Theory of Games and Economic Behavior

PREFACE TO FIRST EDITION 10
PREFACE TO SECOND EDITION 10
PREFACE TO THIRD EDITION 12
TECHNICAL NOTE 14
CONTENTS 16
CHAPTER I FORMULATION OF THE ECONOMIC PROBLEM 26
	1. The Mathematical Method in Economics 26
		1.1. Introductory Remarks 26
		1.2. Difficulties of the Application of the Mathematical Method 27
		1.3. Necessary Limitations of the Objectives 31
		1.4. Concluding Remarks 32
	2. Qualitative Discussion of the Problem of Rational Behavior 33
		2.1. The Problem of Rational Behavior 33
		2.2. \"Robinson Crusoe\" Economy and Social Exchange Economy 34
		2.3. The Number of Variables and the Number of Participants 37
		2.4. The Case of Many Participants : Free Competition 38
		2.5. The \"Lausanne\" Theory 40
	3. The Notion of Utility 40
		3.1. Preferences and Utilities 40
		3.2. Principles of Measurement : Preliminaries 41
		3.3. Probability and Numerical Utilities 42
		3.4. Principles of Measurement : Detailed Discussion 45
		3.6. Conceptual Structure of the Axiomatic Treatment of Numerical Utilities 49
		3.6. The Axioms and Their Interpretation 51
		3.7. General Remarks Concerning the Axioms 53
		3.8. The Role of the Concept of Marginal Utility 54
	4. Structure of the Theory : Solutions and Standards of Behavior 56
		4.1. The Simplest Concept of a Solution for One Participant 56
		4.2. Extension to All Participants 58
		4.3. The Solution as a Set of Imputations 59
		4.4. The Intransitive Notion of \"Superiority\" or \"Domination\" 62
		4.5. The Precise Definition of a Solution 64
		4.6. Interpretation of Our Definition in Terms of \"Standards of Behavior\" 65
		4.7. Games and Social Organizations 68
		4.8. Concluding Remarks 68
CHAPTER II GENERAL FORMAL DESCRIPTION OF GAMES OF STRATEGY 71
	5. Introduction 71
		5.1. Shift of Emphasis from Economics to Games 71
		5.2. General Principles of Classification and of Procedure 71
	6. The Simplified Concept of a Game 73
		6.1. Explanation of the Termini Technici 73
		6.2. The Elements of the Game 74
		6.3. Information and Preliminarity 76
		6.4. Preliminarity, Transitivity, and Signaling 76
	7. The Complete Concept of a Game 80
		7.1. Variability of the Characteristics of Each Move 80
		7.2. The General Description 82
	8. Sets and Partitions 85
		8.1. Desirability of a Set-theoretical Description of a Game 85
		8.2. Sets, Their Properties, and Their Graphical Representation 86
		8.3 Partitions, Their Properties and Their Graphical Representation 88
		8.4. Logistic Interpretation of Sets and Partitions 91
	9. The Set-theoretical Description of a Game 92
		9.1. The Partitions Which Describe a Game 92
		9.2. Discussion of These Partitions and Their Properties 96
	10. Axiomatic Formulation 98
		10.1. The Axioms and Their Interpretations 98
		10.2. Logistic Discussion of the Axioms 101
		10.3. General Remarks Concerning the Axioms 101
		10.4. Graphical Representation 102
	11. Strategies and the Final Simplification of the Description of a Game 104
		11.1. The Concept of a Strategy and Its Formalization 104
		ll.2. The Final Simplification of the Description of a Game 106
		11.3. The Role of Strategies in the Simplified Form of a Game 109
		11.4. The Meaning of the Zero-sum Restriction 109
CHAPTER III ZERO-SUM TWO-PERSON GAMES: THEORY 110
	12. Preliminary Survey 110
		12.1. General Viewpoints 110
		12.2. The One-person Game 110
		12.3. Chance and Probability 112
		12.4. The Next Objective 112
	13. Functional Calculus 113
		13.1. Basic Definitions 113
		13.2. The Operations Max and Min 114
		13.3. Commutativity Questions 116
		13.4. The Mixed case. Saddle Points 118
		13.5. Proofs o! the Main Facts 120
	14. Strictly Determined Games 123
		14.1. Formulation of the Problem 123
		14.2. The Minorant and the Majorant Gaifces 125
		14.3. Discussion of the Auxiliary Games 126
		14.4. Conclusions 130
		14.5. Analysis of Strict Determinateness 131
		14.6. The Interchange of Players. Symmetry 134
		14.7. N on -strictly Determined Games 135
		14.8. Program of a Detailed Analysis of Strict Determinateness 136
