Wed to the Phoenix

Cover; CONTENTS; PREFACE; TECHNICAL NOTE; ACKNOWLEDGMENT; 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 DISCUSSIOIN 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.;This is the classic work upon which modern-day game theory is based. What began more than sixty years ago as a modest proposal that a mathematician and an economist write a short paper together blossomed, in 1944, when Princeton University Press published Theory of Games and Economic Behavior. In it, John von Neumann and Oskar Morgenstern conceived a groundbreaking mathematical theory of economic and social organization, based on a theory of games of strategy. Not only would this revolutionize economics, but the entirely new field of scientific inquiry it yielded--game theory--has si. Cover 1 CONTENTS 18 PREFACE 28 TECHNICAL NOTE 32 ACKNOWLEDGMENT 33 CHAPTER I: FORMULATION OF THE ECONOMIC PROBLEM 34 1. THE MATHEMATICAL METHOD IN ECONOMICS 34 1.1. Introductory remarks 34 1.2. Difficulties of the application of the mathematical method 35 1.3. Necessary limitations of the objectives 39 1.4. Concluding remarks 40 2. QUALITATIVE DISCUSSIOIN OF THE PROBLEM OF RATIONAL BEHAVIOR 41 2.1. The problem of rational behavior 41 2.2. " Robinson Crusoe" economy and social exchange economy 42 2.3. The number of variables and the number of participants 45 2.4. The case of many participants: Free competition 46 2.5. The "Lausanne" theory 48 3. THE NOTION OF UTILITY 48 3.1. Preferences and utilities 48 3.2. Principles of measurement: Preliminaries 49 3.3. Probability and numerical utilities 50 3.4. Principles of measurement: Detailed discussion 53 3.5. Conceptual structure of the axiomatic treatment of numerical utilities 57 3.6. The axioms and their interpretation 59 3.7. General remarks concerning the axioms 61 3.8. The role of the concept of marginal utility 62 4. STRUCTURE OF THE THEORY: SOLUTIONS AND STANDARDS OF BEHAVIOR 64 4.1. The simplest concept of a solution for one participant 64 4.2. Extension to all participants 66 4.3. The solution as a set of imputations 67 4.4. The intransitive notion of "superiority" or "domination" 70 4.5. The precise definition of a solution 72 4.6. Interpretation of our definition in terms of "standards of behavior" 73 4.7. Games and social organizations 76 4.8. Concluding remarks 76 CHAPTER II: GENERAL FORMAL DESCRIPTION OF GAMES OF STRATEGY 79 5. INTRODUCTION 79 5.1. Shift of emphasis from economics to games 79 5.2. General principles of classification and of procedure 79 6. THE SIMPLIFIED CONCEPT OF A GAME 81 6.1. Explanation of the termini technici 81 6.2. The elements of the game 82 6.3. Information and preliminary 84 6.4. Preliminarity, transitivity, and signaling 84 7. THE COMPLETE CONCEPT OF A GAME 88 7.1. Variability of the characteristics of eath move 88 7.2. The general description 90 8. SETS AND PARTITIONS 93 8.1. Desirability of a set-theoretical description of a game 93 8.2. Sets, their properties, and their graphical representation 94 8.3. Partitions, their properties, and their graphical representation 96 8.4. Logistic interpretation of sets and partitions 99 *9. THE SET-THEORETICAL DESCRIPTION OF A GAME 100 *9.1. The partitions which describe a game 100 *9.2. Discussion of these partitions and their properties 104 *10. AXIOMATIC FORMULATION 106 *10.1. The axioms and their interpretations 106 *10.2. Logistic discussion of the axioms 109 *10.3. General remarks concerning the axioms 109 *10.4. Graphical representation 110 11. STRATEGIES AND THE FINAL SIMPLIFICATION OF THE DESCRIPTION OF A GAME 112 11.1. The concept of a strategy and its fomalization 112 11.2. The final simplification of the description of a game 114 11.3. The role of strategies in the simplified form of a game 117 11.4. The meaning of the zero-sum restriction 117 CHAPTER III: ZERO-SUM TWO-PERSON GAMES: THEORY 118 12. PRELIMINARY SURVEY 118 12.1. General viewpoints 118 12.2. The one-person game 118 12.3. Chance and probability 120 12.4. The next objective 120 13. FUNCTIONAL CALCULUS 121 13.1. Basic definitions 121 13.2. The operations Max and Min 122 13.3. Commutativity questions 124 13.4. The mixed case. Saddle points 126 13.5. Proofs of the main facts 128 14. STRICTLY DETERMINED GAMES 131 14.1. Formulation of the problem 131 14.2. The minorant and the majorant games 133 14.3. Discussion of the auxiliary games 134 14.4. Conclusions 138 14.5. Analysis of strict determinateness 139 14.6. The interchange of players. Symmetry 142 14.7. Non strictly determined games 143 14.8. Program of a detailed analysis of strict determinateness 144 *15. GAMES WITH PERFECT INFORMATION 145 *15.1. Statement of purpose. Induction 145 *15.2. The exact condition (First step) 147 *15.3. The exact condition (Entire induction) 149 *15.4. Exact discussion of the inductive step 150 *15.5. Exact discussion of the inductive step (Continuation) 153 *15.6. The result in the case of perfect information 156 *15.7. Application to Chess 157 *15.8. The alternative, verbal discussion 159 16. LINEARITY AND CONVEXITY 161 16.1. Geometrical background 161 16.2. Vector operations 162 16.3. The theorem of the supporting hyperplanes 167 16.4. The theorem of the alternative for matrices 171 17. MIXED STRATEGIES. THE SOLUTION FOR ALL GAMES 176 17.1. Discussion of two elementary examples 176 17.2. Generalization of this viewpoint 178 17.3. Justification of the procedure as applied to an individual play 179 17.4. The minorant