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Archive/John von Neumann and Oskar Morgenstern
Spieltheorie und wirtschaftliches Verhalten

John von Neumann and Oskar Morgenstern · 1961

Spieltheorie und wirtschaftliches Verhalten

344 sections
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About this work

John von Neumann and Oskar Morgenstern, Spieltheorie und wirtschaftliches Verhalten

This German translation of Theory of Games and Economic Behavior presents von Neumann and Morgenstern’s foundational attempt to recast economics as a mathematical theory of strategic interaction.

Dieses Buch enthält eine Darstellung sowie verschiedene Anwendungen einer mathematischen Theorie der Spiele.

English translation: This book contains an exposition, together with various applications, of a mathematical theory of games.

The book’s premise is methodological as much as technical: economics cannot become exact merely by borrowing mathematical language. It must first clarify what its basic objects are—utility, choice, strategy, payoff, coalition, and stability.

Man kann nicht exakte Methoden verwenden, solange keine Klarheit in den Begriffen und Fragen besteht, auf die sie angewendet werden sollen.

English translation: One cannot employ exact methods so long as there is no clarity concerning the concepts and questions to which they are to be applied.

Against models of isolated maximization, the authors define economic action as interdependent choice. Each participant tries to optimize an outcome, but that outcome depends on variables controlled by others. This is why games, not single-agent calculation, become the central formal model.

So versucht jeder Teilnehmer eine Funktion (das oben erwähnte "Resultat") zu maximalisieren, ohne jedoch alle Variablen dieser Funktion zu kontrollieren.

English translation: Thus each participant seeks to maximize a function (the "outcome" mentioned above), without, however, controlling all the variables of this function.

The early chapters build the formal machinery: extensive and normal forms, payoff functions, utility under risk, and strategies understood as complete plans of conduct. In two-person zero-sum games, the decisive result is the minimax theorem. Rational play may require mixed strategies, where probability is not a sign of uncertainty alone but part of optimal strategic design.

Die Strategie des Spielers besteht weder in der Wahl von "Kopf" noch von "Wappen", sondern in der Wahl von "Kopf" mit der Wahrscheinlichkeit 1/2 und der Wahl von "Wappen" mit der Wahrscheinlichkeit 1/2.

English translation: The player's strategy consists neither in the choice of "heads" nor of "tails," but in the choice of "heads" with probability 1/2 and "tails" with probability 1/2.

The later sections extend the theory to n-person games, where coalitions become central. Here the “solution” is no longer necessarily a single outcome, but a stable set of possible imputations: a standard of behavior that resists internal domination while excluding inferior alternatives. This shift lets the authors treat social order, bargaining, and market organization as formal phenomena rather than informal background conditions.

In vielen der weiteren Untersuchungen werden wir sehen, daß diese Vielfalt der Lösungen tatsächlich ein sehr allgemeines Phänomen darstellt.

English translation: In many of the further investigations we shall see that this multiplicity of solutions is in fact a very general phenomenon.

The book’s importance lies in this combination of rigor and scope. It founds modern game theory by showing how economic behavior can be modeled through strategic dependence, randomized play, coalition formation, and stable social standards. Its market discussions connect the abstract theory back to economic life, including competition, monopoly, bargaining blocs, and transfers. Economics is thereby reconstructed as the mathematics of interdependent choice.

Sections

This work was divided into 344 sections when it entered the library's research corpus—an apparatus for search and citation, not necessarily the author's own table of contents. Each title opens its summary.

