2007年新版封面



【资料名称】：Mathematical Methods of Game and Economic Theory (Studies in Mathematics and Its Applications Volume 7)
【资料作者】：Jean-Pierre Aubin
【出版社】：    North-Holland
【简介及目录】：

Product Details  

• Hardcover: 616 pages
• Publisher: North-Holland; Revised edition (July 1, 1982)
• Language: English
• ISBN-10: 0444851844

Product Description  
This book presents a unified treatment of optimization theory, gametheory and a general equilibrium theory in economics in the frameworkof nonlinear functional analysis. It not only provides powerful andversatile tools for solving specific problems in economics and thesocial sciences but also serves as a unifying theme in the mathematicaltheory of these subjects as well as in pure mathematics itself.

Synopsis
Mathematical economics and game theory approachedwith the fundamental mathematical toolbox of nonlinear functionalanalysis are the central themes of this text. Its central applicationis the fundamental economic problem of allocating scarce resourcesamong competing agents, which leads to considerations of theinterrelated applications in game theory and the theory ofoptimization.

From the Publisher
Mathematical economics and game theoryapproached with the fundamental mathematical toolbox of nonlinearfunctional analysis are the central themes of this text. Its centralapplication is the fundamental economic problem of allocating scarceresources among competing agents, which leads to considerations of theinterrelated applications in game theory and the theory ofoptimization.