and the majorant games. (For mixed strategies) 182 17.5. General strict determinateness 183 17.6. Proof of the main theorem 186 17.7. Comparison of the treatment by pure and by mixed strategies 188 17.8. Analysis of general strict determinateness 191 17.9. Further characteristics of good strategies 193 17.10. Mistakes and their consequences. Permanent optimality 195 17.11. The interchange of players. Symmetry 198 CHAPTER IV: ZERO-SUM TWO-PERSON GAMES: EXAMPLES 202 18. SOME ELEMENTARY GAMES 202 18.1. The simplest games 202 18.2. Detailed quantitative discussion of these games 203 18.3. Qualitative characterizations 206 18.4. Discussion of some specific games. (Generalized forms of Matching Pennies) 208 18.5. Discussion of some slightly more complicated games 211 18.6. Chance and imperfect information 215 18.7. Interpretation of this result 218 *19. POKER AND BLUFFING 219 *19.1. Description of Poker 219 *19.2. Bluffing 221 *19.3. Description of Poker (Continued) 222 *19.4. Exact formulation of the rules 223 *19.5. Description of the strategy 224 *19.6. Statement of the problem 228 *19.7. Passage from the discrete to the continuous problem 229 *19.8. Mathematical determination of the solution 232 *19.9. Detailed analysis of the solution 235 *19.10. Interpretation of the solution 237 *19.11. More general forms of Poker 240 *19.12. Discrete hands 241 *19.13. m possible bids 242 *19.14. Alternate bidding 244 *19.15. Mathematical description of all solutions 249 *19.16. Interpretation of the solutions. Conclusions 251 CHAPTER V: ZERO-SUM THREE-PERSON GAMES 253 20. PRELIMINARY SURVEY 253 20.1. General viewpoints 253 20.2. Coalitions 254 21. THE SIMPLE MAJORITY GAME OF THREE PERSONS 255 21.1. Definition of the game 255 21.2. Analysis of the game: Necessity of "understandings" 256 21.3. Analysis of the game: Coalitions. The role of symmetry 257 22. FURTHER EXAMPLES 258 22.1. Unsymmetric distributions. Necessity of compensations 258 22.2. Coalitions of different strength. Discussion 260 22.3. An inequality. Formulae 262 23. THE GENERAL CASE 264 23.1. Detailed discussion. Inessential and essential games 264 23.2. Complete formulae 265 24. DISCUSSION OF AN OBJECTION 266 24.1. The case of perfect information and its significance 266 24.2. Detailed discussion. Necessity of compensations between three or more players 268 CHAPTER VI: FORMULATION OF THE GENERAL THEORY: ZERO-SUM n-PERSON GAMES 271 25. THE CHARACTERISTIC FUNCTION 271 25.1. Motivation and definition 271 25.2. Discussion of the concept 273 25.3. Fundamental properties 274 25.4. Immediate mathematical consequences 275 26. CONSTRUCTION OF A GAME WITH A GIVEN CHARACTERISTIC FUNCTION 276 26.1. The construction 276 26.2. Summary 278 27. STRATEGIC EQUIVALENCE. INESSENTIAL AND ESSENTIAL GAMES 278 27.1. Strategic equivalence. The reduced form 278 27.2. Inequalities. The quantity γ 281 27.3. Inessentiality and essentiality 282 27.4. Various criteria. Non additive utilities 283 27.5. The inequalities in the essential case 285 27.6. Vector operations on characteristic functions 286 28. GROUPS, SYMMETRY AND FAIRNESS 288 28.1. Permutations, their groups and their effect on a game 288 28.2. Symmetry and fairness 291 29. RECONSIDERATION OF THE ZERO-SUM THREE-PERSON GAME 293 29.1. Qualitative discussion 293 29.2. Quantitative discussion 295 30. THE EXACT FORM OF THE GENERAL DEFINITIONS 296 30.1. The definitions 296 30.2. Discussion and recapitulation 298 *30.3. The concept of saturation 299 30.4. Three immediate objectives 304 31. FIRST CONSEQUENCES 305 31.1. Convexity, flatness, and some criteria for domination 305 31.2. The system of all imputations. One element solutions 310 31.3 The isomorphism which corresponds to strategic equivalence 314 32. DETERMINATION OF ALL SOLUTIONS OF THE ESSENTIAL ZERO-SUM THREE-PERSON GAME 315 32.1. Formulation of the mathematical problem. The graphical method 315 32.2. Determination of all solutions 318 33. CONCLUSIONS 321 33.1. The multiplicity of solutions. Discrimination and its meaning 321 33.2. Statics and dynamics 323 CHAPTER VII: ZERO-SUM FOUR-PERSON GAMES 324 34. PRELIMINARY SURVEY 324 34.1. General viewpoints 324 34.2. Formalism of the essential zero sum four person games 324 34.3. Permutations of the players 327 35. DISCUSSION OF SOME SPECIAL POINTS IN THE CUBE Q 328 35.1. The corner I. (and V., VI., VII.) 328 35.2. The corner VIII. (and II., III., IV.,). The three person game and a"Dummy" 332 35.3. Some remarks concerning the interior of Q 335 36. DISCUSSION OF THE MAIN DIAGONALS 337 36.1. The part adjacent to the corner VIII.: Heuristic discussion 337 36.2. The part adjacent to the corner VIII.: Exact discussion 340 *36.3. Other parts of the main diagonals 345 37. THE CENTER AND ITS ENVIRONS 346 37.1. First orientation about the conditions around the center 346 37.2. The two alternatives and the role of symmetry 348 37.3. The first alternative at the center 349 37.4. The second alternative It the center 350 37.5. Comparison of the two central solutions 351 37.6. Unsymmetrical central solutions 352 *38. A FAMILY OF SOLUTIONS FOR A NEIGHBORHOOD OF THE CENTER 354 *38.1. Transformation of the solution belonging to the first alternative at the center 354 *38.2. Exact discussion 355 *38.3. Interpretation of the soIutions 361 CHAPTER VIII: SOME REMARKS CONCERNING n ≧ 5 PARTICIPANTS 363 39. THE NUMBER OF PARAMETERS IN V ARIOUS CI.ASSES OF GAMES 363 39.1. The situation for n = 3, 4 363 39.2. The situation