  1. 1Title Page and Publication Data▾
  2. 2Preface to the First Edition▾
  3. 3Preface to the Second Edition▾
  4. 4Preface to the Third Edition▾
  5. 5Preface to the German Edition▾
  6. 6Notes on the Exposition and Mathematical Prerequisites▾
  7. 7Table of Contents▾
  8. 8Table of Contents: Final Entries, Appendix, Figures, Name Index, Subject Index▾
  9. 9Formulation of the Economic Problem: Mathematical Method in Economics—Introductory Remarks▾
  10. 10Difficulties in Applying Mathematical Methods to Economics▾
  11. 11Necessary Limitation of Aims in Economic Theory▾
  12. 12Concluding Remarks on Mathematical Method in Economics▾
  13. 13Qualitative Study of Rational Behavior: The Problem of Rational Behavior▾
  14. 14Robinson Crusoe Economy versus Social Exchange Economy▾
  15. 15Number of Variables and Number of Participants▾
  16. 16Many Participants, Free Competition, Lausanne School, and Preliminary Utility Measurement▾
  17. 17Probability and Numerical Utility▾
  18. 18Limits of Indifference Curves and the Need to Measure Utility▾
  19. 19Principles of Measurement: Natural Operations and Relations▾
  20. 20Measurement Invariance, Transformation Groups, and Utility▾
  21. 21Formal Axioms for Utility Mixtures▾
  22. 22Interpretation of the Utility Axioms▾
  23. 23General Remarks on Expected Utility, Completeness, and Multidimensional Utility▾
  24. 24Axiomatic Construction of Numerical Utility▾
  25. 25Marginal Utility and Perfect Information▾
  26. 26From Individual Solutions to Imputation for All Participants▾
  27. 27Solutions as Sets of Imputations and Coalitional Possibilities▾
  28. 28Domination and the Intransitivity of Social Superiority▾
  29. 29Exact Definition of a Solution as a Stable Set▾
  30. 30Behavior Standards as Interpretations of Stable Sets▾
  31. 31Games, Social Organization, and Static Theory▾
  32. 32Sets, Partitions, and Information Schemas▾
  33. 33Set-Theoretic Description of a Game▾
  34. 34Axiomatic Formulation and Graphical Representation of Games▾
  35. 35Strategies and the Reduction to Normal Form▾
  36. 36Chapter II Introduction and the Simplified Concept of a Game▾
  37. 37Information, Factual Precedence, and Signaling▾
  38. 38Chapter III Overview: Toward Two-Person Zero-Sum Games▾
  39. 39Function Theory: Basic Definitions▾
  40. 40The Complete Concept of a Game▾
  41. 41Operations Max and Min▾
  42. 42Commutativity Questions for Max and Min▾
  43. 43The Mixed Case and Saddle Points▾
  44. 44Proofs of the Minimax Inequality and Saddle Point Criterion▾
  45. 45Consequences of Saddle Point Existence▾
  46. 46Saddle-Point Existence for Function-Valued Choices▾
  47. 47Uniquely Determined Games: Normal-Form Problem Statement▾
  48. 48Minorant and Majorant Auxiliary Games▾
  49. 49Analysis of the Minorant Game▾
  50. 50Analysis of the Majorant Game and Duality of the Auxiliary Games▾
  51. 51Consequences: Value Bounds and Strict Determinateness▾
  52. 52Strict Determinateness, Player Symmetry, Non-Determined Games, and Complete Information▾
  53. 53Games with Complete Information: Objective and Induction▾
  54. 54The Exact Set-Theoretic Condition for Removing the First Move▾
  55. 55Complete Induction and the Induction Step for Zero-Sum Games▾
  56. 56Exact Examination of the Induction Step (Continuation)▾
  57. 57The Result for Complete Information▾
  58. 58Application to Chess▾
  59. 59The Corresponding Verbal Analysis Begins▾
  60. 60Backward Induction and the Initial Rationality Objection▾
  61. 61Rationality in Zero-Sum Games and the Turn to Convexity▾
  62. 62Geometric Foundations and Basic Vector Operations▾
  63. 63Convex Hulls, Convex Combinations, Simplices, and Positive Orthants▾
  64. 64Euclidean Length and the Supporting Hyperplane Theorem▾
  65. 65The Theorem of the Alternative for Matrices▾
  66. 66Opening of Mixed Strategies: Elementary Examples▾
