PREFACE v

TECHNICAL NOTE v

ACKNOWLEDGMENT x

CHAPTER I: FORMULATION OF THE ECONOMIC PROBLEM

1. THE MATHEMATICAL METHOD IN ECONOMICS 1

1.1. Introductory remarks 1

1.2. Difficulties of the application of the mathematical method 2

1.3. Necessary limitations of the objectives 6

1.4. Concluding remarks 7

2. QUALITATIVE DISCUSSION OF THE PROBLEM OF RATIONAL BEHAVIOR 8

2.1. The problem of rational behavior 8

2.2. "Robinson Crusoe" economy and social exchange economy 9

2.3. The number of variables and the number of participants 12

2.4. The case of many participants: Free competition 13

2.5. The "Lausanne" theory 15

3. THE NOTION OF UTILITY 15

3.1. Preferences and utilities 15

3.2. Principles of measurement: Preliminaries 16

3.3. Probability and numerical utilities 17

3.4. Principles of measurement: Detailed discussion 20

3.5. Conceptual structure of the axiomatic treatment of numerical utilities 24

3.6. The axioms and their interpretation 26

3.7. General remarks concerning the axioms 28

3.8. The role of the concept of marginal utility 29

4. STRUCTURE OF THE THEORY: SOLUTIONS AND STANDARDS OF BEHAVIOR 31

4.1. The simplest concept of a solution for one participant 31

4.2. Extension to all participants 33

4.3. The solution as a set of imputations 34

4.4. The intransitive notion of "superiority" or "domination" 37

4.5. The precise definition of a solution 39

4.6. Interpretation of our definition in terms of "standards of behavior" 40

4.7. Games and social organizations 43

4.8. Concluding remarks 43

CHAPTER II: GENERAL FORMAL DESCRIPTION OF GAMES OF STRATEGY

5. Introduction 46

5.1. Shift of emphasis from economics to games 46

5.2. General principles of classification and of procedure 46

6. THE SIMPLIFIED CONCEPT OF A GAME 48

6.1. Explanation of the *termini technici* 48

6.2. The elements of the game 49

6.3. Information and preliminary 51

6.4. Preliminarity, transitivity, and signaling 51

7. THE COMPLETE CONCEPT OF A GAME 55

7.1. Variability of the characteristics of each move 55

7.2. The general description 57

8. SETS AND PARTITIONS 60

8.1. Desirability of a set-theoretical description of a game 60

8.2. Sets, their properties, and their graphical representation 61

8.3. Partitions, their properties, and their graphical representation 63

8.4. Logistic interpretation of sets and partitions 66

*9. THE SET-THEORETICAL DESCRIPTION OF A CAME 67

*9.1. The partitions which describe a game 67

*9.2. Discussion of these partitions and their properties 71

*10. AXIOMATIC FORMULATION 73

*10.1. The axioms and their interpretations 73

*10.2. Logistic discussion of the axioms 76

*10.3. General remarks concerning the axioms 76

*10.4. Graphical representation 77

11. STRATEGIES AND THE FINAL SIMPLIFICATION OF THE DESCRIPTION OF THE GAME 79

11.1. The concept of a strategy and its formalization 79

11.2. The final simplification of the description of a game 81

11.3. The role of strategies in the simplified form of a game 84

11.4. The meaning of the zero-sum restriction 84

CHAPTER III: ZERO-SUM TWO-PERSON GAMES: THEORY

12. PRELIMINARY SURVEY 85

12.1. General viewpoints 85

12.2. The one-person game 85

12.3. Chance afid probability 87

12.4. The next objective 87

13. FUNCTIONAL CALCULUS 88

13.1. Basic definitions 88

13.2. The operations Max and Min 89

13.3. Commutativity questions 91

13.4. The mixed case. Saddle points 93

13.5. Proofs of the main facts 95

14. STRICTLY DETERMINED GAMES 98

14 1. Formulation of the problem 98

14.2. The minorant and the majorant games 100

14.3. Discussion of the auxiliary games 101

14.4. Conclusions 105

14.5. Analysis of strict determinateness 106

14.6. The interchange of players. Symmetry 109

14.7. Non strictly determined games 110

14.8. Program of a detailed analysis of strict determinateness 111

*15. GAMES WITH PERFECT INFORMATION

*15.1. Statement of purpose. Induction 112

*15.2. The exact condition (First step) 114

*15.3. The exact condition (Entire induction) 116

*15.4. Exact discussion of the inductive step 117

*15.5. Exact discussion of the inductive step (Continuation) 120

*15.6. The result in the case of perfect information 123

*15.7. Application to Chess 124

*15.8. The alternative, verbal discussion 126

16. LINEARITY AND CONVEXITY 128

16.1. Geometrical background 128

16.2. Vector operations 129

16.3. The theorem of the supporting hyperplanes 134

16.4. The theorem of the alternative for matrices 138

17. MIXED STRATEGIES. THE SOLUTION FOR ALL GAMES 143

17.1. Discussion of two elementary examples 143

17.2. Generalization of this viewpoint 145

17.3. Justification of the procedure as applied to an individual play 146

17.4. The minorant and the majorant games. (For mixed strategies) 149

17.5. General strict determinateness 150

17.6. Proof of the main theorem 153

17.7. Comparison of the treatment by pure and by mixed strategies 155

17.8. Analysis of general strict determinateness 158

17.9. Further characteristics of good strategies 160

17.10. Mistakes and their consequences. Permanent optimality 162

