Theory of Games and Economic Behavior

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[read more]