Produktbild: Theory of Games and Economic Behavior

Theory of Games and Economic Behavior 60th Anniversary Commemorative Edition

Name: Theory of Games and Economic Behavior
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60th Anniversary Commemorative Edition

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PREFACE
TECHNICAL NOTE
ACKNOWLEDGMENT
CHAPTER I
FORMULATION OF THE ECONOMIC PROBLEM
1. THE MATHEMATICAL METHOD IN ECONOMICS
- 1.1. Introductory remarks
- 1.2. Difficulties of the application of the mathematical method
- 1.3. Necessary limitations of the objectives
- 1.4. Concluding remarks
2. QUALITATIVE DISCUSSION OF THE PROBLEM OF RATIONAL BEHAVIOR
- 2.1. The problem of rational behavior
- 2.2. “Robinson Crusoe” economy and social exchange economy
- 2.3. The number of variables and the number of participants
- 2.4. The case of many participants: Free competition
- 2.5. The “Lausanne” theory
3. THE NOTION OF UTILITY
- 3.1. Preferences and utilities
- 3.2. Principles of measurement: Preliminaries
- 3.3. Probability and numerical utilities
- 3.4. Principles of measurement: Detailed discussion
- 3.5. Conceptual structure of the axiomatic treatment of numerical utilities
- 3.6. The axioms and their interpretation
- 3.7. General remarks concerning the axioms
- 3.8. The role of the concept of marginal utility
4. STRUCTURE OF THE THEORY: SOLUTIONS AND STANDARDS OF BEHAVIOR
- 4.1. The simplest concept of a solution for one participant
- 4.2. Extension to all participants
- 4.3. The solution as a set of imputations
- 4.4. The intransitive notion of “superiority” or “domination”
- 4.5. The precise definition of a solution
- 4.6. Interpretation of our definition in terms of “standards of behavior”
- 4.7. Games and social organizations
- 4.8. Concluding remarks
CHAPTER II
GENERAL FORMAL DESCRIPTION OF GAMES OF STRATEGY
5. INTRODUCTION
- 5.1. Shift of emphasis from economics to games
- 5.2. General principles of classification and of procedure
6. THE SIMPLIFIED CONCEPT OF A GAME
- 6.1. Explanation of the termini technici
- 6.2. The elements of the game
- 6.3. Information and preliminary
- 6.4. Preliminarity, transitivity, and signaling
7. THE COMPLETE CONCEPT OF A GAME
- 7.1. Variability of the characteristics of each move
- 7.2. The general description
8. SETS AND PARTITIONS
- 8.1. Desirability of a set-theoretical description of a game
- 8.2. Sets, their properties, and their graphical representation
- 8.3. Partitions, their properties, and their graphical representation
- 8.4. Logistic interpretation of sets and partitions
*9. THE SET-THEORETICAL DESCRIPTION OF A GAME
- *9.1. The partitions which describe a game
- *9.2. Discussion of these partitions and their properties
*10. AXIOMATIC FORMULATION
- *10.1. The axioms and their interpretations
- *10.2. Logistic discussion of the axioms
- *10.3. General remarks concerning the axioms
- *10.4. Graphical representation
11. STRATEGIES AND THE FINAL SIMPLIFICATION OF THE DESCRIPTION OF A GAME
- 11.1. The concept of a strategy and its formalization
- 11.2. The final simplification of the description of a game
- 11.3. The role of strategies in the simplified form of a game
- 11.4. The meaning of the zero-sum restriction
CHAPTER III
ZERO-SUM TWO-PERSON GAMES: THEORY
12. PRELIMINARY SURVEY
- 12.1. General viewpoints
- 12.2. The one-person game
- 12.3. Chance and probability
- 12.4. The next objective
13. FUNCTIONAL CALCULUS
- 13.1. Basic definitions
- 13.2. The operations Max and Min
- 13.3 Commutativity questions
- 13.4. The mixed case. Saddle points
- 13.5. Proofs of the main facts
14. STRICTLY DETERMINED GAMES
- 14 1. Formulation of the problem
- 14.2. The minorant and the majorant games
- 14.3. Discussion of the auxiliary games
- 14.4. Conclusions
- 14.5. Analysis of strict determinateness
- 14.6. The interchange of players. Symmetry
- 14.7. Non strictly determined games
- 14.8. Program of a detailed analysis of strict determinateness
*15. GAMES WITH PERFECT INFORMATION
- *15.1. Statement of purpose. Induction
- *15.2. The exact condition (First step)
- *15.3. The exact condition (Entire induction)
- *15.4. Exact discussion of the inductive step
- *15.5. Exact discussion of the inductive step (Continuation)
- *15.6. The result in the case of perfect information
- *15.7. Application to Chess
- *15.8. The alternative, verbal discussion
16. LINEARITY AND CONVEXITY
- 16.1. Geometrical background
- 16.2. Vector operations