for all n ≧ 3 363 40. THE SYMMETRIC FIVE PERSON GAME 365 40.1. Formalism of the symmetric five person game 365 40.2. The two extreme cases 365 40.3. Connection between the symmetric five person game and the 1, 2, 3-symmetric four person game 367 CHAPTER IX: COMPOSITION AND DECOMPOSITION OF GAMES 372 41. COMPOSITION AND DECOMPOSITlON OF GAMES 372 41.1. Search for n-person games for which all solutions can be determined 372 41.2. The first type. Composition and decomposition 373 41.3. Exact definitions 374 41.4. Analysis of decomposability 376 41.5. Desirability of a modification 378 42. MODIFICATION OF THE THEORY 378 42.1. No complete abandonment of the zero sum restriction 378 42.2. Strategic equivalence. Constant sum games 379 42.3. The characteristic function in the new theory 381 42.4. Imputations, domination, solutions in the new theory 383 42.5. Essentiality, inessentiality and decomposability in the new theory 384 43. TUE DECOMPOSITION PARTITION 386 43.1. Splitting sets. Constituents 386 43.2. Properties of the system of all splitting sets 386 43.3. Characterization of the system of all splitting sets. The decomposition partition 387 43.4. Properties of the decomposition partition 390 44. DECOMPOSABLE GAMES. FURTHER EXTENSION OF THE THEORY 391 44.1. Solutions of a (decomposable) game and solutions of its constituents 391 44.2. Composition and decomposition of imputations and of sets of imputations 392 44.3. Composition and decomposition of solutions. The main possibilities and surmises 394 44.4. Extension of the theory. Outside sources 396 44.5. The excess 397 44.6. Limitations of the excess. The non-isolated character of a game in the new setup 399 44.7. Discussion of the new setup. E(e[sub(0)]), F(e[sub(0)]) 400 45. LIMITATIONS OF THE EXCESS. STRUCTURE OF THE EXTENDED THEORY 401 45.1. The lower limit of the excess 401 45.2. The upper limit of the excess. Detached and fully detached imputations 402 45.3. Discussion of the two limits, |г|[sub(1)], |г|[sub(2)]. Their ratio 405 45.4. Detached imputations and various solutions. The theorem connecting E(e[sub(0)]), F(e[sub(0)]) 408 45.5. Proof of the theorem 409 45.6. Summary and conclusions 413 46. DETERMINATION OF ALL SOLUTIONS OF A DECOMPOSABLE GAME 414 46.1. Elementary properties of decompositions 414 46.2. Decomposition and its relation to the solutions: First results concerning F(e[sub(0)]) 417 46.3. Continuation 419 46.4. Continuation 421 46.5. The complete result in E(e[sub(0)]) 423 46.6. The complete result in E(eo) 426 46.7. Graphical representation of a part of the result 427 46.8. Interpretation: Thc normal zone. Heredity of various properties 429 46.9. Dummies 430 46.10. Imbedding of a game 431 46.11 Significance of the normal zone 434 46.12. First occurrence of the phenomenon of transfer: n = 6 435 47. THE ESSENTIAL THREE-PERSON GAME IN THE NEW THEORY 436 47.1. Need for this discussion 436 47.2. Preparatory considerations 436 47.3. The six cases of the discussion. Cases (I)–(III) 439 47.4. Case (IV): First part 440 47.5. Case (IV): Second part 442 47.6. Case (V) 446 47.7. Case (VI) 448 47.8. Interpretation of the result: The curves (one dimensional parts) in the solution 449 47.9. Continuation: The areas (two dimensional parts) in the solution 451 CHAPTER X: SIMPLE GAMES 453 48. WINNING AND LOSING COALITIONS AND GAMES WHERE THEY 453 48.1. The sceond type of 41.1. Decision by coalitions 453 48.2. Winning and Losing Coalitions 454 49. CHARACTERIZATION OF THE SIMPLE GAMES 456 49.1 General concepts of winning and losing coalitions 456 49.2. The special role of one element sets 458 49.3. Characterization of the systems W, L of actual games 459 49.4. Exact definition of simplicity 461 49.5. Some elementary properties of simplicity 461 49.6. Simple games and their W, L. The Minimal winning Coalitions: W[sup(m)] 462 49.7. The solutions of simple games 463 50. THE MAJORITY GAMES AND THE MAIN SOLUTION 464 50.1. Examples of simple games: The majority games 464 50.2. Homogeneity 466 50.3. A more direct use of the concept of imputation in forming solutions 468 50.4. Discussion of this direct approach 469 50.5. Connections with the general theory. Exact formulation 471 50.6. Reformulation of the result 473 50.7. Interpretation of the result 475 50.8. Connection with the Homogeneous Majority game 476 51. METHODS FOR THE ENUMERATION OF ALL SIMPLE GAMES 478 51.1. Preliminary Remarks 478 51.2. The saturation method: Enumeration by means of W 479 51.3. Reasons for passing from W to W[sup(m)]. Difficulties of using W[sup(m)] 481 51.4. Changed Approach: Enumeration by means of W[sup(m)] 483 51.5. Simplicity and decomposition 485 51.6. Inessentiality, Simplicity and Composition. Treatment of the excess 487 51.7. A criterium of decomposability in terms of W[sup(m)] 488 52. THE SIMPLE GAMES FOR SMALL n 490 52.1. Program. n = 1, 2 play no role. Disposal of n = 3 490 52.2. Procedure for n ≧ 4: The two element sets and their role in classifying the W[sup(m)] 491 52.3. Decomposability of cases C*, C[sub(n–2)], C[sub(n–1)] 492 52.4. The simple games other than [1, . . . 