Theory of Games and Economic Behavior

PREFACE TO FIRST EDITION
PREFACE TO SECOND EDITION
PREFACE TO THIRD EDITION
TECHNICAL NOTE
CONTENTS
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
		3.6. 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 Preliminarity
		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
		ll.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 o! the Main Facts
	14. Strictly Determined Games
		14.1. Formulation of the Problem
		14.2. The Minorant and the Majorant Gaifces
		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. N on -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 View Point
		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 Treatments 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 Pennis)
		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
		19.6. Description of the Strategies
		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. Description 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 Distribution. Necessity of Compensations
		22.2. Coalitions of Different Strength. Discussion
		22.3. An Inequality. Formulae
	23. The General Case
		23.1. Exhaustive 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
		26.3. 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 r
		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 Game
		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
		36.3. Other Parts of the Main Diagonals
	37. The Center and Its Environs
		37.1. First Orientation Concerning 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 Abandoning of the Zero-sum Condition
		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
		44.4. Extension of the Theory. Outside Sources
		44.5. The Excess
		44.6. Limitations of the Excess
		44.7. Discussion of the New Setup
	45. Limitations of Excess. Structure of Extended Theory
		45.1 The Lower Limit of the Excesses
		45.2. The Upper Limit of the Excess. Detached and Fully Detached Imputations
		45.3 Discussion of the Two Limits
		45.4. Detached Imputations and Various Solutions
		45.5. Proof of the Theorem
		45.6. Summary and Conclusions
	46. Determination of All Solutions in a Decomposable Game
		46.1. Elementary Properties of Decompositions
		46.2. Decomposition and Its Relation to the Solutions: First Results Concerning F(e )
		46.3. Continuation
		46.4 Continuation
		46.5. The Complete Result in F(e Q )
		46.6. The Complete Result in E(e )
		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 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
		49.6. Some Elementary Properties of Simplicity
		49.6. Simple Games and Their W, L. The Minimal Winning Coalitions : W^m
		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. Connection with the General Theory. Exact Formulation
		60.6. Reformulation of the Result
		50.7. Interpretation of the Result
		50.8. Connection with the Homogeneous Majority Games
	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 W^m. Difficulties of Using W^m
		51.4. Changed Approach : Enumeration by Means of W^m
		51.5. Simplicity and Decomposition
		51.6. Inessentiality, Simplicity and Composition. Treatment of the Excess
		51.7. A Criterion of Decomposability in Terms of W^m
	52. the Simple Games for Small n
		52.1. Program: n 1, 2 Play No Role. Disposal of n = 3
		52.3. Decomposability of the Cases
		52.4. The Simple Games Other than [1, , 1, I - 2]* (with Dummies)
		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
		5S.2. The Six Main Counter-examples (for n 6, 7)
	54. Determination of All Solutions in Suitable Games
		54.1 Reasons to Consider 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]
	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 (III): Determination of V
		55.5. Case (II) : Determination of V
		55.6. Case (II) : a and S+
		55.7. Cases (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
		56.3. Questions Concerning the Character of P
		56.4. Limitations of the Use of f
		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 Where T 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 Forms
		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
	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 68
		61.6. 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
		62.6. 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
		63.6. Algebraic Form of the Result
		68.6. Discussion
	64. The General Market
		64.1. Formulation of the Problem
		64.2. Some Special Properties. Monopoly and Monopsony
CHAPTER XII EXTENSIONS OF THE CONCEPTS OF DOMINATION AND SOLUTION
	65. The Extension. Special Cases
		66.1. Formulation of the Problem
		66.2. General Remarks
		66.3. Orderings, Transitivity, Acyclicity
		65.4. The Solutions : For a Symmetric Relation. For a Complete Ordering
		66.5. The Solutions : For a Partial Ordering
		66.6. Acyclicity and Strict Acyclicity
		65.7. The Solutions : For an Acyclic Relation
		66.8. Uniqueness of the Solutions, Acyclicity and Strict Acyclicity
		66.9. 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
	A.I. Formulation of the Problem
	A.2. Derivation from the Axioms
	A.3. Concluding Remarks
INDEX OF SUBJECTS