercive and semi-coercive functions	p. 76
Functions such that f* is continuous at 0	p. 77
Lower semi-compactness of linear forms	p. 78
Constraint qualification hypothesis	p. 79
Case of infinite dimensional spaces	p. 81
Extension to compact subsets of mixed strategies	p. 82
Proper maps and preimages of compact subsets	p. 83
Proper maps	p. 84
Compactness of some strategy sets	p. 85
Examples where the map L* + 1 is proper	p. 88
Continuous convex functions	p. 90
A characterization of lower semi-continuous convex functions	p. 90
A characterization of continuous convex functions	p. 91
Examples of continuous convex functions	p. 93
Continuity of gL and Lf	p. 94
Continuous convex functions (continuation)	p. 95
Strong continuity of lower semi-continuous convex functions	p. 96
Estimates of lower semi-continuous convex functions	p. 97
Characterization of continuous convex functions	p. 98
Continuity of support functions	p. 99
Maximum of a convex function: extremal points	p. 100
Differentiability and Subdifferentiability: Characterization of Optimal Solutions	p. 103
Subdifferentiability	p. 105
Definitions	p. 105
Examples of subdifferentials	p. 106
Subdifferentiability of continuous convex functions	p. 108
Upper semi-continuity of the subdifferential	p. 109
Characterization of subdifferentiable convex functions	p. 110
Differentiability and variational inequalities	p. 111
Definitions	p. 111
Differentiability and subdifferentiability	p. 112
Legendre transform	p. 113
Interpretation: marginal profit	p. 114
Variational inequalities	p. 114
Differentiability from the right	p. 115
Definition and main inequalities	p. 115
Derivatives from the right and the support function of the subdifferential	p. 117
Derivative of a pointwise supremum	p. 118
Local [epsilon]-subdifferentiability and perturbed minimization problems	p. 120
Approximate optimal solutions in Banach spaces	p. 121
The approximate variational principle	p. 123
Local [epsilon]-subdifferentiability	p. 124
Perturbation of minimization problems	p. 126
Proof of Ekeland-Lebourg's theorem	p. 130
Introduction to Duality Theory	p. 133
Dual problem and Lagrange multipliers	p. 135
Lagrangian	p. 136
Lagrange multipliers and dual problem	p. 137
Marginal interpretation of Lagrange multipliers	p. 139
Example	p. 140
Case of linear constraints: extremality relations	p. 142
Generalized minimization problem	p. 143
Extremality relations	p. 145
The fundamental formula	p. 146
Minimization problem under linear constraints	p. 148
Minimization of a quadratic functional under linear constraints	p. 148
Minimization problem under linear equality constraints	p. 149
Duality and the decomposition principle	p. 150
The decentralization principle	p. 151
Conjugate function of gL	p. 152
Conjugate function of f[subscript 1]+f[subscript 2]	p. 153
Minimization of the projection of a function	p. 154
Minimization on the diagonal of a product	p. 154
Existence of Lagrange multipliers in the case of a finite number of constraints	p. 155
The Fenchel existence theorem	p. 156
Stability properties	p. 157
Applications to subdifferentiability	p. 158
Case of nonlinear constraints: The Uzawa existence theorem	p. 159
Game Theory and the Walras Model of Allocation of Resources	p. 363
Two-Person Games: An Introduction	p. 165
Some solution concepts	p. 167
Description of the game	p. 167
Shadow minimum	p. 367
Conservative solutions and values	p. 168
Non-cooperative equilibrium	p. 169
Pareto minimum	p. 170
Core of a two-person game	p. 171
Selection of strategy of the core	p. 171
Examples: some finite games	p. 172
Example	p. 173
Coordination game	p. 175
Prisoner's dilemma	p. 178
Game of chicken	p. 180
The battle of the sexes	p. 182
Example: Analysis of duopoly	p. 183
The model of a duopoly	p. 184
The set of Pareto minima	p. 185
Conservative solutions	p. 185
Non-cooperative equilibria	p. 186
Stackelberg equilibria	p. 187
Stackelberg disequilibrium	p. 187
