Theory of Games and Economic Behavior 60th Anniversary Commemonceton Edition by John von Neumann , Oskar Morgenstern , Harold Kuhn , Ariel Rubinstein ISBN 0691130612 978-0691130613

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