1, n – 2][sub(h)] (with dummies): The Cases C[sub(k)], k = 0, 1, . . . , n – 3 494 52.5. Disposal of n = 4, 5 495 53. THE NEW POSSIBILITIES OF SIMPLE GAMES FOR n ≧ 6 496 53.1. The Regularities observed for n ≧ 6 496 53.2. The six main counter examples (for n = 6, 7) 497 54. DETERMINATION OF ALL SOLUTIONS IN SUITABLE GAMES 503 54.1. Reasons to consider other solutions than the main solution in simple games 503 54.2. Enumeration of those gamea for which all solutions are known 504 54.3. Reasons to consider the simple game [1, . . . , 1, n – 2][sub(λ)] 505 *55. THE SIMPLE GAME [1, . . . , 1, n – 2][sub(h)] 506 *55.1. Preliminary Remarks 506 *55.2. Domination. The chief player. Cases (I) and (II) 506 *55.3. Disposal of Case (I) 508 *55.4. Case (II): Determination of V 511 *55.5. Case (II): Determination of V 514 *55.6. Case (II): a and S[sub(*)] 517 *55.7. Case (II′) and (II′′). Disposal of Case (II′) 518 *55.8. Case (II′′): a and V′. Domination 520 *55.9. Case (II′′): Determination of V′ 521 *55.10. Disposal of Case (II′′) 527 *55.11. Reformulation of the complete result 530 *55.12. Interpretation of the result 532 CHAPTER XI: GENERAL NON-ZERO-SUM GAMES 537 56. EXTENSION OF THE THEORY 537 56.1. Formulation of the problem 537 56.2. The fictitious player. The zero sum extension Γ 538 56.3. Questions concerning the character of Γ 539 56.4. Limitations of the use of Γ 541 56.5. The two possible procedures 543 56.6. The discriminatory solutions 544 56.7. Alternative possibilities 545 56.8. The new setup 547 56.9. Reconsideration of the case when Γ is a zero sum game 549 56.10. Analysis of the concept of domination 553 56.11. Rigorous discussion 556 56.12. The new definition of a solution 559 57. THE CHARACTERISTIC FUNCTION AND RELATED TOPICS 560 57.1. The characteristic function: The extended and the restricted form 560 57.2. Fundamental properties 561 57.3. Determination of all characteristic functions 563 57.4. Removable sets of players 566 57.5. Strategic equivalence. Zero-sum and constant-sum games 568 58. INTERPRETATION OF THE CHARACTERISTIC FUNCTION 571 58.1. Anslysis of the definition 571 58.2. The desire to make a gain vs. that to inflict a loss 572 58.3. Discussion 574 59. GENERAL CONSIDERATIONS 575 59.1. Discussion of the program 575 59.2. The reduced forms. The inequalities 576 59.3. Various topics 579 60. THE SOLUTIONS OF ALL GENERAL GAMES WITH n ≦ 3 581 60.1. The case n = 1 581 60.2. The case n = 2 582 60.3. The case n = 3 583 60.4. Comparison with the zero sum games 587 61. ECONOMIC INTERPRETATION OF THE RESULTS FOR n = 1, 2 588 61.1. The case n = 1 588 61.2. The Case n = 2. The two person market 588 61.3. Discussion of the two person market and its characteristic function 590 61.4. Justification of the standpoint of 58 592 61.5. Divisible goods. The "marginal pairs" 593 61.6. The price. Discussion 595 62. ECONOMIC INTERPRETATION OF THE RESULTS FOR n = 3: SPECIAL CASE 597 62.1. The case n = 3, special case. The three person market 597 62.2. Preliminary discussion 599 62.3. The solutions: First subcase 599 62.4. The solutions: General form 602 62.5. Algebraical form of the result 603 62.6. Discussion 604 63. ECOMIC INTERPRETATION OF THE RESULTS FOR n = 3: GENERAL CASE 606 63.1. Divisible goods 606 63.2. Analysis of the inequalities 608 63.3. Preliminary discussion 610 63.4. The solutions 610 63.5. Algebraical form of the result 613 63.6. Discussion 614 64. THE GENERAL MARKET 616 64.1. Formulation of the problem 616 64.2. Some special properties. Monopoly and monopsony 617 CHAPTER XII: EXTENSION OF THE CONCEPTS OF DOMINATION AND SOLUTION 620 65. THE EXTENSION. SPECIAL CASES 620 65.1. Formulation of the problem 620 65.2. General remarks 621 65.3. Orderings, transitivity, acyclicity 622 65.4. The solutions: For a symmetric relation. For a complete ordering 624 65.5. The solutions: For a partial ordering 625 65.6. Acyclicity and strict aeyclicity 627 65.7. The solutions: For an acyclic relation 630 65.8. Uniquenss of solutions, acyclicity and strict acyclicity 633 65.9. Application to games: Discreteness and continuity 635 66. GENERALIZATION OF THE CONCEPT OF UTILITY 636 66.1. The generalization. The two phases of the theoretical treatment 636 66.2. Discussion of the first phase 637 66.3. Discussion of the second phase 639 66.4. Desirability of unifying the two phases 640 67. DISCUSSION OF AN EXAMPLE 641 67.1. Description of the example 641 67.2. The solution and its interpretation 644 67.3. Generalization: Different discrete utility scales 647 67.4. Conclusions concerning bargaining 649 APPENDIX: THE AXIOMATIC TREATMENT OF UTILITY 650 Index 760 A 760 B 760 C 760 D 762 E 762 F 763 G 763 H 764 I 765 J 765 K 765 L 766 M 766 N 767 O 767 P 768 Q 769 R 769 S 769 T 771 U 771 V 772 W 772 Y 772 Z 772 Cover......Page 1 CONTENTS......Page 18 PREFACE......Page 28 TECHNICAL NOTE......Page 32 ACKNOWLEDGMENT......Page 33 1.1. Introductory remarks......Page 34 1.2. Difficulties of the application of the mathematical method......Page 35 1.3. Necessary limitations of the objectives......Page 39 1.4. Concluding remarks......Page 40 2.1. The problem of rational behavior......Page 41 2.2. " Robinson Crusoe" economy and social exchange economy......Page 42 2.3. The number of variables and the number of participants......Page 45 2.4. The case of many participants: Free competition......Page 46 3.1. Preferences and utilities......Page 48 3.2. Principles of measurement: Preliminaries......Page 49 3.3. Probability and numerical utilities......Page 50 3.4. Principles of measurement: Detailed discussion......Page 53 3.5. Conceptual structure of the axiomatic treatment of numerical utilities......Page 57 3.6. The axioms and their interpretation......Page 59 3.7. General remarks concerning the axioms......Page 61 3.8. The role of the concept of marginal