Example: Edgeworth economic game	p. 189
The set of feasible allocations	p. 190
The biloss operator	p. 190
The Edgeworth box	p. 192
Pareto minima	p. 193
Core	p. 193
Walras equilibria	p. 194
Two-person zero-sum games	p. 195
Duality gap and value	p. 195
Saddle point	p. 197
Perturbation by linear functions	p. 198
Case of finite strategy sets: Matrix games	p. 200
Two-Person Zero-Sum Games: Existence Theorems	p. 204
The fundamental existence theorems	p. 206
Existence of conservative solutions	p. 208
Decision rules	p. 211
Finite topology on convex subsets	p. 211
Existence of an optimal decision rule	p. 212
The Ky-Fan inequality	p. 213
The Lasry theorem	p. 214
The minisup theorem	p. 216
The Nikaido theorem	p. 217
Existence of saddle points	p. 218
Another existence theorem for saddle points	p. 218
Extension of games without and with exchange of informations	p. 219
Definition of extensions of games	p. 220
Mixed extensions	p. 222
Extensions without exchange of information	p. 223
Sequential extensions	p. 225
Extensions with exchange of information	p. 227
Iterated games	p. 230
Iterated extensions	p. 231
The Moulin theorem	p. 233
Proof of playability of iterated extensions	p. 233
A system of functional equations	p. 236
A lemma on successive approximations	p. 239
Proof of existence of saddle decision rules	p. 240
The Fundamental Economic Model: Walras Equilibria	p. 241
Description of the model	p. 242
The subset of available commodities	p. 242
Appropriation of the economy	p. 244
Demand correspondences	p. 244
Walras equilibrium	p. 245
Examples of subsets of available commodities and of appropriations	p. 245
Example: Quadratic demand functions	p. 247
Existence of a Walras equilibrium	p. 248
Existence of a Walras pre-equilibrium	p. 248
Surjectivity of correspondences: the Debreu-Gale-Nikaido theorem	p. 250
Demand correspondences defined by loss functions	p. 251
Statement of the existence theorem	p. 251
Upper semi-continuity of the demand correspondence	p. 253
Compactification of an economy	p. 254
Proof of the existence of a Walras equilibrium	p. 256
Economies with producers	p. 257
Description of the model	p. 257
Statement of the existence theorem	p. 258
Compactification	p. 259
Proof of the existence of a Walras equilibrium	p. 262
Non-Cooperative n-Person Games	p. 263
Existence of a non-cooperative equilibrium	p. 264
Games described in strategic form	p. 264
Conservative values and multistrategies	p. 265
Non-cooperative equilibria	p. 266
The Nash theorem	p. 267
Stability	p. 268
Associated variational inequalities	p. 269
Case of quadratic loss functions; application to Walras-Cournot equilibria	p. 270
Non-cooperative games with quadratic loss functions	p. 271
Existence of solutions of variational inequalities	p. 272
Examples	p. 274
Multistrategy sets defined by linear constraints	p. 274
Walras-Cournot equilibria	p. 276
Constrained non-cooperative games and fixed point theorems	p. 279
Selection of a fixed point	p. 279
Equilibria of constrained non-cooperative games	p. 282
Fixed-point theorems	p. 283
Non-cooperative Walras equilibria	p. 285
Description of the model	p. 285
Existence of a non-cooperative Walras equilibrium: the Arrow-Debreu theorem	p. 286
Non-cooperative Walras equilibria of economies with producers	p. 289
Main Solution Concepts of Cooperative Games	p. 293
Behavior of the whole set of players: Pareto strategies	p. 295
Pareto strategies	p. 295
Rates of transfer	p. 297
Pareto multipliers	p. 297
Pareto allocations	p. 300
Selection of Pareto strategies and imputations	p. 303
Normalized games	p. 304
Pareto strategies obtained by using selection functions	p. 305
Closest strategy to the shadow minimum	p. 306
The best compromise	p. 307
Existence of Pareto strategies	p. 308
Interpretation: threat functionals	p. 308
Imputations: the Nash bargaining solution	p. 309
Behavior of coalitions of players: the core	p. 310
Coalitions	p. 311
