Mathematical Methods of Game and Economic Theory

ISBN-10: 048646265X
ISBN-13: 9780486462653
Edition: 2007 (Revised)
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Description: Mathematical economics and game theory approached with the fundamental mathematical toolbox of nonlinear functional analysis are the central themes of this text.nbsp;Its central application is the fundamental economic problem of allocating scarce  More...

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Book details

List price: $32.95
Copyright year: 2007
Publisher: Dover Publications, Incorporated
Publication date: 11/2/2007
Binding: Paperback
Pages: 656
Size: 6.25" wide x 9.25" long x 1.50" tall
Weight: 1.782

Mathematical economics and game theory approached with the fundamental mathematical toolbox of nonlinear functional analysis are the central themes of this text.nbsp;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
Preface (1982)
Summary of Results: A Guideline for the Reader
Contents of Other Possible Courses
Notations
Optimization and Convex Analysis
Minimization Problems and Convexity
Strategy sets and loss functions
Optimization problem
Allocation of available commodities
Resource and service operators
Extension of loss functions
Sections and epigraphs
Decomposition principle
Product of a loss function by a linear operator
Example: Inf-convolution of functions
Decomposition principle
Another decomposition principle
Mixed strategies and convexity
Motivation: extension of strategy sets and loss functions
Mixed strategies and linearized loss functions
Interpretation of mixed strategies
Case of finite strategy sets
Representation by infinite sequences of pure strategies
Linearized extension of maps and the barycentric operator
Interpretation of convex functions in terms of risk aversion
Elementary properties of convex subsets and functions
Indicators, support functions and gauges
Indicators and support functions
Reformulation of the Hahn-Banach theorem
The bipolar theorem
Recession cones and barrier cones
Interpretation: production sets and profit functions
Gauges
Existence, Uniqueness and Stability of Optimal Solutions
Existence and uniqueness of an optimal solution
Structure of the optimal set
Existence of an optimal solution
Continuity versus compactness
Lower semi-continuity of convex functions in infinite dimensional spaces
Fundamental property of lower semi-continuous and compact functions
Uniqueness of an optimal solution
Non-satiation property
Minimization of quadratic functionals on convex sets
Hilbert spaces
Existence and uniqueness of the minimal solution
Characterization of the minimal solution
Projectors of best approximation
The duality map from an Hilbert space onto its dual
Minimization of quadratic functionals on subspaces
The fundamental formula
Orthogonal right inverse
Orthogonal left inverse
Another decomposition property
Interpretation
Perturbation by linear forms: conjugate functions
Conjugate functions
Characterization of lower semi-continuous convex functions
Examples of conjugate functions
Elementary properties of conjugate functions
Interpretation: cost and profit functions
Stability properties: an introduction to correspondences
Upper semi-continuous correspondences
Lower semi-continuous correspondences
Closed correspondences
Construction of upper semi-continuous correspondences
Compactness and Continuity Properties
Lower semi-compact functions
Coercive and semi-coercive functions
Functions such that f* is continuous at 0
Lower semi-compactness of linear forms
Constraint qualification hypothesis
Case of infinite dimensional spaces
Extension to compact subsets of mixed strategies
Proper maps and preimages of compact subsets
Proper maps
Compactness of some strategy sets
Examples where the map L* + 1 is proper
Continuous convex functions
A characterization of lower semi-continuous convex functions
A characterization of continuous convex functions
Examples of continuous convex functions
Continuity of gL and Lf
Continuous convex functions (continuation)
Strong continuity of lower semi-continuous convex functions
Estimates of lower semi-continuous convex functions
Characterization of continuous convex functions
Continuity of support functions
Maximum of a convex function: extremal points
Differentiability and Subdifferentiability: Characterization of Optimal Solutions
Subdifferentiability
Definitions
Examples of subdifferentials
Subdifferentiability of continuous convex functions
Upper semi-continuity of the subdifferential
Characterization of subdifferentiable convex functions
Differentiability and variational inequalities
Definitions
Differentiability and subdifferentiability
Legendre transform
Interpretation: marginal profit
Variational inequalities
Differentiability from the right
Definition and main inequalities
Derivatives from the right and the support function of the subdifferential
Derivative of a pointwise supremum
Local [epsilon]-subdifferentiability and perturbed minimization problems
Approximate optimal solutions in Banach spaces
The approximate variational principle
Local [epsilon]-subdifferentiability
Perturbation of minimization problems
Proof of Ekeland-Lebourg's theorem
Introduction to Duality Theory
Dual problem and Lagrange multipliers
Lagrangian
Lagrange multipliers and dual problem
Marginal interpretation of Lagrange multipliers
Example
Case of linear constraints: extremality relations
Generalized minimization problem
Extremality relations