- 16.3. The theorem of the supporting hyperplanes
- 16.4. The theorem of the alternative for matrices
17. MIXED STRATEGIES. THE SOLUTION FOR ALL GAMES
- 17.1. Discussion of two elementary examples
- 17.2. Generalization of this viewpoint
- 17.3. Justification of the procedure as applied to an individual play
- 17.4. The minorant and the majorant games. (For mixed strategies)
- 17.5. General strict determinateness
- 17.6. Proof of the main theorem
- 17.7. Comparison of the treatment by pure and by mixed strategies
- 17.8. Analysis of general strict determinateness
- 17.9. Further characteristics of good strategies
- 17.10. Mistakes and their consequences. Permanent optimality
- 17.11. The interchange of players. Symmetry
CHAPTER IV
ZERO-SUM TWO-PERSON GAMES: EXAMPLES
18. SOME ELEMENTARY GAMES
- 18.1. The simplest games
- 18.2. Detailed quantitative discussion of these games
- 18.3. Qualitative characterizations
- 18.4. Discussion of some specific games. (Generalized forms of Matching Pennies)
- 18.5. Discussion of some slightly more complicated games
- 18.6. Chance and imperfect information
- 18.7. Interpretation of this result
*19. POKER AND BLUFFING
- *19.1. Description of Poker
- *19.2. Bluffing
- *19.3. Description of Poker (Continued)
- *19.4. Exact formulation of the rules
- *19.5. Description of the strategy
- *19.6. Statement of the problem
- *19.7. Passage from the discrete to the continuous problem
- *19.8. Mathematical determination of the solution
- *19.9. Detailed analysis of the solution
- *19.10. Interpretation of the solution
- *19.11. More general forms of Poker
- *19.12. Discrete hands
- *19.13. m possible bids
- *19.14. Alternate bidding
- *19.15. Mathematical description of all solutions
- *19.16. Interpretation of the solutions. Conclusions
CHAPTER V
ZERO-SUM THREE-PERSON GAMES
20. PRELIMINARY SURVEY
- 20.1. General viewpoints
- 20.2. Coalitions
21. THE SIMPLE MAJORITY GAME OF THREE PERSONS
- 21.1. Definition of the game
- 21.2. Analysis of the game: Necessity of “understandings”
- 21.3. Analysis of the game: Coalitions. The role of symmetry
22. FURTHER EXAMPLES
- 22.1. Unsymmetric distributions. Necessity of compensations
- 22.2. Coalitions of different strength. Discussion
- 22.3. An inequality. Formulae
23. THE GENERAL CASE
- 23.1. Detailed discussion. Inessential and essential games
- 23.2. Complete formulae
24. DISCUSSION OF AN OBJECTION
- 24.1. The case of perfect information and its significance
- 24.2. Detailed discussion. Necessity of compensations between three or more players
CHAPTER VI
FORMULATION OF THE GENERAL THEORY: ZERO-SUM n-PERSON GAMES
25. THE CHARACTERISTIC FUNCTION
- 25.1. Motivation and definition
- 25.2. Discussion of the concept
- 25.3. Fundamental properties
- 25.4. Immediate mathematical consequences
26. CONSTRUCTION OF A GAME WITH A GIVEN CHARACTERISTIC FUNCTION
- 26.1. The construction
- 26.2. Summary
27. STRATEGIC EQUIVALENCE. INESSENTIAL AND ESSENTIAL GAMES
- 27.1. Strategic equivalence. The reduced form
- 27.2. Inequalities. The quantity γ
- 27.3. Inessentiality and essentiality
- 27.4. Various criteria. Non additive utilities
- 27.5. The inequalities in the essential case
- 27.6. Vector operations on characteristic functions
28. GROUPS, SYMMETRY AND FAIRNESS
- 28.1. Permutations, their groups and their effect on a game
- 28.2. Symmetry and fairness
29. RECONSIDERATION OF THE ZERO-SUM THREE-PERSON GAME
- 29.1. Qualitative discussion
- 29.2. Quantitative discussion
30. THE EXACT FORM OF THE GENERAL DEFINITIONS
- 30.1. The definitions
- 30.2. Discussion and recapitulation
- *30.3. The concept of saturation
- 30.4. Three immediate objectives
31. FIRST CONSEQUENCES
- 31.1. Convexity, flatness, and some criteria for domination
- 31.2. The system of all imputations. One element solutions
- 31.3. The isomorphism which corresponds to strategic equivalence
32. DETERMINATION OF ALL SOLUTIONS OF THE ESSENTIAL ZERO-SUM THREE-PERSON GAME
- 32.1. Formulation of the mathematical problem. The graphical method
- 32.2. Determination of all solutions
33. CONCLUSIONS
- 33.1. The multiplicity of solutions. Discrimination and its meaning
- 33.2. Statics and dynamics
CHAPTER VII
ZERO-SUM FOUR-PERSON GAMES
34. PRELIMINARY SURVEY
- 34.1. General viewpoints
- 34.2. Formalism of the essential zero sum four person games
- 34.3. Permutations of the players
35. DISCUSSION OF SOME SPECIAL POINTS IN THE CUBE Q
- 35.1. The corner I. (and V., VI., VII.)