  67. 67Carryover Footnote on Index Renaming▾
  68. 68Mixed Strategies in Matching Pennies and Rock-Paper-Scissors▾
  69. 69Generalizing Mixed Strategies to All Zero-Sum Games▾
  70. 70Justification of the Method by Its Applicability to a Single Play▾
  71. 71The Minorant and Majorant Games for Mixed Strategies▾
  72. 72General Determinateness▾
  73. 73Proof of the Main Theorem▾
  74. 74Comparison of Pure and Mixed Strategies▾
  75. 75General Determinacy and the Definition of the Game Value▾
  76. 76Good Play, Saddle Points, and Independence from Rationality Assumptions▾
  77. 77Further Characteristics of Good Strategies: Saddle-Point Support Conditions▾
  78. 78Special Determinateness and Pure Good Strategies▾
  79. 79Errors, Loss Measures, and Optimal Responses▾
  80. 80Permanent Optimality, Complete Information, and Player Interchange▾
  81. 81Symmetric and Fair Games; Skew Symmetry and Craps Example▾
  82. 82Symmetric zero-sum games and protection against loss▾
  83. 83Opening of Chapter IV and the simplest elementary games▾
  84. 84Quantitative and qualitative analysis of elementary 2x2 zero-sum games▾
  85. 85Special games: generalized matching pennies and the Sherlock Holmes example▾
  86. 86More complicated 3×3 games and the limits of simple determinacy criteria▾
  87. 87Replacing chance moves by personal moves under incomplete information▾
  88. 88Interpreting chance replacement: matching pennies, rock-paper-scissors, and card cutting▾
  89. 89Poker and Bluffing: Motivation, Poker Description, and Bluffing Concepts▾
  90. 90Poker Description Continued: Bid Limits, Symmetry, and Simplified Betting Outcomes▾
  91. 91Precise Formal Rules of the Simplified Poker Game▾
  92. 92Strategy Descriptions and the Optimality Problem in the Poker-Bluffing Game▾
  93. 93From Discrete Poker Hands to a Continuous Strength Scale▾
  94. 94Continuous Mixed Strategy Functions and the Continuous Optimality Criterion▾
  95. 95Mathematical Determination of the Poker Solution▾
  96. 96Detailed Verification of the Poker Strategy▾
  97. 97Interpretation of the Poker Solution: Bluffing and Deviations▾
  98. 98Generalized Forms of Poker▾
  99. 99Discrete Hands▾
  100. 100m Possible Bids▾
  101. 101Alternating Bidding▾
  102. 102Mathematical Description of All Solutions▾
  103. 103Interpreting Solutions: Bluffing in the Sequential Poker Variant▾
  104. 104Opening of Three-Person Zero-Sum Games: From Antagonism to Parallel Interests▾
  105. 105Coalitions and the Simple Three-Person Majority Game▾
  106. 106Analysis of the Simple Majority Game: Necessity of Agreements, Opening▾
  107. 107Footnotes to the Definition of the Simple Majority Game▾
  108. 108Analysis of the Simple Majority Game: Extra-Game Agreements and Single-Play Theory▾
  109. 109Analysis of the Simple Majority Game: Coalitions and the Role of Symmetry▾
  110. 110Further Examples: Asymmetrical Distribution and the Necessity of Compensation▾
  111. 111Coalitions of Different Strength: General Setup and Initial Feasibility Argument▾
  112. 112Footnotes on Heuristic Arguments and the Homo Oeconomicus Assumption▾
  113. 113Coalitions of Different Strength: Bounds Alpha, Beta, and Gamma▾
  114. 114An Inequality and Formulas for Coalition Incentives▾
  115. 115General Case: Essential and Inessential Three-Person Zero-Sum Games▾
  116. 116Complete Formulas for Coalition Values in the General Three-Person Case▾
  117. 117Objection from Games of Complete Information▾
  118. 118Detailed Critique: Compensation and Coalitions with Three or More Players▾
  119. 119Characteristic Function: Motivation, Definition, and Conceptual Discussion▾
  120. 120Characteristic Function: Fundamental Properties, Consequences, and Transition to Game Construction▾
  121. 121Construction of a Game with a Prescribed Characteristic Function▾