	15. Games with Perfect Information 137
		15.1. Statement of Purpose. Induction 137
		15.2. The Exact Condition (First Step) 139
		15.3. The Exact Condition (Entire Induction) 141
		15.4. Exact Discussion of the Inductive Step 142
		15.5. Exact Discussion of the Inductive Step (Continuation) 145
		15.6. The Result in the Case of Perfect Information 148
		15.7. Application to Chess 149
		15.8. The Alternative, Verbal Discussion 151
	16. Linearity and Convexity 153
		16.1. Geometrical Background 153
		16.2. Vector Operations 154
		16.3. The Theorem of the Supporting Hyperplanes 159
		16.4. The Theorem of the Alternative for Matrices 163
	17. Mixed Strategies. The Solution for All Games 168
		17.1. Discussion of Two Elementary Examples 168
		17.2. Generalization of This View Point 170
		17.3. Justification of the Procedure As Applied to an Individual Play 171
		17.4. The Minorant and the Majorant Games (For Mixed Strategies) 174
		17.5. General Strict Determinateness 175
		17.6 Proof of the Main Theorem 178
		17.7. Comparison of the Treatments by Pure and by Mixed Strategies 180
		17.8. Analysis of General Strict Determinateness 183
		17.9. Further Characteristics of Good Strategies 185
		17.10. Mistakes and Their Consequences. Permanent Optimality 187
		17.11. The Interchange of Players. Symmetry 190
CHAPTER IV ZERO-SUM TWO-PERSON GAMES: EXAMPLES 194
	18. Some Elementary Games 194
		18.1. The Simplest Games 194
		18.2. Detailed Quantitative Discussion of These Games 195
		18.3. Qualitative Characterizations 198
		18.4. Discussion of Some Specific Games (Generalized Forms of Matching Pennis) 200
		18.5. Discussion of Some Slightly More Complicated Games 203
		18.6. Chance and Imperfect information 207
		18.7. Interpretation of This Result 210
	19. Poker and Bluffing 211
		19.1. Description of Poker 211
		19.2. Bluffing 213
		19.3. Description of Poker (Continued) 214
		19.4. Exact Formulation of the Rules 215
		19.6. Description of the Strategies 216
		19.6. Statement of the Problem 220
		19.7. Passage from the Discrete to the Continuous Problem 221
		19.8. Mathematical Determination of the Solution 224
		19.9. Detailed Analysis of the Solution 227
		19.10. Interpretation of the Solution 229
		19.11. More General Forms of Poker 232
		19.12. Discrete Hands 233
		19.13. m possible Bids 234
		19.14. Alternate Bidding 236
		19.15. Mathematical Description of All Solutions 241
		19.16. Interpretation of the Solutions. Conclusions 243
CHAPTER V ZERO-SUM THREE-PERSON GAMES 245
	20. Preliminary Survey 245
		20.1. General Viewpoints 245
		20.2. Coalitions 246
	21. The Simple Majority Game of Three Persons 247
		21.1. Description of the Game 247
		21.2. Analysis of the Game. Necessity of \"Understandings\" 248
		21.3. Analysis of the Game : Coalitions. The Role of Symmetry 249
	22. Further Examples 250
		22.1. Unsymmetric Distribution. Necessity of Compensations 250
		22.2. Coalitions of Different Strength. Discussion 252
		22.3. An Inequality. Formulae 254
	23. The General Case 256
		23.1. Exhaustive Discussion. Inessential and Essential Games 256
		23.2. Complete Formulae 257
	24. Discussion of an Objection 258
		24.1. The Case of Perfect Information and Its Significance 258
		24.2. Detailed Discussion. Necessity of Compensations between Three or More Players 260
CHAPTER VI FORMULATION OF THE GENERAL THEORY: ZERO-SUM n-PERSON GAMES 263
	25. The Characteristic Function 263
		25.1. Motivation and Definition 263
		25.2. Discussion of the Concept 265
		26.3. Fundamental Properties 266
		25.4. Immediate Mathematical Consequences 267
	26. Construction of a Game with a Given Characteristic Function 268
		26.1. The Construction 268
		26.2. Summary 270
	27. Strategic Equivalence. Inessential and Essential Games 270
		27.1. Strategic Equivalence. The Reduced Form 270
		27.2. Inequalities. The Quantity r 273
		27.3. Inessentiality and Essentiality 274
		27.4. Various Criteria. Non-additive Utilities 275
		27.5. The Inequalities in the Essential Case 277
		27.6. Vector Operations on Characteristic Functions 278