utility......Page 62 4.1. The simplest concept of a solution for one participant......Page 64 4.2. Extension to all participants......Page 66 4.3. The solution as a set of imputations......Page 67 4.4. The intransitive notion of "superiority" or "domination"......Page 70 4.5. The precise definition of a solution......Page 72 4.6. Interpretation of our definition in terms of "standards of behavior"......Page 73 4.8. Concluding remarks......Page 76 5.2. General principles of classification and of procedure......Page 79 6.1. Explanation of the termini technici......Page 81 6.2. The elements of the game......Page 82 6.4. Preliminarity, transitivity, and signaling......Page 84 7.1. Variability of the characteristics of eath move......Page 88 7.2. The general description......Page 90 8.1. Desirability of a set-theoretical description of a game......Page 93 8.2. Sets, their properties, and their graphical representation......Page 94 8.3. Partitions, their properties, and their graphical representation......Page 96 8.4. Logistic interpretation of sets and partitions......Page 99 *9.1. The partitions which describe a game......Page 100 *9.2. Discussion of these partitions and their properties......Page 104 *10.1. The axioms and their interpretations......Page 106 *10.3. General remarks concerning the axioms......Page 109 *10.4. Graphical representation......Page 110 11.1. The concept of a strategy and its fomalization......Page 112 11.2. The final simplification of the description of a game......Page 114 11.4. The meaning of the zero-sum restriction......Page 117 12.2. The one-person game......Page 118 12.4. The next objective......Page 120 13.1. Basic definitions......Page 121 13.2. The operations Max and Min......Page 122 13.3. Commutativity questions......Page 124 13.4. The mixed case. Saddle points......Page 126 13.5. Proofs of the main facts......Page 128 14.1. Formulation of the problem......Page 131 14.2. The minorant and the majorant games......Page 133 14.3. Discussion of the auxiliary games......Page 134 14.4. Conclusions......Page 138 14.5. Analysis of strict determinateness......Page 139 14.6. The interchange of players. Symmetry......Page 142 14.7. Non strictly determined games......Page 143 14.8. Program of a detailed analysis of strict determinateness......Page 144 *15.1. Statement of purpose. Induction......Page 145 *15.2. The exact condition (First step)......Page 147 *15.3. The exact condition (Entire induction)......Page 149 *15.4. Exact discussion of the inductive step......Page 150 *15.5. Exact discussion of the inductive step (Continuation)......Page 153 *15.6. The result in the case of perfect information......Page 156 *15.7. Application to Chess......Page 157 *15.8. The alternative, verbal discussion......Page 159 16.1. Geometrical background......Page 161 16.2. Vector operations......Page 162 16.3. The theorem of the supporting hyperplanes......Page 167 16.4. The theorem of the alternative for matrices......Page 171 17.1. Discussion of two elementary examples......Page 176 17.2. Generalization of this viewpoint......Page 178 17.3. Justification of the procedure as applied to an individual play......Page 179 17.4. The minorant and the majorant games. (For mixed strategies)......Page 182 17.5. General strict determinateness......Page 183 17.6. Proof of the main theorem......Page 186 17.7. Comparison of the treatment by pure and by mixed strategies......Page 188 17.8. Analysis of general strict determinateness......Page 191 17.9. Further characteristics of good strategies......Page 193 17.10. Mistakes and their consequences. Permanent optimality......Page 195 17.11. The interchange of players. Symmetry......Page 198 18.1. The simplest games......Page 202 18.2. Detailed quantitative discussion of these games......Page 203 18.3. Qualitative characterizations......Page 206 18.4. Discussion of some specific games. (Generalized forms of Matching Pennies)......Page 208 18.5. Discussion of some slightly more complicated games......Page 211 18.6. Chance and imperfect information......Page 215 18.7. Interpretation of this result......Page 218 *19.1. Description of Poker......Page 219 *19.2. Bluffing......Page 221 *19.3. Description of Poker (Continued)......Page 222 *19.4. Exact formulation of the rules......Page 223 *19.5. Description of the strategy......Page 224 *19.6. Statement of the problem......Page 228 *19.7. Passage from the discrete to the continuous problem......Page 229 *19.8. Mathematical determination of the solution......Page 232 *19.9. Detailed analysis of the solution......Page 235 *19.10. Interpretation of the solution......Page 237 *19.11. More general forms of Poker......Page 240 *19.12. Discrete hands......Page 241 *19.13. m possible bids......Page 242 *19.14. Alternate bidding......Page 244 *19.15. Mathematical description of all solutions......Page 249 *19.16. Interpretation of the solutions. Conclusions......Page 251 20.1. General viewpoints......Page 253 20.2. Coalitions......Page 254 21.1. Definition of the game......Page 255 21.2. Analysis of the game: Necessity of "understandings"......Page 256 21.3. Analysis of the game: Coalitions. The role of symmetry......Page 257 22.1. Unsymmetric distributions. Necessity of compensations......Page 258 22.2. Coalitions of different strength. Discussion......Page 260 22.3. An inequality. Formulae......Page 262 23.1. Detailed discussion. Inessential and essential games......Page 264 23.2. Complete