The fundamental formula
Minimization problem under linear constraints
Minimization of a quadratic functional under linear constraints
Minimization problem under linear equality constraints
Duality and the decomposition principle
The decentralization principle
Conjugate function of gL
Conjugate function of f[subscript 1]+f[subscript 2]
Minimization of the projection of a function
Minimization on the diagonal of a product
Existence of Lagrange multipliers in the case of a finite number of constraints
The Fenchel existence theorem
Stability properties
Applications to subdifferentiability
Case of nonlinear constraints: The Uzawa existence theorem
Game Theory and the Walras Model of Allocation of Resources
Two-Person Games: An Introduction
Some solution concepts
Description of the game
Shadow minimum
Conservative solutions and values
Non-cooperative equilibrium
Pareto minimum
Core of a two-person game
Selection of strategy of the core
Examples: some finite games
Example
Coordination game
Prisoner's dilemma
Game of chicken
The battle of the sexes
Example: Analysis of duopoly
The model of a duopoly
The set of Pareto minima
Conservative solutions
Non-cooperative equilibria
Stackelberg equilibria
Stackelberg disequilibrium
Example: Edgeworth economic game
The set of feasible allocations
The biloss operator
The Edgeworth box
Pareto minima
Core
Walras equilibria
Two-person zero-sum games
Duality gap and value
Saddle point
Perturbation by linear functions
Case of finite strategy sets: Matrix games
Two-Person Zero-Sum Games: Existence Theorems
The fundamental existence theorems
Existence of conservative solutions
Decision rules
Finite topology on convex subsets
Existence of an optimal decision rule
The Ky-Fan inequality
The Lasry theorem
The minisup theorem
The Nikaido theorem
Existence of saddle points
Another existence theorem for saddle points
Extension of games without and with exchange of informations
Definition of extensions of games
Mixed extensions
Extensions without exchange of information
Sequential extensions
Extensions with exchange of information
Iterated games
Iterated extensions
The Moulin theorem
Proof of playability of iterated extensions
A system of functional equations
A lemma on successive approximations
Proof of existence of saddle decision rules
The Fundamental Economic Model: Walras Equilibria
Description of the model
The subset of available commodities
Appropriation of the economy
Demand correspondences
Walras equilibrium
Examples of subsets of available commodities and of appropriations
Example: Quadratic demand functions
Existence of a Walras equilibrium
Existence of a Walras pre-equilibrium
Surjectivity of correspondences: the Debreu-Gale-Nikaido theorem
Demand correspondences defined by loss functions
Statement of the existence theorem
Upper semi-continuity of the demand correspondence
Compactification of an economy
Proof of the existence of a Walras equilibrium
Economies with producers
Description of the model
Statement of the existence theorem
Compactification
Proof of the existence of a Walras equilibrium
Non-Cooperative n-Person Games
Existence of a non-cooperative equilibrium
Games described in strategic form
Conservative values and multistrategies
Non-cooperative equilibria
The Nash theorem
Stability
Associated variational inequalities
Case of quadratic loss functions; application to Walras-Cournot equilibria
Non-cooperative games with quadratic loss functions
Existence of solutions of variational inequalities
Examples
Multistrategy sets defined by linear constraints
Walras-Cournot equilibria
Constrained non-cooperative games and fixed point theorems
Selection of a fixed point
Equilibria of constrained non-cooperative games
Fixed-point theorems
Non-cooperative Walras equilibria
Description of the model
Existence of a non-cooperative Walras equilibrium: the Arrow-Debreu theorem
Non-cooperative Walras equilibria of economies with producers
Main Solution Concepts of Cooperative Games
Behavior of the whole set of players: Pareto strategies
Pareto strategies
Rates of transfer
Pareto multipliers
Pareto allocations
Selection of Pareto strategies and imputations
Normalized games
Pareto strategies obtained by using selection functions
Closest strategy to the shadow minimum
The best compromise
Existence of Pareto strategies
Interpretation: threat functionals
Imputations: the Nash bargaining solution
Behavior of coalitions of players: the core
Coalitions
Cooperative game described in strategic form and its core
The multiloss operator F[superscript A]# of the coalition A
Examples of multistrategy sets X(A)
Economic games and core of an economy
Cooperative game described in characteristic form and its core
Behavior of fuzzy coalitions: the fuzzy core
Fuzzy coalitions
Extension of a family of coalitions
Debreu-Scarf coalitions
Fuzzy coalitions on a continuum of players
Fuzzy games described in characteristic form
Characterization of the core of a (fuzzy) game
Fuzzy economic games and fuzzy core of an economy
Fuzzy games described in strategic form and fuzzy core
Selection of elements of the core: cooperative equilibrium and nucleolus
Canonical cooperative equilibrium
Least-core
Nucleolus
Games With Side-Payments
Core of a fuzzy game with side-payments
Core of a game with side-payments
Linear games
Non-emptiness of the core of fuzzy games with side-payments
Core of fuzzy market games
Core of a game with side-payments
Convex cover of a game
Non-emptiness of the core of a balanced game