- 35.2. The corner VIII. (and II., III., IV.,). The three person game and a “Dummy”
- 35.3. Some remarks concerning the interior of Q
36. DISCUSSION OF THE MAIN DIAGONALS
- 36.1. The part adjacent to the corner VIII.: Heuristic discussion
- 36.2. The part adjacent to the corner VIII.: Exact discussion
- *36.3. Other parts of the main diagonals
37. THE CENTER AND ITS ENVIRONS
- 37.1. First orientation about the conditions around the center
- 37.2. The two alternatives and the role of symmetry
- 37.3. The first alternative at the center
- 37.4. The second alternative at the center
- 37.5. Comparison of the two central solutions
- 37.6. Unsymmetrical central solutions
*38. A FAMILY OF SOLUTIONS FOR A NEIGHBORHOOD OF THE CENTER
- *38.1. Transformation of the solution belonging to the first alternative at the center
- *38.2. Exact discussion
- *38.3. Interpretation of the solutions
CHAPTER VIII
SOME REMARKS CONCERNING n ≥ 5 PARTICIPANTS
39. THE NUMBER OF PARAMETERS IN VARIOUS CLASSES OF GAMES
- 39.1. The situation for n = 3, 4
- 39.2. The situation for all n ≥ 3
40. THE SYMMETRIC FIVE PERSON GAME
- 40.1. Formalism of the symmetric five person game
- 40.2. The two extreme cases
- 40.3. Connection between the symmetric five person game and the 1, 2, 3-symmetric four person game
CHAPTER IX
COMPOSITION AND DECOMPOSITION OF GAMES
41. COMPOSITION AND DECOMPOSITION
- 41.1. Search for n-person games for which all solutions can be determined
- 41.2. The first type. Composition and decomposition
- 41.3. Exact definitions
- 41.4. Analysis of decomposability
- 41.5. Desirability of a modification
42. MODIFICATION OF THE THEORY
- 42.1. No complete abandonment of the zero sum restriction
- 42.2. Strategic equivalence. Constant sum games
- 42.3. The characteristic function in the new theory
- 42.4. Imputations, domination, solutions in the new theory
- 42.5. Essentiality, inessentiality and decomposability in the new theory
43. THE DECOMPOSITION PARTITION
- 43.1. Splitting sets. Constituents
- 43.2. Properties of the system of all splitting sets
- 43.3. Characterization of the system of all splitting sets. The decomposition partition
- 43.4. Properties of the decomposition partition
44. DECOMPOSABLE GAMES. FURTHER EXTENSION OF THE THEORY
- 44.1. Solutions of a (decomposable) game and solutions of its constituents
- 44.2. Composition and decomposition of imputations and of sets of imputations
- 44.3. Composition and decomposition of solutions. The main possibilities and surmises
- 44.4. Extension of the theory. Outside sources
- 44.5. The excess
- 44.6. Limitations of the excess. The non-isolated character of a game in the new setup
- 44.7. Discussion of the new setup. E(e0), F(e0)
45. LIMITATIONS OF THE EXCESS. STRUCTURE OF THE EXTENDED THEORY
- 45.1. The lower limit of the excess
- 45.2. The upper limit of the excess. Detached and fully detached imputations
- 45.3. Discussion of the two limits, |Γ|1. |Γ|2. Their ratio
- 45.4. Detached imputations and various solutions. The theorem connecting E(e0), F(e0)
- 45.5. Proof of the theorem
- 45.6. Summary and conclusions
46. DETERMINATION OF ALL SOLUTIONS OF A DECOMPOSABLE GAME
- 46.1. Elementary properties of decompositions
- 46.2. Decomposition and its relation to the solutions: First results concerning F(e0)
- 46.3. Continuation
- 46.4. Continuation
- 46.5. The complete result in F(e0)
- 46.6. The complete result in E(e0)
- 46.7. Graphical representation of a part of the result