formulae......Page 265 24.1. The case of perfect information and its significance......Page 266 24.2. Detailed discussion. Necessity of compensations between three or more players......Page 268 25.1. Motivation and definition......Page 271 25.2. Discussion of the concept......Page 273 25.3. Fundamental properties......Page 274 25.4. Immediate mathematical consequences......Page 275 26.1. The construction......Page 276 27.1. Strategic equivalence. The reduced form......Page 278 27.2. Inequalities. The quantity γ......Page 281 27.3. Inessentiality and essentiality......Page 282 27.4. Various criteria. Non additive utilities......Page 283 27.5. The inequalities in the essential case......Page 285 27.6. Vector operations on characteristic functions......Page 286 28.1. Permutations, their groups and their effect on a game......Page 288 28.2. Symmetry and fairness......Page 291 29.1. Qualitative discussion......Page 293 29.2. Quantitative discussion......Page 295 30.1. The definitions......Page 296 30.2. Discussion and recapitulation......Page 298 *30.3. The concept of saturation......Page 299 30.4. Three immediate objectives......Page 304 31.1. Convexity, flatness, and some criteria for domination......Page 305 31.2. The system of all imputations. One element solutions......Page 310 31.3 The isomorphism which corresponds to strategic equivalence......Page 314 32.1. Formulation of the mathematical problem. The graphical method......Page 315 32.2. Determination of all solutions......Page 318 33.1. The multiplicity of solutions. Discrimination and its meaning......Page 321 33.2. Statics and dynamics......Page 323 34.2. Formalism of the essential zero sum four person games......Page 324 34.3. Permutations of the players......Page 327 35.1. The corner I. (and V., VI., VII.)......Page 328 35.2. The corner VIII. (and II., III., IV.,). The three person game and a"Dummy"......Page 332 35.3. Some remarks concerning the interior of Q......Page 335 36.1. The part adjacent to the corner VIII.: Heuristic discussion......Page 337 36.2. The part adjacent to the corner VIII.: Exact discussion......Page 340 *36.3. Other parts of the main diagonals......Page 345 37.1. First orientation about the conditions around the center......Page 346 37.2. The two alternatives and the role of symmetry......Page 348 37.3. The first alternative at the center......Page 349 37.4. The second alternative It the center......Page 350 37.5. Comparison of the two central solutions......Page 351 37.6. Unsymmetrical central solutions......Page 352 *38.1. Transformation of the solution belonging to the first alternative at the center......Page 354 *38.2. Exact discussion......Page 355 *38.3. Interpretation of the soIutions......Page 361 39.2. The situation for all n ≧ 3......Page 363 40.2. The two extreme cases......Page 365 40.3. Connection between the symmetric five person game and the 1, 2, 3-symmetric four person game......Page 367 41.1. Search for n-person games for which all solutions can be determined......Page 372 41.2. The first type. Composition and decomposition......Page 373 41.3. Exact definitions......Page 374 41.4. Analysis of decomposability......Page 376 42.1. No complete abandonment of the zero sum restriction......Page 378 42.2. Strategic equivalence. Constant sum games......Page 379 42.3. The characteristic function in the new theory......Page 381 42.4. Imputations, domination, solutions in the new theory......Page 383 42.5. Essentiality, inessentiality and decomposability in the new theory......Page 384 43.2. Properties of the system of all splitting sets......Page 386 43.3. Characterization of the system of all splitting sets. The decomposition partition......Page 387 43.4. Properties of the decomposition partition......Page 390 44.1. Solutions of a (decomposable) game and solutions of its constituents......Page 391 44.2. Composition and decomposition of imputations and of sets of imputations......Page 392 44.3. Composition and decomposition of solutions. The main possibilities and surmises......Page 394 44.4. Extension of the theory. Outside sources......Page 396 44.5. The excess......Page 397 44.6. Limitations of the excess. The non-isolated character of a game in the new setup......Page 399 44.7. Discussion of the new setup. E(e[sub(0)]), F(e[sub(0)])......Page 400 45.1. The lower limit of the excess......Page 401 45.2. The upper limit of the excess. Detached and fully detached imputations......Page 402 45.3. Discussion of the two limits, |г|[sub(1)], |г|[sub(2)]. Their ratio......Page 405 45.4. Detached imputations and various solutions. The theorem connecting E(e[sub(0)]), F(e[sub(0)])......Page 408 45.5. Proof of the theorem......Page 409 45.6. Summary and conclusions......Page 413 46.1. Elementary properties of decompositions......Page 414 46.2. Decomposition and its relation to the solutions: First results concerning F(e[sub(0)])......Page 417 46.3. Continuation......Page 419 46.4. Continuation......Page 421 46.5. The complete result in E(e[sub(0)])......Page 423 46.6. The complete result in E(eo)......Page 426 46.7. Graphical representation of a part of the result......Page 427 46.8. Interpretation: Thc normal zone. Heredity of various properties......Page 429 46.9. Dummies......Page 430 46.10. Imbedding of a game......Page 431 46.11 Significance of the normal zone......Page 434 46.12. First occurrence of the phenomenon of transfer: n = 6......Page 435 47.2. Preparatory considerations......Page 