Cooperative game described in strategic form and its core	p. 312
The multiloss operator F[superscript A]# of the coalition A	p. 313
Examples of multistrategy sets X(A)	p. 313
Economic games and core of an economy	p. 314
Cooperative game described in characteristic form and its core	p. 314
Behavior of fuzzy coalitions: the fuzzy core	p. 316
Fuzzy coalitions	p. 316
Extension of a family of coalitions	p. 317
Debreu-Scarf coalitions	p. 318
Fuzzy coalitions on a continuum of players	p. 319
Fuzzy games described in characteristic form	p. 320
Characterization of the core of a (fuzzy) game	p. 320
Fuzzy economic games and fuzzy core of an economy	p. 321
Fuzzy games described in strategic form and fuzzy core	p. 324
Selection of elements of the core: cooperative equilibrium and nucleolus	p. 329
Canonical cooperative equilibrium	p. 329
Least-core	p. 331
Nucleolus	p. 333
Games With Side-Payments	p. 336
Core of a fuzzy game with side-payments	p. 338
Core of a game with side-payments	p. 338
Linear games	p. 340
Non-emptiness of the core of fuzzy games with side-payments	p. 341
Core of fuzzy market games	p. 343
Core of a game with side-payments	p. 344
Convex cover of a game	p. 345
Non-emptiness of the core of a balanced game	p. 346
Balanced family of multistrategy sets	p. 347
Balanced characteristic functions and convex loss functions	p. 348
Further properties of convex functions and balances	p. 351
Values of fuzzy games	p. 353
The diagonal property	p. 354
Sequence of fuzzy values	p. 355
Existence and uniqueness of a sequence of fuzzy values	p. 356
Relations between core and fuzzy value	p. 359
Best approximation property of fuzzy values	p. 358
Generalized solution to locally Lipschitz games	p. 359
Shapley value and nucleolus of games with side-payments	p. 360
The Shapley value	p. 360
Existence and uniqueness of a Shapley value	p. 361
Simple games	p. 367
Nucleolus of games with side-payments	p. 367
Games Without Side-Payments	p. 370
Equivalence between the fuzzy core and the set of equilibria	p. 370
Representation of a game	p. 371
Equilibrium of a representation	p. 373
Cover associated with a representation	p. 374
Fuzzy core of a representation	p. 376
The equivalence theorem	p. 376
Non-emptiness of the fuzzy core of a balanced game	p. 378
Statement of theorems of non-emptiness of the fuzzy core	p. 379
Upper semi-continuity of the associated side-payment games	p. 382
Existence of approximate cooperative equilibria	p. 384
Proof of the non-emptiness of the core	p. 386
Equivalence between the fuzzy core of an economy and the set of Walras allocations	p. 386
Representation of economic games	p. 386
Fuzzy core and Walras allocations	p. 389
The equivalence theorem	p. 390
Non-Linear Analysis and Optimal Control Theory	p. 391
Minimax Type Inequalities, Monotone Correspondences and [gamma]-Convex Functions	p. 393
Relaxation of compactness assumptions	p. 395
Existence of a conservative solution	p. 395
Proof of existence of a conservative solution	p. 397
Existence of optimal decision rules and minisup under weaker compactness assumptions	p. 399
Relaxation of continuity assumptions: variational inequalities for monotone correspondences	p. 405
Variational inequalities	p. 406
Existence of a solution to variational inequalities for completely upper semi-continuous correspondences	p. 408
Pseudo-monotone functions: the Brezis-Nirenberg-Stampacchia theorem	p. 410
Existence of a solution to variational inequalities for pseudo-monotone maps	p. 413
Pseudo-monotonicity of monotone maps	p. 414
Monotone and cyclically monotone correspondences	p. 416
Maximal monotone correspondences	p. 417
Relaxation of convexity assumptions	p. 423
Definition of [gamma]-convex functions	p. 424
The fundamental characteristic property of families of [gamma]-convex functions	p. 424
The minisup theorem for [gamma subscript x]-convex-[gamma subscript y]-concave functions	p. 426
