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Spieltheorie und wirtschaftliches Verhalten
1953
by
Morgenstern and Neumann
Game Theory
John von Neumann
Oskar Morgenstern
Mathematical Economics
Paul Samuelson
Utility
Market Structure
Lausanne School
Methodology
Marginal Utility
Monopoly
Competition
Oligopoly
Price Theory
Profit and Loss
Business Cycle Theory
International Trade
Austrian School
Eugen von Bohm-Bawerk
Indifference Curves
Equilibrium
Vilfredo Pareto
Carl Menger
Zurechnung
Anthropology
Capitalism
Empiricism
Uncertainty
Expectations
Stabilization
Planned Economy
Price Formation
Table of Contents · 543 segments
1
Title Page and Publication Data
essay
2
Preface to the First Edition
essay
3
Preface to the Second Edition
essay
4
Preface to the Third Edition: Revisions and Bibliographic Scope
essay
5
Third-Edition Selective Bibliography and Literature Survey
bibliography
6
Third-Edition Methodological Note on Utility Axioms
theoretical
7
Preface to the German Edition: Translation and Acknowledgments
essay
8
German-Edition Bibliography of Post-1953 Game-Theory Literature
bibliography
9
German-Edition Preface: Applications, Cooperative n-Person Games, and Memorial to von Neumann
essay
10
On the Technique of Presentation
theoretical
11
Opening of the Table of Contents
chapter
12
Table of Contents: Economic Problem, Strategic Games, and Two-Person Zero-Sum Game Theory
chapter
13
Contents: Elementary Games; Poker and Bluffing
chapter
14
Contents: Three-Person Zero-Sum Games
chapter
15
Contents: Formulation of the General Theory of n-Person Zero-Sum Games
chapter
16
Contents: Four-Person Zero-Sum Games
chapter
17
Contents: Remarks for n ≥ 5 Participants
chapter
18
Contents: Composition and Decomposition of Games
chapter
19
Table of Contents: Modification, Decomposition, and Simple Games
chapter
20
Table of Contents: General Non-Zero-Sum Games
chapter
21
Table of Contents: Extensions of Domination, Utility, Appendix, Indexes, and Opening of Chapter I
chapter
22
Mathematical Method in Economics: Introductory Remarks
theoretical
23
Difficulties in Applying Mathematical Methods to Economics
theoretical
24
Necessary Limitation of Aims
theoretical
25
Concluding Remarks on the Scope and Development of Economic Theory
theoretical
26
The Problem of Rational Behavior
theoretical
27
Robinson Crusoe Economy and Social Exchange Economy
theoretical
28
Variables, Participants, and Player Number
theoretical
29
Many Participants and Free Competition
theoretical
30
The Lausanne School and Equilibrium Theory
theoretical
31
The Concept of Utility: Preferences, Measurement, and Probability
theoretical
32
Numerical Utility, Probability Mixtures, and Principles of Measurement
theoretical
33
Natural Operations, Measurement Scales, and Utility Transformations
theoretical
34
Numerical Utility, Natural Operations, and Linear Transformations
theoretical
35
Criteria for Choosing Utility Axioms
theoretical
36
Formal Axioms for Abstract Utility
theoretical
37
Analysis and Interpretation of the Utility Axioms
theoretical
38
General Remarks on Expected Utility, Comparability, and the Marginal Utility Transition
theoretical
39
The Role of Marginal Utility
theoretical
40
The Simplest Concept of a Solution for One Participant
theoretical
41
Extension to All Participants and the Problem of Imputation
theoretical
42
Limits of a Single-Imputation Solution and the Two-Person Zero-Sum Exception
theoretical
43
The Solution as a Set of Imputations
theoretical
44
The Intransitive Concept of Superiority or Domination
theoretical
45
The Exact Definition of a Solution
theoretical
46
Solution Postulates and Implicit Definition
theoretical
47
Behavioral Standards as Sets of Imputations
theoretical
48
Stability of Behavioral Standards and Intransitive Domination
theoretical
49
Existence of Solutions as an Open Problem
theoretical
50
Non-Uniqueness and Multiple Stable Behavioral Standards
theoretical
51
Social Doctrine, A Priori Principles, and Internal Stability
theoretical
52
Games as Models for Social Organizations
theoretical
53
Formal Remarks on Solution Sets, Behavioral Standards, and Strategies
theoretical
54
Static Theory, Dynamic Reasoning, and Equilibrium
theoretical
55
Social Theory and Its Possible Difference from Mathematical Physics
theoretical
56
Conclusion on Static and Dynamic Theory in Social Phenomena
theoretical
57
Chapter II: General Formal Description of Strategic Games