436 47.3. The six cases of the discussion. Cases (I)–(III)......Page 439 47.4. Case (IV): First part......Page 440 47.5. Case (IV): Second part......Page 442 47.6. Case (V)......Page 446 47.7. Case (VI)......Page 448 47.8. Interpretation of the result: The curves (one dimensional parts) in the solution......Page 449 47.9. Continuation: The areas (two dimensional parts) in the solution......Page 451 48.1. The sceond type of 41.1. Decision by coalitions......Page 453 48.2. Winning and Losing Coalitions......Page 454 49.1 General concepts of winning and losing coalitions......Page 456 49.2. The special role of one element sets......Page 458 49.3. Characterization of the systems W, L of actual games......Page 459 49.5. Some elementary properties of simplicity......Page 461 49.6. Simple games and their W, L. The Minimal winning Coalitions: W[sup(m)]......Page 462 49.7. The solutions of simple games......Page 463 50.1. Examples of simple games: The majority games......Page 464 50.2. Homogeneity......Page 466 50.3. A more direct use of the concept of imputation in forming solutions......Page 468 50.4. Discussion of this direct approach......Page 469 50.5. Connections with the general theory. Exact formulation......Page 471 50.6. Reformulation of the result......Page 473 50.7. Interpretation of the result......Page 475 50.8. Connection with the Homogeneous Majority game......Page 476 51.1. Preliminary Remarks......Page 478 51.2. The saturation method: Enumeration by means of W......Page 479 51.3. Reasons for passing from W to W[sup(m)]. Difficulties of using W[sup(m)]......Page 481 51.4. Changed Approach: Enumeration by means of W[sup(m)]......Page 483 51.5. Simplicity and decomposition......Page 485 51.6. Inessentiality, Simplicity and Composition. Treatment of the excess......Page 487 51.7. A criterium of decomposability in terms of W[sup(m)]......Page 488 52.1. Program. n = 1, 2 play no role. Disposal of n = 3......Page 490 52.2. Procedure for n ≧ 4: The two element sets and their role in classifying the W[sup(m)]......Page 491 52.3. Decomposability of cases C*, C[sub(n–2)], C[sub(n–1)]......Page 492 52.4. The simple games other than [1, . . . 1, n – 2][sub(h)] (with dummies): The Cases C[sub(k)], k = 0, 1, . . . , n – 3......Page 494 52.5. Disposal of n = 4, 5......Page 495 53.1. The Regularities observed for n ≧ 6......Page 496 53.2. The six main counter examples (for n = 6, 7)......Page 497 54.1. Reasons to consider other solutions than the main solution in simple games......Page 503 54.2. Enumeration of those gamea for which all solutions are known......Page 504 54.3. Reasons to consider the simple game [1, . . . , 1, n – 2][sub(λ)]......Page 505 *55.2. Domination. The chief player. Cases (I) and (II)......Page 506 *55.3. Disposal of Case (I)......Page 508 *55.4. Case (II): Determination of V......Page 511 *55.5. Case (II): Determination of V......Page 514 *55.6. Case (II): a and S[sub(*)]......Page 517 *55.7. Case (II′) and (II′′). Disposal of Case (II′)......Page 518 *55.8. Case (II′′): a and V′. Domination......Page 520 *55.9. Case (II′′): Determination of V′......Page 521 *55.10. Disposal of Case (II′′)......Page 527 *55.11. Reformulation of the complete result......Page 530 *55.12. Interpretation of the result......Page 532 56.1. Formulation of the problem......Page 537 56.2. The fictitious player. The zero sum extension Γ......Page 538 56.3. Questions concerning the character of Γ......Page 539 56.4. Limitations of the use of Γ......Page 541 56.5. The two possible procedures......Page 543 56.6. The discriminatory solutions......Page 544 56.7. Alternative possibilities......Page 545 56.8. The new setup......Page 547 56.9. Reconsideration of the case when Γ is a zero sum game......Page 549 56.10. Analysis of the concept of domination......Page 553 56.11. Rigorous discussion......Page 556 56.12. The new definition of a solution......Page 559 57.1. The characteristic function: The extended and the restricted form......Page 560 57.2. Fundamental properties......Page 561 57.3. Determination of all characteristic functions......Page 563 57.4. Removable sets of players......Page 566 57.5. Strategic equivalence. Zero-sum and constant-sum games......Page 568 58.1. Anslysis of the definition......Page 571 58.2. The desire to make a gain vs. that to inflict a loss......Page 572 58.3. Discussion......Page 574 59.1. Discussion of the program......Page 575 59.2. The reduced forms. The inequalities......Page 576 59.3. Various topics......Page 579 60.1. The case n = 1......Page 581 60.2. The case n = 2......Page 582 60.3. The case n = 3......Page 583 60.4. Comparison with the zero sum games......Page 587 61.2. The Case n = 2. The two person market......Page 588 61.3. Discussion of the two person market and its characteristic function......Page 590 61.4. Justification of the standpoint of 58......Page 592 61.5. Divisible goods. The "marginal pairs"......Page 593 61.6. The price. Discussion......Page 595 62.1. The case n = 3, special case. The three person market......Page 597 62.3. The solutions: First subcase......Page 599 62.4. The solutions: General form......Page 602 62.5. Algebraical form of the result......Page 603 62.6. Discussion......Page 604 63.1. Divisible goods......Page 606 63.2. Analysis of the inequalities......Page 608 63.4. The solutions......Page 610 63.5. Algebraical form of the result......Page 613 63.6. Discussion......Page 614 64.1. Formulation of the