	28. Groups, Symmetry and Fairness 280
		28.1. Permutations, Their Groups, and Their Effect on a Game 280
		28.2. Symmetry and Fairness 283
	29. Reconsideration of the Zero-sum Three-person Game 285
		29.1. Qualitative Discussion 285
		29.2. Quantitative Discussion 287
	30. The Exact Form of the General Definitions 288
		30.1. The Definitions 288
		30.2. Discussion and Recapitulation 290
		30.3 The Concept of Saturation 291
		30.4. Three Immediate Objectives 296
	31. First Consequences 297
		31.1. Convexity, Flatness, and Some Criteria for Domination 297
		31.2. The System of All Imputations. One -element Solutions 302
		31.3. The Isomorphism Which Corresponds to Strategic Equivalence 306
	32. Determination of all Solutions of the Essential Zero-sum Three-person Game 307
		32.1. Formulation of the Mathematical Problem. The Graphical Method 307
		32.2 Determination of ALL Solutions 310
	33. Conclusions 313
		33.1. The Multiplicity of Solutions. Discrimination and Its Meaning 313
		33.2. Statics and Dynamics 315
CHAPTER VII ZERO-SUM FOUR-PERSON GAMES 316
	34. Preliminary Survey 316
		34.1. General Viewpoints 316
		34.2. Formalism of the Essential Zero -sum Four-person Game 316
		34.3. Permutations of the Players 319
	35. Discussion of Some Special Points in the Cube Q 320
		35.1 The Corner I (and V,VI, VII) 320
		35.2. The Corner VIII (and II, III, IV). The Three-person Game and a \"Dummy 324
		35.3. Some Remarks Concerning the Interior of Q 327
	36. Discussion of the Main Diagonals 329
		36.1. The Part Adjacent to the Corner VIII.: Heuristic Discussion 329
		36.2. The Part Adjacent to the Corner VIII. : Exact 332
		36.3. Other Parts of the Main Diagonals 337
	37. The Center and Its Environs 338
		37.1. First Orientation Concerning the Conditions around the Center 338
		37.2. The Two Alternatives and the Role of Symmetry 340
		37.3. The First Alternative at the Center 341
		37.4. The Second Alternative at the Center 342
		37.5. Comparison of the Two Central Solutions 343
		37.6. Unsymmetrical Central Solutions 344
	38. A Family of Solutions for a Neighborhood of the Center 346
		38.1. Transformation of the Solution Belonging to the First Alternative at the Center 346
		38.2. Exact Discussion 347
		38.3. Interpretation of The Solutions 353
CHAPTER VIII SOME REMARKS CONCERNING n >=5 PARTICIPANTS 355
	39. The Number of Parameters in Various Classes of Games 355
		39.1. The Situation for n = 3,4 355
		39.2. The Situation for All n>=3 355
	40. The Symmetric Five -person Game 357
		40.1. Formalism of the Symmetric Five-person Game 357
		40.2. The Two Extreme Cases 357
		40.3. Connection between the Symmetric Five-person Game and the 1,2,3-symmetric Four-person Game 359
CHAPTER IX COMPOSITION AND DECOMPOSITION OF GAMES 364
	41. Composition and Decomposition 364
		41.1. Search for n-person Games for Which All Solutions Can Be Determined 364
		41.2. The First Type. Composition and Decomposition 365
		41.3. Exact Definitions 366
		41.4. Analysis of Decomposability 368
		41.5. Desirability of a Modification 370
	42. Modification of the Theory 370
		42.1. No Complete Abandoning of the Zero-sum Condition 370
		42.2. Strategic Equivalence. Constant-sum Games 371
		42.3. The Characteristic Function in the New Theory 373
		42.4. Imputations, Domination, Solutions in the New Theory 375
		42.5. Essentiality, Inessentiality, and Decomposability in the New Theory 376
	43. The Decomposition Partition 378
		43.1. Splitting Sets. Constituents 378
		43.2. Properties of the System of All Splitting Sets 378
		43.3. Characterization of the System of All Splitting Sets. The Decomposition Partition 379
		43.4. Properties of the Decomposition Partition 382
	44. Decomposable Games. Further Extension of the Theory 383
		44.1. Solutions of a (Decomposable) Game and Solutions of Its Constituents 383
		44.2. Composition and Decomposition of Imputations and of Sets of Imputations 384
		44.3. Composition and Decomposition of Solutions. 386
		44.4. Extension of the Theory. Outside Sources 388