chapter
58
Introduction: Shift from Economics to Games
theoretical
59
General Principles: Zero-Sum Games, Strategy, and Formal Description
theoretical
60
Simplified Concept of a Game: Terms, Elements, and Information
theoretical
61
Factual Precedence, Transitivity, and Signaling
theoretical
62
Complete Concept of a Game: Variability of Move Characteristics
theoretical
63
General Description of Information and Variable Game Length
theoretical
64
Chapter 8 Introduction: Set-Theoretic Description of Games
chapter
65
Sets, Set Operations, Empty Sets, Singletons, and Diagrams
theoretical
66
Partitions, Subpartitions, Superposition, and Tree Diagrams
theoretical
67
Logical and Informational Interpretation of Sets and Partitions
theoretical
68
Set-Theoretic Game Description: Partitions, Information, and Successive Histories
theoretical
69
Double-Blind Chess and Apparently Illegal Moves
theoretical
70
Outcome Functions for Players
theoretical
71
Axiomatic Formulation: Axioms and Interpretation
theoretical
72
Logical Status and Methodological Remarks on the Axioms
theoretical
73
Graphical Representation of Extensive Game Structures
theoretical
74
Strategies and the Normal-Form Reduction of a Game
theoretical
75
No Higher-Order Strategies, the Zero-Sum Restriction, and Transition to Two-Person Theory
theoretical
76
Preliminary overview: general considerations on game forms and classification
theoretical
77
The one-person game and the limits of pure maximization
theoretical
78
Chance, probability, and the next target of the theory
theoretical
79
Functional Calculus: Basic Definitions of Functions
theoretical
80
The Max and Min Operations
theoretical
81
Commutativity of Max and Min Operations
theoretical
82
The Mixed Max-Min Case, Saddle Points, and Proofs
theoretical
83
Formulating the Two-Person Zero-Sum Problem
theoretical
84
Matrix Representation of the Payoff Function
theoretical
85
Minorant and Majorant Games
theoretical
86
Purpose of the Minorant and Majorant Games as Bounds
theoretical
87
Analysis of the Minorant Game Γ1 and the Maximin Value v1
theoretical
88
Methodological Role of the Simple Auxiliary-Game Analysis
theoretical
89
Beginning the Analysis of the Majorant Game Γ3
theoretical
90
Value of the Majorant Game Γ2 and Guarantees from Good Play
theoretical
91
Duality of Γ1 and Γ2 and the Heuristic Extension of Game Value
theoretical
92
Consequences for Γ: v1, v2, and Strategic Insight
theoretical
93
Necessary Bounds for the Value of Γ
theoretical
94
Determinate Games: Equality of Maximin and Minimax Values
theoretical
95
Investigation of Determinateness: Value and Security Guarantees
theoretical
96
Good Strategies and Saddle Points in Determinate Games
theoretical
97
Interchanging Players and Symmetry
theoretical
98
Player Interchange and Examples of Non-Determined Games
theoretical
99
The Core Difficulty of Non-Determined Games
theoretical
100
Program for Studying Determinateness in Extensive Form
theoretical
101
Transition to Complete-Information Games and Strict Determinateness
theoretical
102
Complete-Information Games: Objective and Induction
chapter
103
Exact Set-Theoretic Condition for the First Move
theoretical
104
First-Move Alternatives and the Restricted Game
theoretical
105
Reformulating the Restriction on Information Partitions
theoretical
106
Exact Condition and Complete Induction for Two-Person Zero-Sum Games
theoretical
107
Backward Construction from the Terminal Subgame
theoretical
108
Complete Information Criterion and Setup of the Induction Step
theoretical
109
Induction Step for a Chance First Move
theoretical
110
Interpreting 13.5.3 Through Minorant Games and Complete Information
theoretical
111
Exact Induction Step for a Personal Move by Player 1
theoretical
112
Personal Move by Player 2 and Transition to the Summary of Induction Formulas
theoretical
113
Recursive Move Operators for Game Values
theoretical
114
Inductive Proof of Strict Determinacy under Complete Information
theoretical
115
Explicit Complete-Information Value Formula and Chess Application
theoretical
116
Chess, Complete Theory, and the Practical Limits of Game Determination
theoretical
117
A Verbal Examination of Determinate Two-Person Zero-Sum Games
theoretical
118
Backward Induction, Rationality Objection, and Transition to Linearity and Convexity
theoretical
119
Critique of Rationality Assumptions in Two-Person Zero-Sum Games
theoretical
120