problem......Page 616 64.2. Some special properties. Monopoly and monopsony......Page 617 65.1. Formulation of the problem......Page 620 65.2. General remarks......Page 621 65.3. Orderings, transitivity, acyclicity......Page 622 65.4. The solutions: For a symmetric relation. For a complete ordering......Page 624 65.5. The solutions: For a partial ordering......Page 625 65.6. Acyclicity and strict aeyclicity......Page 627 65.7. The solutions: For an acyclic relation......Page 630 65.8. Uniquenss of solutions, acyclicity and strict acyclicity......Page 633 65.9. Application to games: Discreteness and continuity......Page 635 66.1. The generalization. The two phases of the theoretical treatment......Page 636 66.2. Discussion of the first phase......Page 637 66.3. Discussion of the second phase......Page 639 66.4. Desirability of unifying the two phases......Page 640 67.1. Description of the example......Page 641 67.2. The solution and its interpretation......Page 644 67.3. Generalization: Different discrete utility scales......Page 647 67.4. Conclusions concerning bargaining......Page 649 APPENDIX: THE AXIOMATIC TREATMENT OF UTILITY......Page 650 C......Page 760 E......Page 762 G......Page 763 H......Page 764 K......Page 765 M......Page 766 O......Page 767 P......Page 768 S......Page 769 U......Page 771 Z......Page 772 This is the classic work upon which modern-day game theory is based. What began more than sixty years ago as a modest proposal that a mathematician and an economist write a short paper together blossomed, in 1944, when Princeton University Press published "Theory of Games and Economic Behavior". In it, John von Neumann and Oskar Morgenstern conceived a groundbreaking mathematical theory of economic and social organization, based on a theory of games of strategy. Not only would this revolutionize economics, but the entirely new field of scientific inquiry, it yielded - game theory - which has since been widely used to analyze a host of real-world phenomena from arms races to optimal policy choices of presidential candidates, from vaccination policy to major league baseball salary negotiations. And it is today established throughout both the social sciences and a wide range of other sciences. This sixtieth anniversary edition includes not only the original text but also an introduction by Harold Kuhn, an afterword by Ariel Rubinstein, and reviews and articles on the book that appeared at the time of its original publication in the "New York Times", the "American Economic Review", and a variety of other publications. Together, these writings provide readers a matchless opportunity to more fully appreciate a work whose influence will yet resound for generations to come "This is the classic work upon which modernday game theory is based. What began more than sixty years ago as a modest proposal that a mathematician and an economist write a short paper together blossomed, in 1944, when Princeton University Press published Theory of Games and Economic Behavior. In it, John von Neumann and Oskar Morgenstern conceived a groundbreaking mathematical theory of economic and social organization, based on a theory of games of strategy. Not only would this revolutionize economics, but the entirely new field of scientific inquiry it yielded -- game theory -- has since been widely used to analyze a host of real-world phenomena from arms races to optimal policy choices of presidential candidates, from vaccination policy to major league baseball salary negotiations. And it is today established throughout both the social sciences and a wide range of other sciences." "This sixtieth-anniversary edition includes not only the original text but also an introduction by Harold Kuhn, an afterword by Ariel Rubinstein, and contemporary reviews and articles on the book that appeared in the New York Times, The American Economic Review, and a variety of other publications. Together, these writings provide readers a matchless opportunity to more fully appreciate a work whose influence will yet resound for generations to come. Book jacket."--BOOK JACKET This is the classic work upon which modern-day game theory is based. What began more than sixty years ago as a modest proposal that a mathematician and an economist write a short paper together blossomed, in 1944, when Princeton University Press published Theory of Games and Economic Behavior . In it, John von Neumann and Oskar Morgenstern conceived a groundbreaking mathematical theory of economic and social organization, based on a theory of games of strategy. Not only would this revolutionize economics, but the entirely new field of scientific inquiry it yielded—game theory—has since been widely used to analyze a host of real-world phenomena from arms races to optimal policy choices of presidential candidates, from vaccination policy to major league baseball salary negotiations. And it is today established throughout both the social sciences and a wide range of other sciences. This sixtieth anniversary edition includes not only the original text but also an introduction by Harold Kuhn, an afterword by Ariel Rubinstein, and reviews and articles on the book that appeared at the time of its original publication in the New York Times , tthe American Economic Review , and a variety of other publications. Together, these writings provide readers a matchless opportunity to more fully appreciate a work whose influence will yet resound for generations to come.