- 46.8. Interpretation: The normal zone. Heredity of various properties
- 46.9. Dummies
- 46.10. Imbedding of a game
- 46.11. Significance of the normal zone
- 46.12. First occurrence of the phenomenon of transfer: n = 6
47. THE ESSENTIAL THREE-PERSON GAME IN THE NEW THEORY
- 47.1. Need for this discussion
- 47.2. Preparatory considerations
- 47.3. The six cases of the discussion. Cases (I)−(III)
- 47.4. Case (IV): First part
- 47.5. Case (IV): Second part
- 47.6. Case (V)
- 47.7. Case (VI)
- 47.8. Interpretation of the result: The curves (one dimensional parts) in the solution
- 47.9. Continuation: The areas (two dimensional parts) in the solution
CHAPTER X
SIMPLE GAMES
48. WINNING AND LOSING COALITIONS AND GAMES WHERE THEY OCCUR
- 48.1. The second type of 41.1. Decision by coalitions
- 48.2. Winning and Losing Coalitions
49. CHARACTERIZATION OF THE SIMPLE GAMES
- 49.1. General concepts of winning and losing coalitions
- 49.2. The special role of one element sets
- 49.3. Characterization of the systems W, L of actual games
- 49.4. Exact definition of simplicity
- 49.5. Some elementary properties of simplicity
- 49.6. Simple games and their W, L. The Minimal winning coalitions: Wm
- 49.7. The solutions of simple games
50. THE MAJORITY GAMES AND THE MAIN SOLUTION
- 50.1. Examples of simple games: The majority games
- 50.2. Homogeneity
- 50.3. A more direct use of the concept of imputation in forming solutions
- 50.4. Discussion of this direct approach
- 50.5. Connections with the general theory. Exact formulation
- 50.6. Reformulation of the result
- 50.7. Interpretation of the result
- 50.8. Connection with the Homogeneous Majority game.
51. METHODS FOR THE ENUMERATION OF ALL SIMPLE GAMES
- 51.1. Preliminary Remarks
- 51.2. The saturation method: Enumeration by means of W
- 51.3. Reasons for passing from W to Wm. Difficulties of using Wm
- 51.4. Changed Approach: Enumeration by means of Wm
- 51.5. Simplicity and decomposition
- 51.6. Inessentiality, Simplicity and Composition. Treatment of the excess
- 51.7. A criterium of decomposability in terms of Wm
52. THE SIMPLE GAMES FOR SMALL n
- 52.1. Program. n = 1, 2 play no role. Disposal of n = 3
- 52.2. Procedure for n ≥ 4: The two element sets and their role in classifying the Wm
- 52.3. Decomposability of cases C*, Cn−2, Cn−1
- 52.4. The simple games other than [1, • • •, 1, n − 2]h (with dummies): The Cases Ck, k = 0, 1, • • •, n − 3
- 52.5. Disposal of n = 4, 5
53. THE NEW POSSIBILITIES OF SIMPLE GAMES FOR n ≥ 6
- 53.1. The Regularities observed for n ≥ 6
- 53.2. The six main counter examples (for n = 6, 7)
54. DETERMINATION OF ALL SOLUTIONS IN SUITABLE GAMES
- 54.1. Reasons to consider other solutions than the main solution in simple games
- 54.2. Enumeration of those games for which all solutions are known
- 54.3. Reasons to consider the simple game [1, • • •, 1, n – 2]h
*55. THE SIMPLE GAME [1, • • •, 1, n – 2]h
- *55.1. Preliminary Remarks
- *55.2. Domination. The chief player. Cases (I) and (II)
- *55.3. Disposal of Case (I)
- *55.4. Case (II): Determination of Ṿ
- *55.5. Case (II): Determination of Ṿ
- *55.6. Case (II): a and S*
- *55.7. Case (II′) and (II″). Disposal of Case (II′)
- *55.8. Case (II″): a and V′. Domination
- *55.9. Case (II″): Determination of V′
- *55.10. Disposal of Case (II″)
- *55.11. Reformulation of the complete result
- *55.12. Interpretation of the result
CHAPTER XI
GENERAL NON-ZERO-SUM GAMES