		44.5. The Excess 389
		44.6. Limitations of the Excess. 391
		44.7. Discussion of the New Setup 392
	45. Limitations of Excess. Structure of Extended Theory 393
		45.1 The Lower Limit of the Excesses 393
		45.2. The Upper Limit of the Excess. Detached and Fully Detached Imputations 394
		45.3 Discussion of the Two Limits 397
		45.4. Detached Imputations and Various Solutions. 400
		45.5. Proof of the Theorem 401
		45.6. Summary and Conclusions 405
	46. Determination of All Solutions in a Decomposable Game 406
		46.1. Elementary Properties of Decompositions 406
		46.2. Decomposition and Its Relation to the Solutions: First Results Concerning F(e ) 409
		46.3. Continuation 411
		46.4 Continuation 413
		46.5. The Complete Result in F(e Q ) 415
		46.6. The Complete Result in E(e ) 418
		46.7 Graphical Representation of a Part of the Result 419
		46.8. Interpretation : The Normal Zone. Heredity of Various Properties 421
		46.9. Dummies 422
		46.10 Imbedding a Game 423
		46.11. Significance of the Normal Zone 426
		46.12. First Occurrence of the Phenomenon of Transfer: n - 6 427
	47. The Essential Three-person Game in the New Theory 428
		47.1. Need for This Discussion 428
		47.2. Preparatory Considerations 428
		47.3. The Six Cases of the Discussion. Cases (I)-(III) 431
		47.4. Case (IV) : First Part 432
		47.5 Case (IV) : Second Part 434
		47.6. Case (V) 438
		47.7 Case (VI) 440
		47.8. Interpretation of the Result: The Curves (One Dimensional Parts) in the Solution 441
		47.9. Continuation : The Areas (Two-dimensional Parts) in the Solution 443
CHAPTER X SIMPLE GAMES 445
	48. Winning and Losing Coalitions and Games Where They Occur 445
		48.1. The Second Type of 41.1. Decision by Coalitions 445
		48.2. Winning and Losing Coalitions 446
	49. Characterization of the Simple Games 448
		49.1. General Concepts of Winning and Losing Coalitions 448
		49.2. The Special Role of One-element Sets 450
		49.3. Characterization of the Systems W, L of Actual Games 451
		49.4. Exact Definition of Simplicity 453
		49.6. Some Elementary Properties of Simplicity 453
		49.6. Simple Games and Their W, L. The Minimal Winning Coalitions : W^m 454
		49.7. The Solutions of Simple Games 455
	50. The Majority Games and the Main Solution 456
		50.1. Examples of Simple Games : The Majority Games 456
		50.2. Homogeneity 458
		50.3. A More Direct Use of the Concept of Imputation in Forming Solutions 460
		50.4. Discussion of This Direct Approach 461
		50.5. Connection with the General Theory. Exact Formulation 463
		60.6. Reformulation of the Result 465
		50.7. Interpretation of the Result 467
		50.8. Connection with the Homogeneous Majority Games 468
	51. Methods for the Enumeration of All Simple Games 470
		51.1. Preliminary Remarks 470
		51.2. The Saturation Method : Enumeration by Means of W 471
		51.3. Reasons for Passing from W to W^m. Difficulties of Using W^m 473
		51.4. Changed Approach : Enumeration by Means of W^m 475
		51.5. Simplicity and Decomposition 477
		51.6. Inessentiality, Simplicity and Composition. Treatment of the Excess 479
		51.7. A Criterion of Decomposability in Terms of W^m 480
	52. the Simple Games for Small n 482
		52.1. Program: n 1, 2 Play No Role. Disposal of n = 3 482
		52.3. Decomposability of the Cases 484
		52.4. The Simple Games Other than [1, , 1, I - 2]* (with Dummies) 486
		52.5. Disposal of n = 4, 5 487
	53. The New Possibilities of Simple Games for n>=6 488
		53.1. The Regularities Observed for n < 6 488
		5S.2. The Six Main Counter-examples (for n 6, 7) 489
	54. Determination of All Solutions in Suitable Games 495
		54.1 Reasons to Consider Solutions than the Main Solution in Simple Games 495
		54.2. Enumeration of Those Games for Which All Solutions Are Known 496
		54.3. Reasons to Consider the Simple Game [1, - , 1, n 2]* 497
	55. The Simple Game [1, , 1, n - 2] h 498
		55.1. Preliminary Remarks 498
		55.2. Domination. The Chief Player. Cases (I) and (II) 498
		55.3. Disposal of Case (I) 500
		55.4  Case (III): Determination of V 503
