Introduction to the Theory of Games

Approach your problems from the right It isn't that they can't see the solution. end and begin with the answers. Then It is that they can't see the problem. one day, perhaps you will find the final question. G. K. Chesterton. The Scandal of Father Brown 'The Point of a Pin'. 'The Hermit Clad in Crane Feathers' in R. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Min­ kowsky lemma, coding theory and the structure of water meet one another in packing and covering theory: quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces.

Contenu

1. On equilibrium of systems.- 1.1. Basic ideas.- 1.2. Chains and traversable regions.- 1.3. Equilibrium point, stability set, equilibrium set.- 1.4. "Equilibrium properties" of equilibrium points and equilibrium sets.- 1.5. On the existence of an equilibrium point.- 1.6. On the existence of stability sets.- 2. The n-person game.- 3. Existence theorems of equilibrium points.- 4. Special n-person games and methods to solve them.- 4.1. Mathematical programming methods for the solution of n-person concave games.- 4.2. Generalized polyhedral games.- 4.3. Solution of n-person zero-sum concave-convex games.- 4.4. Concave games with unique equilibrium points.- 5. The Scarf-Hansen algorithm for approximating an equilibrium point of a finite n-person game.- 6. The oligopoly game.- 6.1. The reduction principle.- 6.2. The general multiproduct case.- 6.3. The general linear case.- 6.4. The single-product case.- 7. Two-person games.- 8. Bimatrix games.- 8.1. Basic definitions and some simple properties of bimatrix games.- 8.2. Methods for solving bimatrix games.- 8.3. Examples.- 9. Matrix games.- 9.1. Equilibrium and the minimax principle.- 9.2. The set of equilibrium strategies.- 10. Symmetric games.- 11. Connection between matrix games and linear programming.- 12. Methods for solving general matrix games.- 12.1 Solution of matrix games by linear programming.- 12.2. Method of fictitious play.- 12.3. von Neumann's method.- 13. Some Special Games and methods.- 13.1. Matrices with saddle-points.- 13.2. Dominance relations.- 13.3. 2 x n games.- 13.4. Convex (concave) matrix games.- 14. Decomposition of matrix games.- 15. Examples of matrix games.- 15.1. Example 1.- 15.2. Example 2.- 16. Games played over the unit square.- 17. Some special classes of games on the unit square.- 18. Approximate solution of two- person zero-sum games played over the unit square.- 19. Two-person zero-sum games over metric spaces sequential games.- 20. Sequential games.- 20.1. Shapley's stochastic game.- 20.2. Recursive games.- 21. Games against nature.- 22. Cooperative games in characteristic function form.- 23. Solution concepts for n-person cooperative games.- 23.1. The von Neumann-Morgenstern solution.- 23.2. The core.- 23.3. The strong e-core.- 23.4. The kernel.- 23.5. The nucleolus.- 23.6. The Shapley-value.- 24. Stability of pay-off configurations.- 25. A bargaining model of cooperative games.- 26. The solution concept of nash for n-person cooperative games.- 27. Examples of cooperative games.- 27.1. A linear production game.- 27.2. A market game.- 27.3. The cooperative oligopoly game.- 27.4. A game theoretic approach for cost allocation: a case.- 27.5. Committee decision making as a game.- 28. Game theoretical treatment of multicriteria decision making.- 29. Games with incomplete information.- 29.1. The Harsanyi-model.- 29.2. The Selten-model.- 29.3. Dynamic processes and games with limited information about the pay-off function.- Epilogue.- References.- Name Index.