Mathematical Methods of Game and Economic Theory Revised Edition

Mathematical economics and game theory approached with the fundamental mathematical toolbox of nonlinear functional analysis are the central themes of this text. Both optimization and equilibrium theories are covered in full detail. The book's 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. Mathematicians, mathematical economists, and operations research specialists will find that it provides a solid foundation in nonlinear functional analysis.
This text begins by developing linear and convex analysis in the context of optimization theory. The treatment includes results on the existence and stability of solutions to optimization problems as well as an introduction to duality theory. The second part explores a number of topics in game theory and mathematical economics, including two-person games, which provide the framework to study theorems of nonlinear analysis. The text concludes with an introduction to non-linear analysis and optimal control theory, including an array of fixed point and subjectivity theorems that offer powerful tools in proving existence theorems.

Preface to the Dover EditionPreface (1982)Summary of ResultsContents of Other Possible CoursesNotationsI. Optimization and convex analysis1. Minimization problems and convexity2. Existence, uniqueness and stability of optimal solutions3. Compactness and continuity properties4. Differentiability and subdifferentiability: characterization of optimal solutions5. Introduction to duality theoryII. Game theory and the Walras model of allocation of resources6. Two-person games: an introduction7. Two-person zero-sum games: existence theorems8. The fundamental economic model: Walras equilibria9. Non-cooperative n-person games10. Main solution concepts of cooperative games11. Games with side-payments12. Games without side-paymentsIII. Non-linear analysis and optimal control theory13. Minimax type inequalities, monotone correspondences and gamma-convex functions14. Introduction to calculus of variations and optimal control15. Fixed point theorems, quasi-variational inequalities and correspondencesAppendix A. Summary of linear functional analysisAppendix B. The Knaster-Kuratowski-Mazurkiewicz lemmaAppendix C. Lyapunov's theorem on the range of a vector valued measureCommentsReferencesSubject index