  122. 122Summary of the Characterization of Characteristic Functions▾
  123. 123Strategic Equivalence and the Reduced Form of a Characteristic Function▾
  124. 124Inequalities for Reduced Characteristic Functions and the Quantity Gamma▾
  125. 125Inessential and Essential Games▾
  126. 126Strategic Equivalence: Essentiality, Nonadditive Utility, Inequalities, and Vector Operations▾
  127. 127Groups, Symmetry, Fairness, and Permutations in n-Person Games▾
  128. 128Symmetry and Fairness▾
  129. 129Qualitative Reconsideration of the Three-Person Zero-Sum Game▾
  130. 130Quantitative Stability Analysis of the Three-Person Solution▾
  131. 131Exact Form of the General Definitions▾
  132. 132Definitions of Imputation, Domination, and Solution▾
  133. 133Discussion and Recapitulation of the Solution Concept▾
  134. 134Formal Definition of Saturation▾
  135. 135Examples, Symmetry, and Maximality of Saturation▾
  136. 136Why Saturation Methods Do Not Directly Prove Existence of Game Solutions▾
  137. 137Three Immediate Questions and Transition to First Consequences▾
  138. 138First consequences: domination, side conditions, and necessary coalitions▾
  139. 139Convexity and flat coalitions as domination criteria▾
  140. 140The system of imputations and one-element solutions▾
  141. 141Isomorphism Corresponding to Strategic Equivalence▾
  142. 142Essential Three-Person Zero-Sum Game: Formulation and Graphical Method▾
  143. 143Determination of All Solutions of the Essential Three-Person Zero-Sum Game▾
  144. 144Conclusions▾
  145. 145Diversity of Solutions: Discrimination and Its Significance▾
  146. 146Statics and Dynamics in Coalition Theory▾
  147. 147Four-Person Zero-Sum Games: Preliminary Overview and Formalism▾
  148. 148Permutations of Players and Geometric Symmetries of the Cube▾
  149. 149The Corner I and Equivalent Vertices: A Strategically Favored Player▾
  150. 150The Corner VIII and the Straw Man: Reduction to a Three-Person Game▾
  151. 151Summary of the Corner VIII Four-Person Zero-Sum Game▾
  152. 152Strohman Exclusion, Symmetric Center of Q, and the Main Diagonal▾
  153. 153Interior Points of Q, Perturbations, and Limits of Barycentric Heuristics▾
  154. 154Heuristic Analysis Near Corner VIII on the Main Diagonal▾
  155. 155Exact solution set near corner VIII: conjecture and criterion (36:A)▾
  156. 156Proof of criterion (36:A) and further intervals on the main diagonal▾
  157. 157Chapter 37 heading: the midpoint and its surroundings▾
  158. 158Initial Orientation around the Center▾
  159. 159The Two Alternatives and the Role of Symmetry▾
  160. 160First and Second Alternatives at the Center▾
  161. 161Comparison of the Two Center Solutions▾
  162. 162Asymmetric Center Solutions▾
  163. 163A Family of Solutions for a Neighborhood of the Center▾
  164. 164Chapter VIII: Some Remarks for n ≥ 5 Participants▾
  165. 165Parameter Counts for Classes of n-Person Zero-Sum Games▾
  166. 166Formal Setup of the Symmetric Five-Person Zero-Sum Game▾
  167. 167Extreme Cases of the Symmetric Five-Person Game▾
  168. 168Relating the Symmetric Five-Person Game to a 1,2,3-Symmetric Four-Person Game▾
  169. 169Chapter IX Opening and Search for Solvable n-Person Games▾
  170. 170First Type: Composition and Decomposition of Games▾
  171. 171Exact Definitions of Composition and Decomposability▾
  172. 172Investigation of Decomposability Conditions▾
  173. 173Need for a Modification of the Theory▾
  174. 174Modifying the Theory and Introducing Constant-Sum Strategic Equivalence▾
  175. 175Characteristic Functions, Imputations, Domination, and Solutions in Constant-Sum Games▾
  176. 176Essentiality and Decomposability in the Constant-Sum Theory▾
  177. 177The Decomposition Partition: Split Sets and Elementary Properties▾
  178. 178Closure of Split Sets under Intersection and Union▾