		55.5. Case (II) : Determination of V 506
		55.6. Case (II) : a and S+ 509
		55.7. Cases (II\') and (II\"). Disposal of Case (II\') 510
		55.8. Case (II\") : a and V. Domination 512
		55.9  Case (II): Determination of V 513
		55.10. Disposal of Case (II\") 519
		55.11.Reformulation of the Complete Result 522
		55.12. Interpretation of the Result 524
CHAPTER XI GENERAL NON-ZERO-SUM GAMES 529
	56. Extension of the Theory 529
		56.1. Formulation of the Problem 529
		56.2. The Fictitious Player. The Zero-sum Extension 530
		56.3. Questions Concerning the Character of P 531
		56.4. Limitations of the Use of f 533
		56.5. The Two Possible Procedures 535
		56.6. The Discriminatory Solutions 536
		56.7. Alternative Possibilities 537
		56.8. The New Setup 539
		56.9. Reconsideration of the Case Where T is a Zero-sum Game 541
		56.10. Analysis of the Concept of Domination 545
		56.11. Rigorous Discussion 548
		56.12 The New Definition of a Solution 551
	57. The Characteristic Function and Related Topics 552
		57.1. The Characteristic Function : The Extended and the Restricted Forms 552
		57.2. Fundamental Properties 553
		57.3. Determination of All Characteristic Functions 555
		57.4. Removable Sets of Players 558
		57.5. Strategic Equivalence. Zero-sum and Constant-sum Games 560
	58. Interpretation of the Characteristic Function 563
		58.1. Analysis of the Definition 563
		58.2 The Desire to make a Gain vs That to inflict a loss 564
		58.3. Discussion 566
	59. General Considerations 567
		59.1. Discussion of the Program 567
		59.2. The Reduced Forms. The Inequalities 568
		59.3. Various Topics 571
	60. The Solutions of All General Games with n^3 573
		60.1. The Case n-1 573
		60.2 The Case n=2 574
		60.3 The case n=3 575
	61. Economic Interpretation of the Results for n = 1,2 580
		61.1. The Case n-1 580
		61.2. The Case n = 2. The Two-person Market 580
		61.3. Discussion of the Two-person Market and Its Characteristic Function 582
		61.4. Justification of the Standpoint of 68. 584
		61.6. Divisible Goods. The \"Marginal Pairs\" 585
		61.6. The Price. Discussion 587
	62. Economic Interpretation of the Results for n = 3 : Special Case 589
		62.1. The Case n 3, Special Case. The Three-person Market 589
		62.2. Preliminary Discussion 591
		62.3. The Solutions : First Subcase 591
		62.4. The Solutions : General Form 594
		62.6. Algebraical Form of the Result 595
		62.6. Discussion 596
	63. Economic Interpretation of the Results for n = 3 : General Case 598
		63.1. Divisible Goods 598
		63.2. Analysis of the Inequalities 600
		63.3. Preliminary Discussion 602
		63.4. The Solutions 602
		63.6. Algebraic Form of the Result 605
		68.6. Discussion 606
	64. The General Market 608
		64.1. Formulation of the Problem 608
		64.2. Some Special Properties. Monopoly and Monopsony 609
CHAPTER XII EXTENSIONS OF THE CONCEPTS OF DOMINATION AND SOLUTION 612
	65. The Extension. Special Cases 612
		66.1. Formulation of the Problem 612
		66.2. General Remarks 613
		66.3. Orderings, Transitivity, Acyclicity 614
		65.4. The Solutions : For a Symmetric Relation. For a Complete Ordering 616
		66.5. The Solutions : For a Partial Ordering 617
		66.6. Acyclicity and Strict Acyclicity 619
		65.7. The Solutions : For an Acyclic Relation 622
		66.8. Uniqueness of the Solutions, Acyclicity and Strict Acyclicity 625
		66.9. Application to Games : Discreteness and Continuity 627
	66. Generalization of the Concept of Utility 628
		66.1. The Generalization. The Two Phases of the Theoretical Treatment 628
		66.2. Discussion of the First Phase 629
		66.3. Discussion of the Second Phase 631
		66.4. Desirability of Unifying the Two Phases 632
	67. Discussion of an Example 633
		67.1. Description of the Example 633
		67.2. The Solution and Its Interpretation 636
		67.3. Generalization : Different Discrete Utility Scales 639
		67.4. Conclusions Concerning Bargaining 641
APPENDIX. THE AXIOMATIC TREATMENT OF UTILITY 642
	A.I. Formulation of the Problem 642
	A.2. Derivation from the Axioms 643
	A.3. Concluding Remarks 653
INDEX OF SUBJECTS 660