Table of Contents
Preface to the Dover Edition     iii
Preface (1982)     vii
Summary of Results: A Guideline for the Reader     xxi
Contents of Other Possible Courses     xxvii
Notations     xxix
Optimization and Convex Analysis     1
Minimization Problems and Convexity     3
Strategy sets and loss functions     4
Optimization problem     4
Allocation of available commodities     5
Resource and service operators     6
Extension of loss functions     8
Sections and epigraphs     10
Decomposition principle     11
Product of a loss function by a linear operator     11
Example: Inf-convolution of functions     12
Decomposition principle     13
Another decomposition principle     15
Mixed strategies and convexity     17
Motivation: extension of strategy sets and loss functions     18
Mixed strategies and linearized loss functions     19
Interpretation of mixed strategies     21
Case of finite strategy sets     21
Representation by infinite sequences of pure strategies     22
Linearized extension of maps and the barycentric operator     24
Interpretation of convex functions in terms of risk aversion     25
Elementary properties of convex subsets and functions     25
Indicators, support functions and gauges     27
Indicators and support functions     28
Reformulation of the Hahn-Banach theorem     31
The bipolar theorem     32
Recession cones and barrier cones     34
Interpretation: production sets and profit functions     35
Gauges     38
Existence, Uniqueness and Stability of Optimal Solutions     42
Existence and uniqueness of an optimal solution     43
Structure of the optimal set     43
Existence of an optimal solution     45
Continuity versus compactness     45
Lower semi-continuity of convex functions in infinite dimensional spaces     45
Fundamental property of lower semi-continuous and compact functions     46
Uniqueness of an optimal solution     47
Non-satiation property     48
Minimization of quadratic functionals on convex sets     48
Hilbert spaces     49
Existence and uniqueness of the minimal solution     49
Characterization of the minimal solution     50
Projectors of best approximation      51
The duality map from an Hilbert space onto its dual     52
Minimization of quadratic functionals on subspaces     54
The fundamental formula     54
Orthogonal right inverse     56
Orthogonal left inverse     57
Another decomposition property     58
Interpretation     59
Perturbation by linear forms: conjugate functions     60
Conjugate functions     60
Characterization of lower semi-continuous convex functions     61
Examples of conjugate functions     62
Elementary properties of conjugate functions     64
Interpretation: cost and profit functions     65
Stability properties: an introduction to correspondences     66
Upper semi-continuous correspondences     66
Lower semi-continuous correspondences     68
Closed correspondences     70
Construction of upper semi-continuous correspondences     73
Compactness and Continuity Properties     75
Lower semi-compact functions     76
Coercive and semi-coercive functions     76
Functions such that f* is continuous at 0     77
Lower semi-compactness of linear forms     78
Constraint qualification hypothesis     79
Case of infinite dimensional spaces     81
Extension to compact subsets of mixed strategies     82
Proper maps and preimages of compact subsets     83
Proper maps     84
Compactness of some strategy sets     85
Examples where the map L* + 1 is proper     88
Continuous convex functions     90
A characterization of lower semi-continuous convex functions     90
A characterization of continuous convex functions     91
Examples of continuous convex functions     93
Continuity of gL and Lf     94
Continuous convex functions (continuation)     95
Strong continuity of lower semi-continuous convex functions     96
Estimates of lower semi-continuous convex functions     97
Characterization of continuous convex functions     98
Continuity of support functions     99
Maximum of a convex function: extremal points     100
Differentiability and Subdifferentiability: Characterization of Optimal Solutions     103
Subdifferentiability     105
Definitions     105
Examples of subdifferentials     106
Subdifferentiability of continuous convex functions     108
Upper semi-continuity of the subdifferential     109
Characterization of subdifferentiable convex functions     110
Differentiability and variational inequalities     111
Definitions     111
Differentiability and subdifferentiability     112
Legendre transform     113
Interpretation: marginal profit     114
Variational inequalities     114
Differentiability from the right     115
Definition and main inequalities     115
Derivatives from the right and the support function of the subdifferential     117
Derivative of a pointwise supremum     118
Local [epsilon]-subdifferentiability and perturbed minimization problems     120
Approximate optimal solutions in Banach spaces     121
The approximate variational principle     123
Local [epsilon]-subdifferentiability     124
Perturbation of minimization problems     126
Proof of Ekeland-Lebourg's theorem     130
Introduction to Duality Theory     133
Dual problem and Lagrange multipliers     135
Lagrangian     136
Lagrange multipliers and dual problem     137
Marginal interpretation of Lagrange multipliers     139
Example      140
Case of linear constraints: extremality relations     142
Generalized minimization problem     143
Extremality relations     145
The fundamental formula     146
Minimization problem under linear constraints     148
Minimization of a quadratic functional under linear constraints     148
Minimization problem under linear equality constraints     149
Duality and the decomposition principle     150
The decentralization principle     151
Conjugate function of gL     152
Conjugate function of f[subscript 1]+f[subscript 2]     153
Minimization of the projection of a function     154
Minimization on the diagonal of a product     154
Existence of Lagrange multipliers in the case of a finite number of constraints     155
The Fenchel existence theorem     156
Stability properties     157
Applications to subdifferentiability     158
Case of nonlinear constraints: The Uzawa existence theorem     159
Game Theory and the Walras Model of Allocation of Resources     363
Two-Person Games: An Introduction     165
Some solution concepts     167
Description of the game     167
Shadow minimum     367
Conservative solutions and values     168
Non-cooperative equilibrium     169
Pareto minimum     170
Core of a two-person game     171
Selection of strategy of the core     171
Examples: some finite games     172
Example     173
Coordination game     175
Prisoner's dilemma     178
Game of chicken     180
The battle of the sexes     182
Example: Analysis of duopoly     183
The model of a duopoly     184
The set of Pareto minima     185
Conservative solutions     185
Non-cooperative equilibria     186
Stackelberg equilibria     187
Stackelberg disequilibrium     187
Example: Edgeworth economic game     189
The set of feasible allocations     190
The biloss operator     190
The Edgeworth box     192
Pareto minima     193
Core     193
Walras equilibria     194
Two-person zero-sum games     195
Duality gap and value     195
Saddle point     197
Perturbation by linear functions      198
Case of finite strategy sets: Matrix games     200
Two-Person Zero-Sum Games: Existence Theorems     204
The fundamental existence theorems     206
Existence of conservative solutions     208
Decision rules     211
Finite topology on convex subsets     211