17.11. The interchange of players. Symmetry 165

CHAPTER IV: ZERO-SUM TWO-PERSON GAMES: EXAMPLES

18. SOME ELEMENTARY GAMES 169

18.1. The simplest games 169

18.2. Detailed quantitative discussion of these games 170

18.3. Qualitative characterizations 173

18.4. Discussion of some specific games. (Generalized forms of Matching Pennies) 175

18.5. Discussion of some slightly more complicated games 178

18.6. Chance and imperfect information 182

18.7. Interpretation of this result 185

*19. POKER AND BLUFFING 186

*19.1. Description of Poker 186

*19.2. Bluffing 188

*19.3. Description of Poker (Continued) 189

*19.4. Exact formulation of the rules 190

*19.5. Description of the strategy 191

*19.6. Statement of the problem 195

*19.7. Passage from the discrete to the continuous problem 196

*19.8. Mathematical determination of the solution 199

*19.9. Detailed analysis of the solution 202

*19.10. Interpretation of the solution 204

*19.11. More general forms of Poker 207

*19.12. Discrete hands 208

*19.13. *m* possible bids 209

*19.14. Alternate bidding 211

*19.15. Mathematical description of all solutions 216

*19.16. Interpretation of the solutions. Conclusions 218

CHAPTER V: ZERO-SUM THREE-PERSON GAMES

20. PRELIMINARY SURVEY 220

20.1. General viewpoints 220

20.2. Coalitions 221

21. THE SIMPLE MAJORITY GAME OF THREE PERSONS 222

21.1. Definition of the game 222

21.2. Analysis of the game: Necessity of "understandings" 223

21.3. Analysis of the game: Coalitions. The role of symmetry 224

22. FURTHER EXAMPLES 225

22.1. Unsymmetric distributions. Necessity of compensations 225

22.2. Coalitions of different strength. Discussion 227

22.3. An inequality. Formulae 229

23. THE GENERAL CASE 231

23.1. Detailed discussion. Inessential and essential games 231

23.2. Complete formulae 232

24. DISCUSSION OF AN OBJECTION 233

24.1. The case of perfect information and its significance 233

24.2. Detailed discussion. Necessity of compensations between three or more players 235

CHAPTER VI: FORMULATION OF THE GENERAL THEORY: ZERO-SUM *n*-PERSON GAMES

25. THE CHARACTERISTIC FUNCTION 238

25.1. Motivation and definition 238

25.2. Discussion of the concept 240

25.3. Fundamental properties 241

25.4. Immediate mathematical consequences 242

26. CONSTRUCTION OF A GAME WITH A GIVEN CHARACTERISTIC FUNCTION 243

26.1. The construction 243

26.2. Summary 245

27. STRATEGIC EQUIVALENCE. INESSENTIAL AND ESSENTIAL GAMES 245

27.1. Strategic equivalence. The reduced form 245

27.2. Inequalities. The quantity [gamma] 248

27.3. Inessentiality and essentiality 249

27.4. Various criteria. Non additive utilities 250

27.5. The inequalities in the essential case 252

27.6. Vector operations on characteristic functions 253

28. GROUPS, SYMMETRY AND FAIRNESS 255

28.1. Permutations, their groups and their effect on a game 255

28.2. Symmetry and fairness 258

29. RECONSIDERATION OF THE ZERO-SUM THREE-PERSON GAME 260

29.1. Qualitative discussion 260

29.2. Quantitative discussion 262

30. THE EXACT FORM OF THE GENERAL DEFINITIONS 263

30.1. The definitions 263

30.2. Discussion and recapitulation 265

*30.3. The concept of saturation 266

30.4. Three immediate objectives 271

31. FIRST CONSEQUENCES 272

31.1. Convexity, flatness, and some criteria for domination 272

31.2. The system of all imputations. One element solutions 277

31.3. The isomorphism which corresponds to strategic equivalence 281

32. DETERMINATION OF ALL SOLUTIONS OF THE ESSENTIAL ZERO-SUM THREE-PERSON GAME 282

32.1. Formulation of the mathematical problem. The graphical method 282

32.2. Determination of all solutions 285

33. CONCLUSIONS 288

33.1. The multiplicity of solutions. Discrimination and its meaning 288

33.2. Statics and dynamics 290

CHAPTER VII: ZERO-SUM FOUR-PERSON GAMES

34. PRELIMINARY SURVEY 291

34.1. General viewpoints 291

34.2. Formalism of the essential zero sum four person games 291

34.3. Permutations of the players 294

35. DISCUSSION OF SOME SPECIAL POINTS IN THE CUBE *Q* 295

35.1. The corner *I*. (and *V., VI., VII.*) 295

35.2. The corner *VIII*. (and *II., III., IV.,*). The three person game and a "Dummy" 299

35.3. Some remarks concerning the interior of *Q* 302

36. DISCUSSION OF THE MAIN DIAGONALS 304

36.1. The part adjacent to the corner *VIII*.: Heuristic discussion 304

36.2. The part adjacent to the corner *VIII*.: Exact discussion 307

*36.3. Other parts of the main diagonals 312

37. THE CENTER AND ITS ENVIRONS 313

37.1. First orientation about the conditions around the center 313

37.2. The two alternatives and the role of symmetry 315

37.3. The first alternative at the center 316

37.4. The second alternative at the center 317

37.5. Comparison of the two central solutions 318

37.6. Unsymmetrical central solutions 319

*38. A FAMILY OF SOLUTIONS FOR A NEIGHBORHOOD OF THE CENTER 321

*38.1. Transformation of the solution belonging to the first alternative at the center 321

*38.2. Exact discussion 322

*38.3. Interpretation of the solutions 327

CHAPTER VIII: SOME REMARKS CONCERNING *n* [equal to or greater than] 5 PARTICIPANTS

