Optima and Equilibria: An Introduction to Nonlinear AnalysisProgress in the theory of economic equilibria and in game theory has proceeded hand in hand with that of the mathematical tools used in the field, namely nonlinear analysis and, in particular, convex analysis. Jean-Pierre Aubin, one of the leading specialists in nonlinear analysis and its application to economics, has written a rigorous and concise - yet still elementary and self-contained - textbook providing the mathematical tools needed to study optima and equilibria, as solutions to problems, arising in economics, management sciences, operations research, cooperative and non-cooperative games, fuzzy games etc. It begins with the foundations of optimization theory, and mathematical programming, and in particular convex and nonsmooth analysis. Nonlinear analysis is then presented, first game-theoretically, then in the framework of set valued analysis. These results are then applied to the main classes of economic equilibria. The book contains numerous exercises and problems: the latter allow the reader to venture into areas of nonlinear analysis that lie beyond the scope of the book and of most graduate courses. |
Contents
II | 9 |
III | 10 |
IV | 11 |
VI | 13 |
VII | 15 |
VIII | 17 |
IX | 21 |
X | 24 |
XCIII | 303 |
XCIV | 304 |
XCVI | 305 |
XCVIII | 306 |
C | 307 |
CII | 310 |
CIII | 312 |
CIV | 313 |
XI | 25 |
XII | 27 |
XIII | 31 |
XIV | 35 |
XV | 37 |
XVI | 39 |
XVII | 43 |
XVIII | 48 |
XIX | 52 |
XX | 57 |
XXI | 61 |
XXII | 64 |
XXIII | 66 |
XXIV | 67 |
XXV | 70 |
XXVI | 75 |
XXVII | 76 |
XXVIII | 80 |
XXIX | 82 |
XXX | 84 |
XXXI | 87 |
XXXII | 91 |
XXXIII | 95 |
XXXIV | 97 |
XXXV | 99 |
XXXVI | 101 |
XXXVII | 102 |
XXXVIII | 104 |
XXXIX | 105 |
XL | 106 |
XLI | 108 |
XLII | 110 |
XLIII | 112 |
XLIV | 116 |
XLV | 125 |
XLVI | 130 |
XLVII | 135 |
XLVIII | 137 |
XLIX | 143 |
L | 144 |
LI | 148 |
LII | 149 |
LIII | 150 |
LIV | 152 |
LV | 154 |
LVI | 155 |
LVII | 157 |
LVIII | 159 |
LIX | 160 |
LX | 162 |
LXI | 167 |
LXII | 168 |
LXIII | 169 |
LXIV | 173 |
LXV | 174 |
LXVI | 179 |
LXVII | 184 |
LXVIII | 187 |
LXIX | 189 |
LXX | 190 |
LXXI | 192 |
LXXII | 193 |
LXXIII | 196 |
LXXIV | 197 |
LXXV | 205 |
LXXVI | 211 |
LXXVII | 216 |
LXXVIII | 218 |
LXXIX | 226 |
LXXX | 235 |
LXXXII | 237 |
LXXXIII | 242 |
LXXXIV | 247 |
LXXXV | 256 |
LXXXVI | 263 |
LXXXVII | 270 |
LXXXVIII | 277 |
LXXXIX | 282 |
XC | 287 |
XCI | 292 |
XCII | 299 |
CVI | 315 |
CVII | 316 |
CVIII | 317 |
CIX | 318 |
CX | 319 |
CXI | 321 |
CXII | 322 |
CXIII | 323 |
CXIV | 324 |
CXV | 325 |
CXVI | 326 |
CXVII | 327 |
CXVIII | 328 |
CXIX | 329 |
CXXI | 331 |
CXXIII | 332 |
CXXIV | 333 |
CXXV | 334 |
CXXVI | 335 |
CXXVIII | 336 |
CXXIX | 337 |
CXXX | 338 |
CXXXI | 339 |
CXXXII | 340 |
CXXXIII | 341 |
CXXXV | 342 |
CXXXVI | 343 |
CXXXVIII | 345 |
CXXXIX | 346 |
CXL | 349 |
CXLI | 350 |
CXLIII | 352 |
CXLV | 353 |
CXLVII | 354 |
CXLIX | 358 |
CL | 360 |
CLI | 361 |
CLII | 362 |
CLIII | 364 |
CLIV | 365 |
CLV | 367 |
CLVI | 368 |
CLVII | 369 |
CLVIII | 370 |
CLIX | 371 |
CLXII | 372 |
CLXIII | 374 |
CLXIV | 375 |
CLXV | 376 |
CLXVII | 377 |
CLXVIII | 378 |
CLXIX | 379 |
CLXX | 380 |
CLXXI | 381 |
CLXXII | 383 |
CLXXIII | 384 |
CLXXIV | 385 |
CLXXV | 386 |
CLXXVI | 387 |
CLXXVII | 388 |
CLXXVIII | 389 |
CLXXIX | 390 |
CLXXX | 391 |
CLXXXI | 392 |
CLXXXII | 393 |
CLXXXV | 395 |
CLXXXVI | 397 |
CLXXXVII | 399 |
CLXXXVIII | 401 |
CLXXXIX | 403 |
CXC | 405 |
CXCI | 406 |
CXCII | 407 |
CXCIII | 410 |
CXCIV | 411 |
CXCV | 413 |
CXCVI | 415 |
CXCVII | 417 |
CXCVIII | 423 |
| 429 | |
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Common terms and phrases
associate assumptions belongs Brouwer's Fixed-point Theorem closed values coalition conjugate function consider continuous linear operator continuous mapping converges convex and closed convex and lower convex closed cone convex closed subset convex compact subset convex function Corollary decision rules Deduce defined denote differentiable Domf Emil equivalent Exercise exists a solution f(xn f(xo finite fixed point function f fuzzy Gâteaux differentiable generalised graph Graph(F Hilbert space implies inf f(x inf sup K₁ locally Lipschitz loss functions lower semi-continuous function map F mathematics minimisation problem multilosses non-empty nontrivial function normal cone Pareto Pareto optimal partition of unity players problem h(y Proof Proposition satisfying Shapley value Show strategy pair subdifferential sup f(x sup inf Suppose that f tangent tangential condition theory topology upper semi-continuous variational inequality VxEK Walrasian equilibrium whence
Popular passages
Page 2 - The next subject to be mentioned concerns the static or dynamic nature of the theory. We repeat most emphatically that our theory is thoroughly static. A dynamic theory would unquestionably be more complete and therefore preferable. But there is ample evidence from other branches of science that it is futile to try to build one as long as the static side is not thoroughly understood. On the other hand, the reader may object to some definitely dynamic arguments which were made in the course of our...
Page 2 - Finally let us note a point at which the theory of social phenomena will presumably take a very definite turn away from the existing patterns of mathematical physics.
Page 2 - Our static theory specifies equilibria— ie solutions . . . which are sets of imputations. A dynamic theory— when one is found— will probably describe the changes in terms of simpler concepts: of a single imputation — valid at the moment under consideration— or something similar. This indicates that the formal structure of this part of the theory — the relationship between statics and dynamics — may be generically different from that of the classical physical theories.