## Theory of Games and Economic Behavior |

PREFACE v 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 44651.3. Reasons for passing from W to W. Difficulties of using ^{m}W 448^{m}51.4. Changed Approach: Enumeration by means of W 450^{m}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 455^{m}52.THE SIMPLE GAMES FOR SMALL n 45752.1. Program. n = 1, 2 play no role. Disposal of n = 3 45752.2. Procedure for n [equal to or greater than] 4: The two element sets and their role in classify ing the W 458^{m}52.3. Decomposability of cases C*, C_{n-2}, C_{n-1} 45952.4. The simple games other than [1, ^{. . .} , 1, n - 2]_{h}, (with dummies): The Cases C_{k}, k = 0, 1, ^{. . .} , n - 3 46152.5. Disposal of n = 4, 5 46253.THE NEW POSSIBILITIES OF SIMPLE GAMES FOR n [equal to or greater than] 6 46353.1. The Regularities observed for n [equal to or greater than] 6 46353.2. The six main counter examples (for n = 6, 7) 46454.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 54860.1. The case n = 1 54860.2. The case n = 2 54960.3. The case n = 3 55060.4. Comparison with the zero sum games 554 61.ECONOMIC INTERPRETATION OF THE RESULTS FOR n = 1, 2 55561.1. The case n = 1 55561.2. The case n = 2. The two person market 55561.3. 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 56462.1. The case n = 3, special case. The three person market 56462.2. 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 57363.1. 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
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