39. THE NUMBER OF PARAMETERS IN VARIOUS CLASSES OF GAMES 330

39.1. The situation for n = 3, 4 330

39.2. The situation for all *n* [equal to or greater than] 3 330

40. THE SYMMETRIC FIVE PERSON GAME 332

40.1. Formalism of the symmetric five person game 332

40.2. The two extreme cases 332

40.3. Connection between the symmetric five person game and the 1, 2, 3 symmetric four person game 334

CHAPTER IX: COMPOSITION AND DECOMPOSITION OF GAMES

41. COMPOSITION AND DECOMPOSITION 339

41.1. Search for *n*-person games for which all solutions can be determined 339

41.2. The first type. Composition and decomposition 340

41.3. Exact definitions 341

41.4. Analysis of decomposability 343

41.5. Desirability of a modification 345

42. MODIFICATION OF THE THEORY 345

42.1. No complete abandonment of the zero sum restriction 345

42.2. Strategic equivalence. Constant sum games 346

42.3. The characteristic function in the new theory 348

42.4. Imputations, domination, solutions in the new theory 350

42.5. Essentiality, inessentiality and decomposability in the new theory 351

43. THE DECOMPOSITION PARTITION 353

43.1. Splitting sets. Constituents 353

43.2. Properties of the system of all splitting sets 353

43.3. Characterization of the system of all splitting sets. The decomposition partition 354

43.4. Properties of the decomposition partition 357

44. DECOMPOSABLE GAMES. FURTHER EXTENSION OF THE THEORY 358

44.1. Solutions of a (decomposable) game and solutions of its constituents 358

44.2. Composition and decomposition of imputations and of sets of imputations 359

44.3. Composition and decomposition of solutions. The main possibilities and surmises 361

44.4. Extension of the theory. Outside sources 363

44.5. The excess 364

44.6. Limitations of the excess. The non-isolated character of a game in the new setup 366

44.7. Discussion of the new setup. *E*(*e*_{0}), *F*(*e*_{0}) 367

45. LIMITATIONS OF THE EXCESS. STRUCTURE OF THE EXTENDED THEORY 378

45.1. The lower limit of the excess 368

45.2. The upper limit of the excess. Detached and fully detached imputations 369

45.3. Discussion of the two limits, |[Gamma]|_{1}, |[Gamma]|_{2}. Their ratio 372

45.4. Detached imputations and various solutions. The theorem connecting *E*(*e*_{0}), *F*(*e*_{0}) 375

45.5. Proof of the theorem 376

45.6. Summary and conclusions 380

46. DETERMINATION OF ALL SOLUTIONS OF A DECOMPOSABLE GAME 381

46.1. Elementary properties of decompositions 381

46.2. Decomposition and its relation to the solutions: First results concerning *F*(*e*_{0}) 384

46.3. Continuation 386

46.4. Continuation 388

46.5. The complete result in *F*(*e*_{0}) 390

46.6. The complete result in *E*(*e*_{0}) 393

46.7. Graphical representation of a part of the result 394

46.8. Interpretation: The normal zone. Heredity of various properties 396

46.9. Dummies 397

46.10. Imbedding of a game 398

46.11. Significance of the normal zone 401

46.12. First occurrence of the phenomenon of transfer: *n* = 6 402

47. THE ESSENTIAL THREE-PERSON GAME IN THE NEW THEORY 403

47.1. Need for this discussion 403

47.2. Preparatory considerations 403

47.3. The six cases of the discussion. Cases (I)-(III) 406

47.4. Case (IV): First part 407