Balanced family of multistrategy sets
Balanced characteristic functions and convex loss functions
Further properties of convex functions and balances
Values of fuzzy games
The diagonal property
Sequence of fuzzy values
Existence and uniqueness of a sequence of fuzzy values
Relations between core and fuzzy value
Best approximation property of fuzzy values
Generalized solution to locally Lipschitz games
Shapley value and nucleolus of games with side-payments
The Shapley value
Existence and uniqueness of a Shapley value
Simple games
Nucleolus of games with side-payments
Games Without Side-Payments
Equivalence between the fuzzy core and the set of equilibria
Representation of a game
Equilibrium of a representation
Cover associated with a representation
Fuzzy core of a representation
The equivalence theorem
Non-emptiness of the fuzzy core of a balanced game
Statement of theorems of non-emptiness of the fuzzy core
Upper semi-continuity of the associated side-payment games
Existence of approximate cooperative equilibria
Proof of the non-emptiness of the core
Equivalence between the fuzzy core of an economy and the set of Walras allocations
Representation of economic games
Fuzzy core and Walras allocations
The equivalence theorem
Non-Linear Analysis and Optimal Control Theory
Minimax Type Inequalities, Monotone Correspondences and [gamma]-Convex Functions
Relaxation of compactness assumptions
Existence of a conservative solution
Proof of existence of a conservative solution
Existence of optimal decision rules and minisup under weaker compactness assumptions
Relaxation of continuity assumptions: variational inequalities for monotone correspondences
Variational inequalities
Existence of a solution to variational inequalities for completely upper semi-continuous correspondences
Pseudo-monotone functions: the Brezis-Nirenberg-Stampacchia theorem
Existence of a solution to variational inequalities for pseudo-monotone maps
Pseudo-monotonicity of monotone maps
Monotone and cyclically monotone correspondences
Maximal monotone correspondences
Relaxation of convexity assumptions
Definition of [gamma]-convex functions
The fundamental characteristic property of families of [gamma]-convex functions
The minisup theorem for [gamma subscript x]-convex-[gamma subscript y]-concave functions
Existence of optimal decision rules for functions [gamma subscript y]-concave with respect to y
Example: Image of a cone of convex functions by [pi]*
Relations between convexity and [gamma]-convexity
Example: [beta]-convex set functions
Example: Convex functions of atomless vector measures
Introduction to Calculus of Variations and Optimal Control
Duality in infinite dimensional spaces
Lagrangian of a minimization problem under linear constraints
Extremality relations
Existence of a Lagrange multiplier under the Slater condition
Relaxation of the Slater condition
Generalized Lagrangian of a minimization problem
Characterization of a Lagrangian by perturbations of the minimization problem
Duality in the case of non-convex integral criterion and contraints
Modulus of non-convexity of a function
Estimate of the duality gap
The Shapley-Folkman theorem
Sharp estimate of the duality gap
Applications
Extremality relations
The Aumann-Perles duality theorem
The approximation procedure
Duality in calculus of variations
The Green formula
Abstract problem of calculus of variations
The Hamiltonian system
Lagrangian of a problem of calculus of variations
Existence of a Lagrange multiplier
Example: the Dirichlet variational problem
The maximum principle for optimal control problems
Optimal control and impulsive control problems
The Hamilton-Jacobi-Bellman equation of a control problem
Construction of the closed loop control
The principle of optimality
The quadratic case: Riccati equations
The Bensoussan-Lions variational inequalities of a stopping time problem
Construction of the optimal stopping time
The Bensoussan-Lions quasi-variational inequalities of an impulsive control problem
Construction of the optimal impulsive control
Fixed Point Theorems, Quasi-Variational Inequalities and Correspondences
Fixed point and surjectivity theorems for correspondences
The Browder-Ky-Fan existence theorem for critical points
Properties of inward and outward correspondences
Critical points of homotopic correspondences
Other existence theorems for critical points
Quasi-variational inequalities
Selection of fixed point by pseudo-monotone functions
Fixed point theorem for increasing maps
Quasi-variational inequalities for increasing correspondences
Other properties and examples of upper and lower semi-continuous correspondences
Lower semi-continuity of preimages of linear operators
Lower semi-continuity of correspondences defined by constraints
Continuous selection theorem
Weak Hausdorff topology on the family of closed subsets of topological vector spaces
Relations between hemi-continuity and semi-continuity
Summary of Linear Functional Analysis
Hahn-Banach theorems
Paired spaces
Topologies of uniform convergence
Topologies associated with a duality pairing
The Banach-Steinhauss theorem
The Knaster-Kuratowski-Mazurkiewicz Lemma
Barycentric subdivision of simplexes
Sequence of barycentric subdivisions
The Sperner lemma
The Knaster-Kuratowski-Mazurkiewicz lemma
The Brouwer theorem
Lyapunov's Theorem on the Range of A Vector Valued Measure
Comments
References
Subject Index

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