  179. 179Minimal Split Sets and the Decomposition Partition▾
  180. 180Extremes and Player-Count Consequences of the Decomposition Partition▾
  181. 181Solutions of a Decomposable Game and of Its Components▾
  182. 182Composition and Decomposition of Imputations and Sets of Imputations▾
  183. 183Composition and Decomposition of Solutions: Main Possibilities and Conjectures▾
  184. 184Extension of the Theory: Outside Sources▾
  185. 185The Excess▾
  186. 186Limits on Excess and the Non-Isolated Character of the Game▾
  187. 187The Extended Sets E(e0) and F(e0) and Solutions▾
  188. 188The Lower Bound for Excess▾
  189. 189The Upper Bound for Excess: Separated and Completely Separated Imputations▾
  190. 190Relation Between the Two Excess Bounds |Γ|1 and |Γ|2▾
  191. 191Separated Imputations and the Relation Between E(e0) and F(e0)▾
  192. 192Auxiliary Lemmas on Domination and Separated Imputations▾
  193. 193Unique Decomposition of F(e0) Solutions and Solution Status of V▾
  194. 194Converse Construction of F(e0) Solutions and Proof of Theorem 45:I▾
  195. 195Summary of Threshold Results for Excess▾
  196. 196Excess Range in Nonempty F(e0) Solutions and Opening of Decomposable Games Chapter▾
  197. 197Elementary properties of decompositions and additive essentiality▾
  198. 198Dominating coalitions can be localized in decomposable games▾
  199. 199Initial setup for solutions of a decomposable game under F(e0)▾
  200. 200Component mixing criterion and product characterization of U▾
  201. 201Component solution properties and extremal excess identities▾
  202. 202Constructing all full-game solutions from compatible component solutions▾
  203. 203Deriving Explicit Cases for the Complete F(e0) Result▾
  204. 204Theorem H: Explicit Conditions for F(e0) and Transition to E(e0)▾
  205. 205Complete E(e0) and F(e0) Solutions, Graphical Representation, and Normal-Zone Interpretation▾
  206. 206Schema (44:D) and the Zero-Excess Special Case▾
  207. 207Straw Men and Inessential Component Games▾
  208. 208Embedding a Game and the Justification of the New Theory▾
  209. 209Meaning of the Normal Zone and Tribute Transfers▾
  210. 210First Appearance of the Transfer Phenomenon at Six Players▾
  211. 211Need to Analyze the Essential Three-Person Game in the New Theory▾
  212. 212Preparatory Framework: Quasi-Components, Fundamental Triangle, and Domination▾
  213. 213Cases I–III: Empty, Singleton, and Scaled Old-Theory Solutions▾
  214. 214Case IV: Inner-Triangle Solutions and Curve Families▾
  215. 215Case V: Moderate Excess and Solution Curves▾
  216. 216Case VI and Opening of the Interpretation: Large Excess▾
  217. 217Interpretation of the Result: Curves in the Solution▾
  218. 218Continuation: Surfaces in the Solution▾
  219. 219Chapter X: Simple Games▾
  220. 22048. Winning and Losing Coalitions and the Games in Which They Occur▾
  221. 22149.1 General Concepts of Winning and Losing Coalitions▾
  222. 22249.2 The Special Role of Singleton Coalitions▾
  223. 22349.3 Characterization of Winning and Losing Systems of Actual Games▾
  224. 224Exact Definition of Simplicity▾
  225. 225Elementary Properties and Characterization of Simple Games▾
  226. 226Solutions of Simple Games and Direct Majority Games▾
  227. 227Weighted Majority Games and Homogeneity▾
  228. 228Direct Use of Imputation in Forming Solutions▾
  229. 229Direct Economic Procedure for Simple Games and Its Exact Theoretical Formulation▾
  230. 230Reformulating the Solution Criterion with U-Plus and Feasible Imputations▾
  231. 231Equality Cases and Indifferent Players in the Solution Criterion▾
  232. 232Interpretation of the Solution Criterion and the Choice of U▾
  233. 233Simple Main Solutions and Homogeneous Weighted Majority Games▾
  234. 234Preliminary Remarks on Enumerating All Simple Games▾
  235. 235The Saturation Method for Specifying Winning Coalitions▾