47.5. Case (IV): Second part 409

47.6. Case (V) 413

47.7. Case (VI) 415

47.8. Interpretation of the result: The curves (one dimensional parts) in the solution 416

47.9. Continuation: The areas (two dimensional parts) in the solution 418

CHAPTER X: SIMPLE GAMES

48. WINNING AND LOSING COALITIONS AND GAMES WHERE THEY OCCUR 420

48.1. The second type of 41.1. Decision by coalitions 420

48.2. Winning and Losing Coalitions 421

49. CHARACTERIZATION OF THE SIMPLE GAMES 423

49.1. General concepts of winning and losing coalitions 423

49.2. The special role of one element sets 425

49.3. Characterization of the systems *W, L* of actual games 426

49.4. Exact definition of simplicity 428

49.5. Some elementary properties of simplicity 428

49.6. Simple games and their *W, L*. The Minimal winning coalitions: *W*^{m} 429

49.7. The solutions of simple games 430

50. THE MAJORITY GAMES AND THE MAIN SOLUTION 431

50.1. Examples of simple games: The majority games 481

50.2. Homogeneity 433

50.3. A more direct use of the concept of imputation in forming solutions 435

50.4. Discussion of this direct approach 436

50.5. Connections with the general theory. Exact formulation 438

50.6. Reformulation of the result 440

50.7. Interpretation of the result 442

50.8. Connection with the Homogeneous Majority game 443

51. METHODS FOR THE ENUMERATION OF ALL SIMPLE GAMES 445

51.1. Preliminary Remarks 445

51.2. The saturation method: Enumeration by means of *W* 446

51.3. Reasons for passing from *W* to *W*^{m}. Difficulties of using *W*^{m} 448

51.4. Changed Approach: Enumeration by means of *W*^{m} 450

51.5. Simplicity and decomposition 452

51.6. Inessentiality, Simplicity and Composition. Treatment of the excess 454

51.7. A criterium of decomposability in terms of *W*^{m} 455

52. THE SIMPLE GAMES FOR SMALL *n* 457

52.1. Program. *n* = 1, 2 play no role. Disposal of *n* = 3 457

52.2. Procedure for *n* [equal to or greater than] 4: The two element sets and their role in classify ing the *W*^{m} 458

52.3. Decomposability of cases *C*, C*_{n-2}, *C*_{n-1} 459

52.4. The simple games other than [1, ^{. . .}, 1, *n* - 2]_{h}, (with dummies): The Cases *C*_{k}, *k* = 0, 1, ^{. . .}, *n* - 3 461

52.5. Disposal of *n* = 4, 5 462

53. THE NEW POSSIBILITIES OF SIMPLE GAMES FOR *n* [equal to or greater than] 6 463

53.1. The Regularities observed for *n* [equal to or greater than] 6 463

53.2. The six main counter examples (for *n* = 6, 7) 464

54. DETERMINATION OF ALL SOLUTIONS IN SUITABLE GAMES 470

54.1. Reasons to consider other solutions than the main solution in simple games 470

54.2. Enumeration of those games for which all solutions are known 471

54.3. Reasons to consider the simple game [1, ^{. . .}, 1, *n* - 2]_{h}, 472

*55. THE SIMPLE GAME [1, ^{. . .}, 1, *n* - 2]_{h} 473

*55.1. Preliminary Remarks 473

*55.2. Domination. The chief player. Cases (I) and (11) 473

*55.3. Disposal of Case (I) 475

*55.4. Case (II): Determination of V [above horizontal bar] 478

*55.5. Case (II): Determination of V [below horizontal bar] 481

*55.6. Case (II): [alpha] and *S*_{*} 484

*55.7. Case (II') and (II''). Disposal of Case (II') 485

*55.8. Case (II''): [alpha] and V'. Domination 488

*55.9. Case (II''): Determination of V'

