Table of Contents
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
141. 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(e0), F(e0) 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(e0), F(e0) 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(e0) 384
46.3. Continuation 386
46.4. Continuation 388
46.5. The complete result in F(e0) 390
46.6. The complete result in E(e0) 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: Wm 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 Wm. Difficulties of using Wm 448
51.4. Changed Approach: Enumeration by means of Wm 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 Wm 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 Wm 458
52.3. Decomposability of cases C*, Cn-2, Cn-1 459
52.4. The simple games other than [1, . . . , 1, n - 2]h, (with dummies): The Cases Ck, 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