56. EXTENSION OF THE THEORY
- 56.1. Formulation of the problem
- 56.2. The fictitious player. The zero sum extension Γ
- 56.3. Questions concerning the character of Γ
- 56.4. Limitations of the use of Γ
- 56.5. The two possible procedures
- 56.6. The discriminatory solutions
- 56.7. Alternative possibilities
- 56.8. The new setup
- 56.9. Reconsideration of the case when Γ is a zero sum game
- 56.10. Analysis of the concept of domination
- 56.11. Rigorous discussion
- 56.12. The new definition of a solution
57. THE CHARACTERISTIC FUNCTION AND RELATED TOPICS
- 57.1. The characteristic function: The extended and the restricted form
- 57.2. Fundamental properties
- 57.3. Determination of all characteristic functions
- 57.4. Removable sets of players
- 57.5. Strategic equivalence. Zero-sum and constant-sum games
58. INTERPRETATION OF THE CHARACTERISTIC FUNCTION
- 58.1. Analysis of the definition
- 58.2. The desire to make a gain vs. that to inflict a loss
- 58.3. Discussion
59. GENERAL CONSIDERATIONS
- 59.1. Discussion of the program
- 59.2. The reduced forms. The inequalities
- 59.3. Various topics
60. THE SOLUTIONS OF ALL GENERAL GAMES WITH n ≦ 3
- 60.1. The case n = 1
- 60.2. The case n = 2
- 60.3. The case n = 3
- 60.4. Comparison with the zero sum games
61. ECONOMIC INTERPRETATION OF THE RESULTS FOR n = 1, 2
- 61.1. The case n = 1
- 61.2. The case n = 2. The two person market
- 61.3. Discussion of the two person market and its characteristic function
- 61.4. Justification of the standpoint of 58
- 61.5. Divisible goods. The “marginal pairs”
- 61.6. The price. Discussion
62. ECONOMIC INTERPRETATION OF THE RESULTS FOR n = 3: SPECIAL CASE
- 62.1. The case n = 3, special case. The three person market
- 62.2. Preliminary discussion
- 62.3. The solutions: First subcase
- 62.4. The solutions: General form
- 62.5. Algebraical form of the result
- 62.6. Discussion
63. ECONOMIC INTERPRETATION OF THE RESULTS FOR n = 3: GENERAL CASE
- 63.1. Divisible goods
- 63.2. Analysis of the inequalities
- 63.3. Preliminary discussion
- 63.4. The solutions
- 63.5. Algebraical form of the result
- 63.6. Discussion
64. THE GENERAL MARKET
- 64.1. Formulation of the problem
- 64.2. Some special properties. Monopoly and monopsony
CHAPTER XII
EXTENSION OF THE CONCEPTS OF DOMINATION AND SOLUTION
65. THE EXTENSION. SPECIAL CASES
- 65.1. Formulation of the problem
- 65.2. General remarks
- 65.3. Orderings, transitivity, acyclicity
- 65.4. The solutions: For a symmetric relation. For a complete ordering
- 65.5. The solutions: For a partial ordering
- 65.6. Acyclicity and strict acyclicity
- 65.7. The solutions: For an acyclic relation
- 65.8. Uniqueness of solutions, acyclicity and strict acyclicity
- 65.9. Application to games: Discreteness and continuity
66. GENERALIZATION OF THE CONCEPT OF UTILITY
- 66.1. The generalization. The two phases of the theoretical treatment
- 66.2. Discussion of the first phase
- 66.3. Discussion of the second phase
- 66.4. Desirability of unifying the two phases
67. DISCUSSION OF AN EXAMPLE
- 67.1. Description of the example
- 67.2. The solution and its interpretation
- 67.3. Generalization: Different discrete utility scales
- 67.4. Conclusions concerning bargaining
APPENDIX: THE AXIOMATIC TREATMENT OF UTILITY

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Theory of Games and Economic Behavior 60th Anniversary Commemorative Edition

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