*55.10. Disposal of Case (II'') 488

*55.11. Reformulation of the complete result 497

*55.12. Interpretation of the result 499

CHAPTER XI: GENERAL NON-ZERO-SUM GAMES

56. EXTENSION OF THE THEORY 504

56.1. Formulation of the problem 504

56.2. The fictitious player. The zero sum extension [Gamma] 505

56.3. Questions concerning the character of [Gamma below horizontal bar] 506

56.4. Limitations of the use of [Gamma above horizontal bar] 508

56.5. The two possible procedures 510

56.6. The discriminatory solutions 511

56.7. Alternative possibilities 512

56.8. The new setup 514

56.9. Reconsideration of the case when [Gamma] is a zero sum game 516

56.10. Analysis of the concept of domination 520

56.11. Rigorous discussion 523

56.12. The new definition of a solution 526

57. THE CHARACTERISTIC FUNCTION AND RELATED TOPICS 527

57.1. The characteristic function: The extended and the restricted form 527

57.2. Fundamental properties 528

57.3. Determination of all characteristic functions 530

57.4. Removable sets of players 533

57.5. Strategic equivalence. Zero-sum and constant-sum games 535

58. INTERPRETATION OF THE CHARACTERISTIC FUNCTION 538

58.1. Analysis of the definition 538

58.2. The desire to make a gain vs. that to inflict a loss 539

58.3. Discussion 541

59. GENERAL CONSIDERATIONS 542

59.1. Discussion of the program 542

59.2. The reduced forms. The inequalities 543

59.3. Various topics 546

60. THE SOLUTIONS OF ALL GENERAL GAMES WITH *n* [equal to or less than] 3 548

60.1. The case *n* = 1 548

60.2. The case *n* = 2 549

60.3. The case *n* = 3 550

60.4. Comparison with the zero sum games 554

61. ECONOMIC INTERPRETATION OF THE RESULTS FOR *n* = 1, 2 555

61.1. The case *n* = 1 555

61.2. The case *n* = 2. The two person market 555

61.3. Discussion of the two person market and its characteristic function 557

61.4. Justification of the standpoint of 58 559

61.5. Divisible goods. The "marginal pairs" 560

61.6. The price. Discussion 562

62. ECONOMIC INTERPRETATION OF THE RESULTS FOR *n* = 3: SPECIAL CASE 564

62.1. The case *n* = 3, special case. The three person market 564

62.2. Preliminary discussion 566

62.3. The solutions: First subcase 566

62.4. The solutions: General form 569

62.5. Algebraical form of the result 570

62.6. Discussion 571

63. ECONOMIC INTERPRETATION OF THE RESULTS FOR *n* = 3: GENERAL CASE 573

63.1. Divisible goods 573

63.2. Analysis of the inequalities 575

63.3. Preliminary discussion 577

63.4. The solutions 577

63.5. Algebraical form of the result 580

63.6. Discussion 581

64. THE GENERAL MARKET 583

64.1. Formulation of the problem 583

64.2. Some special properties. Monopoly and monopsony 584

CHAPTER XII: EXTENSION OF THE CONCEPTS OF DOMINATION AND SOLUTION

65. THE EXTENSION. SPECIAL CASES 587

65.1. Formulation of the problem 587

65.2. General remarks 588

65.3. Orderings, transitivity, acyclicity 589

65.4. The solutions: For a symmetric relation. For a complete ordering 591

65.5. The solutions: For a partial ordering 592

65.6. Acyclicity and strict acyclicity 594

65.7. The solutions: For an acyclic relation 597

65.8. Uniqueness of solutions, acyclicity and strict acyclicity 600

65.9. Application to games: Discreteness and continuity 602

66. GENERALIZATION OF THE CONCEPT OF UTILITY 603

66.1. The generalization. The two phases of the theoretical treatment 603

66.2. Discussion of the first phase 604

66.3. Discussion of the second phase 606

66.4. Desirability of unifying the two phases 607

67. DISCUSSION OF AN EXAMPLE 608

67.1. Description of the example 608

67.2. The solution and its interpretation 611

67.3. Generalization: Different discrete utility scales 614

67.4. Conclusions concerning bargaining 616

APPENDIX: THE AXIOMATIC TREATMENT OF UTILITY 617

INDEX OF FIGURES 633

INDEX OF NAMES 634

INDEX OF SUBJECTS 635