Existence of optimal decision rules for functions [gamma subscript y]-concave with respect to y	p. 428
Example: Image of a cone of convex functions by [pi]*	p. 429
Relations between convexity and [gamma]-convexity	p. 431
Example: [beta]-convex set functions	p. 434
Example: Convex functions of atomless vector measures	p. 436
Introduction to Calculus of Variations and Optimal Control	p. 438
Duality in infinite dimensional spaces	p. 441
Lagrangian of a minimization problem under linear constraints	p. 443
Extremality relations	p. 446
Existence of a Lagrange multiplier under the Slater condition	p. 447
Relaxation of the Slater condition	p. 449
Generalized Lagrangian of a minimization problem	p. 451
Characterization of a Lagrangian by perturbations of the minimization problem	p. 456
Duality in the case of non-convex integral criterion and contraints	p. 458
Modulus of non-convexity of a function	p. 459
Estimate of the duality gap	p. 461
The Shapley-Folkman theorem	p. 463
Sharp estimate of the duality gap	p. 465
Applications	p. 468
Extremality relations	p. 470
The Aumann-Perles duality theorem	p. 472
The approximation procedure	p. 474
Duality in calculus of variations	p. 476
The Green formula	p. 480
Abstract problem of calculus of variations	p. 482
The Hamiltonian system	p. 484
Lagrangian of a problem of calculus of variations	p. 486
Existence of a Lagrange multiplier	p. 487
Example: the Dirichlet variational problem	p. 488
The maximum principle for optimal control problems	p. 492
Optimal control and impulsive control problems	p. 497
The Hamilton-Jacobi-Bellman equation of a control problem	p. 498
Construction of the closed loop control	p. 502
The principle of optimality	p. 503
The quadratic case: Riccati equations	p. 505
The Bensoussan-Lions variational inequalities of a stopping time problem	p. 508
Construction of the optimal stopping time	p. 511
The Bensoussan-Lions quasi-variational inequalities of an impulsive control problem	p. 511
Construction of the optimal impulsive control	p. 515
Fixed Point Theorems, Quasi-Variational Inequalities and Correspondences	p. 518
Fixed point and surjectivity theorems for correspondences	p. 518
The Browder-Ky-Fan existence theorem for critical points	p. 519
Properties of inward and outward correspondences	p. 527
Critical points of homotopic correspondences	p. 530
Other existence theorems for critical points	p. 532
Quasi-variational inequalities	p. 534
Selection of fixed point by pseudo-monotone functions	p. 535
Fixed point theorem for increasing maps	p. 538
Quasi-variational inequalities for increasing correspondences	p. 539
Other properties and examples of upper and lower semi-continuous correspondences	p. 542
Lower semi-continuity of preimages of linear operators	p. 543
Lower semi-continuity of correspondences defined by constraints	p. 547
Continuous selection theorem	p. 548
Weak Hausdorff topology on the family of closed subsets of topological vector spaces	p. 552
Relations between hemi-continuity and semi-continuity	p. 554
Summary of Linear Functional Analysis	p. 558
Hahn-Banach theorems	p. 558
Paired spaces	p. 559
Topologies of uniform convergence	p. 561
Topologies associated with a duality pairing	p. 563
The Banach-Steinhauss theorem	p. 565
The Knaster-Kuratowski-Mazurkiewicz Lemma	p. 566
Barycentric subdivision of simplexes	p. 566
Sequence of barycentric subdivisions	p. 569
The Sperner lemma	p. 570
The Knaster-Kuratowski-Mazurkiewicz lemma	p. 572
The Brouwer theorem	p. 575
Lyapunov's Theorem on the Range of A Vector Valued Measure	p. 577
Comments	p. 579
References	p. 587
Subject Index	p. 614
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Mathematical Methods of Game and Economic Theory Revised Edition

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Mathematical Methods of Game and Economic Theory Revised Edition

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