  236. 236Direct Characterization of Minimal Winning Systems and the Limits of an Asymmetric Saturation Relation▾
  237. 237Symmetric Saturation Construction of Minimal Winning Coalitions and Simplicity under Decomposition▾
  238. 238Inessentiality, Simplicity, Composition, and Excess▾
  239. 239A Criterion for Decomposability Using Minimal Winning Coalitions▾
  240. 240The Decomposition Partition of a Simple Game▾
  241. 241Indecomposable Simple Games and the Core▾
  242. 242Simple Games for Small n: Program and the n = 3 Case▾
  243. 243Classifying Minimal Winning Coalitions by Two-Element Sets▾
  244. 244Decomposability of the C* Case▾
  245. 245Case C n minus 1 and the Unique Homogeneous Weighted Majority Game▾
  246. 246Completion of Simple Game Classification for Cases C n minus 2, General l-Player Forms, and n = 4,5▾
  247. 247Patterns from n < 6 and First n = 6 Multiplicity Example▾
  248. 248n = 6 Counterexamples: Weighted Nonhomogeneous Game and Nonweighted Game with Principal Solution▾
  249. 249n = 6 Game without Weights or Principal Solution and n = 7 Fano-Plane Game▾
  250. 250Reasons for Seeking Solutions Beyond the Main Solution in Simple Games▾
  251. 251Known Classes of Games for Which All Solutions Are Available▾
  252. 252Motivation for the Simple Game [1,...,1,n-2]▾
  253. 253Preliminary Remarks on Solving the Game [1,...,1,n-2]A▾
  254. 254Domination, the Main Player, and the Two Fundamental Cases▾
  255. 255Resolution of Case I: The Separated Main Player▾
  256. 256Case (II): determination of lower and upper solution parts and the alpha set▾
  257. 257Cases Pi prime and Pi double-prime: beginning the completion of case Pi prime▾
  258. 258Completion of the Nondiscriminatory Case II Prime and Setup of the Discriminatory Case▾
  259. 259Stability of V Prime in Alpha and the Simplified Domination Criterion▾
  260. 260Geometry of Alpha and the Maximal Element Alpha Star in Case II Double Prime▾
  261. 261Parametrization of V Prime by the Main Player Component▾
  262. 262Properties of the Alpha Functions and Complete Description of V in Case II Double Prime▾
  263. 263Proof That All Sets Described by L Prime Are Solutions in Case II Double Prime▾
  264. 264Reformulation of the Complete Result for the Simple Game▾
  265. 265Interpretation of the Solutions for the Simple Game▾
  266. 266Chapter XI: General Non-Zero-Sum Games▾
  267. 267Extending the Theory Beyond Zero-Sum Games▾
  268. 268The Fictitious Player and the Zero-Sum Extension▾
  269. 269Limits on Treating the Zero-Sum Extension as the Original Game▾
  270. 270Example Showing the Limits of the Fictitious-Player Construction▾
  271. 271Two Possible Procedures for Handling the Difficulty▾
  272. 272Discriminating Solutions for the Fictitious Player▾
  273. 273Alternative Possibilities for the Fictitious Player’s Payoff▾
  274. 274New Imputation Notation and Consistency with the Zero-Sum Case▾
  275. 275Analysis of Domination in the Non-Zero-Sum Extension▾
  276. 276Rigorous Examination of the New Domination Concept▾
  277. 277The New Definition of a Solution▾
  278. 278Characteristic Function: Extended and Restricted Forms▾
  279. 279Basic Properties of Restricted and Extended Characteristic Functions▾
  280. 280Sufficiency Problem for Characteristic Functions with a Fictitious Player▾
  281. 281Construction Proving Sufficiency for Restricted Characteristic Functions▾
  282. 282Extended Characteristic Functions and Eliminability of Individual Players▾
  283. 283Limits of Eliminability, Extreme Games, and Strategic Equivalence▾
  284. 284Zero-Sum, Constant-Sum, and the Fictitious Player as Strawman▾
  285. 285Interpretation of the Characteristic Function: Analysis and Definition▾
  286. 286Winning versus Inflicting Losses on the Opponent▾
  287. 287Discussion: Threats, Compensation, and Applications▾
