Theory of Games and Economic Behavior (Commemorative Edition) by Rubinstein, Ariel, Morgenstern, Oskar, Kuhn, Harold William, von Neumann, John -- Read -- Imperial Library of Trantor

Index

Cover Half title Title Copyright Introduction Contents 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 V *55.5. Case (II): Determination of V *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 ease 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 Afterword Reviews

Heads I Win, and Tails, You Lose Big D Mathematics of Games and Economics Theory of Games Mathematical Theory of Poker Is Applied to Business Problems A Theory of Strategy The Collaboration between Oskar Morgenstern and John von Neumann on the Theory of Games

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