  288. 288General Considerations: Program for Applications▾
  289. 289Reduced Forms and Inequalities for Characteristic Functions▾
  290. 290Miscellaneous Extensions to General Non-Zero-Sum Games▾
  291. 291Solutions for One- and Two-Person General Games▾
  292. 292Reduced Form of Essential Three-Person General Games▾
  293. 293Imputation Geometry and Domination in Three-Person General Games▾
  294. 294Case Distinctions for Three-Person General Game Solutions▾
  295. 295Comparison with Zero-Sum Games▾
  296. 296Economic Interpretation: The One-Person Case▾
  297. 297The Two-Person Market and Bilateral Monopoly▾
  298. 298Characteristic Function of the Two-Person Market▾
  299. 299Justification of the Characteristic Function Method▾
  300. 300Divisible Goods and Böhm-Bawerk’s Marginal Pairs▾
  301. 301Price determination and discussion in the two-person market▾
  302. 302Special n=3 case: the three-person market model▾
  303. 303Introductory comparison of three-person game solutions and ordinary market reasoning▾
  304. 304Solutions for the first subcase of the three-person market▾
  305. 305General form of the three-person market solutions▾
  306. 306Algebraic Form of the Solution for the Special Three-Person Market▾
  307. 307Discussion of the Special Three-Person Market and Opening of the General Case▾
  308. 308Divisible Goods Market: Characteristic Function for Seller and Two Buyers▾
  309. 309Interpreting Equality and Strictness in the Divisible-Goods Inequalities▾
  310. 310Case with One Absolutely Stronger Buyer▾
  311. 311Strict Nondegenerate Case and Diminishing Utility Inequality▾
  312. 312Section 63.3 Introductory Discussion of the General Non-Zero-Sum Case▾
  313. 313Section 63.4 Solutions for the General Three-Person Case▾
  314. 314Section 63.5 Algebraic Form of the Solution▾
  315. 315Section 63.6.1 Economic Interpretation: Seller, Two Buyers, and Transfer Quantities▾
  316. 316Footnote Continuation on Excluding a Duplicate Boundary Point▾
  317. 317Section 63.6.2 Economic Interpretation: Prices, Competition, and Monopoly-Duopoly Structure▾
  318. 318Section 64.1.1 General Market Setup▾
  319. 319Section 64.1.2 Characteristic Function of the General Market▾
  320. 320Special Properties of the General Market: Monopoly and Monopsony▾
  321. 321Chapter XII: Extending Domination and Solution Concepts—Problem Statement and General Remarks▾
  322. 322Orders, Transitivity, and Acyclic Relations▾
  323. 323Solutions for Symmetric Relations and Complete Orders▾
  324. 324Solutions for Partial Orders▾
  325. 325Acyclicity, Strict Acyclicity, and Maximal Elements▾
  326. 326Constructing the Unique Solution for an Acyclic Relation▾
  327. 327Uniqueness of Solutions, Acyclicity, and Strict Acyclicity▾
  328. 328Application to Games: Discreteness and Continuity▾
  329. 329Generalizing Utility and the Two Phases of Game-Theoretic Treatment▾
  330. 330Analysis of the First Phase: Numerical Utility and Values▾
  331. 331Investigation of the Second Phase▾
  332. 332The Desire to Unify the Two Phases▾
  333. 333Discussion of an Example: Indivisible Utility and Bargaining▾
  334. 334Generalization: Different discrete utility scales▾
  335. 335Conclusions for bargaining from discrete utility scales▾
  336. 336Appendix problem statement for the axiomatic treatment of utility▾
  337. 337Axiomatic derivation I: interval mappings and uniqueness lemmas▾
  338. 338Axiomatic derivation II: fitting interval mappings into a global utility function▾
  339. 339Axiomatic derivation III: existence and uniqueness up to positive affine transformation▾
  340. 340Concluding Remarks on the Axiomatic Treatment of Utility▾
  341. 341List of Figures▾
  342. 342Name Index▾
  343. 343Subject Index, A through V▾
  344. 344Subject Index (V–Z, continued)▾

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