Geometric Foundations of Linearity and Convexity
theoretical
121
Vector Operations, Convex Hulls, Simplexes, and Norms
theoretical
122
The Supporting Hyperplane Theorem for Convex Sets
theoretical
123
Matrix Alternative Theorems and the Skew-Symmetric Corollary
theoretical
124
Mixed Strategies: Motivation from Elementary Indeterminate Games
theoretical
125
Statistical Strategies in Matching Pennies and Rock-Paper-Scissors
theoretical
126
Generalizing Mixed Strategies in Normal Form
theoretical
127
Mixed Strategies and Freedom to Choose Pure Strategies
theoretical
128
Pure Strategies as Basis Vectors and Mixed Strategies as Probability Vectors
theoretical
129
Justification of Mixed Strategies for a Single Play
theoretical
130
The Minorant and Majorant Games for Mixed Strategies
theoretical
131
Completion of the Expected Payoff Formulation for Mixed Strategies
theoretical
132
Majorant and Minorant Games for Mixed Strategies
theoretical
133
Special and General Determinacy
theoretical
134
Lemma Reducing Mixed-Strategy Optimization to Pure Responses
theoretical
135
Completion of the Inequality Between Pure and Mixed Values
theoretical
136
Proof of the Minimax Theorem for Mixed Strategies
theoretical
137
Comparison of Pure and Mixed Strategies
theoretical
138
Mixed Strategies, Bilinear Payoffs, and General Determinacy
theoretical
139
Interpreting the Game Value and Good Play
theoretical
140
Good Mixed Strategies as Saddle Points and Mutual Optima
theoretical
141
Special Unique Determination and Pure Good Strategies
theoretical
142
Errors, Deviation Measures, and Worst-Case Risk
theoretical
143
Limits of Alpha and Beta as Loss Measures
theoretical
144
Permanent Optimality, Offensive Play, and Player Interchange
theoretical
145
Symmetric Games and Skew-Symmetric Payoff Forms
theoretical
146
Symmetric Games, Fairness, and Self-Optimal Mixed Strategies
theoretical
147
Avoiding Loss in Symmetric Games
theoretical
148
Chapter IV: Two-Person Zero-Sum Games—Examples
chapter
149
The Simplest 2x2 Zero-Sum Games and Label Symmetries
theoretical
150
Quantitative Classification by Maximum and Minimum Matrix Entries
theoretical
151
Case A: Opposite Maximum and Minimum Fields Yield a Saddle Point
theoretical
152
Case B and Subcase B1: Same-Column Extremes and a Saddle Point
theoretical
153
Subcase B2: Constructing Good Mixed Strategies by Equalization
theoretical
154
Completion of the 2x2 Mixed-Strategy Solution
theoretical
155
Qualitative Criterion: Separation of Matrix Diagonals
theoretical
156
Unique Positive Good Strategies in Non-Strictly Determined 2x2 Games
theoretical
157
Qualitative Criterion: Row and Column Majorization
theoretical
158
Interpretation and Limits of the Majorization Criterion
theoretical
159
Applications: Ordinary Matching Pennies
theoretical
160
Modified Matching Pennies with Double Reward for Matching Heads
theoretical
161
Modified Matching Pennies with Double Gain and Triple Loss
theoretical
162
Sherlock Holmes and Moriarty as a Two-Person Zero-Sum Game
theoretical
163
Beginning of the Study of More Complicated Games
theoretical
164
Limits of the 2x2 simplification and the 3x3 cancellation schema
theoretical
165
Failure of earlier 2x2 results and an optimal-strategy counterexample
theoretical
166
Generalized majorization by mixed strategies
theoretical
167
Beginning of a determined 3x3 game without mixed-strategy majorization
theoretical
168
Saddle Point and Unique Pure Good Strategy in the Figure 34 Game
theoretical
169
Chance and Incomplete Information: Replacing Random Moves with Personal Moves
theoretical
170
Interpreting the Elimination of Chance Moves
theoretical
171
Opening of the Generalized Delta Construction
theoretical
172
Footnote on Formalizing the Equivalence Argument
footnotes
173
Balance Conditions and the Card-Cutting Example
theoretical
174
Poker and Bluffing: From Simple Moves to Strategies
chapter
175
Residual Footnotes on Randomization and Card Cutting
footnotes
176
Simplifying Poker into a Two-Person Stud Poker Model
theoretical
177
Poker Betting Actions: Pass, Showdown, and Raise
theoretical
178
Bluffing as a False Signal of Strength
theoretical
179
Continuation of Bluffing: Motives and Misleading Signals
theoretical
180
Indirect Motives, Risk, and the Static Interpretation of Bluffing
theoretical
181
Description of Poker Continued: Bid Limits, Symmetry, and Passing
theoretical
182
Precise Formulation of the Simplified Poker Rules
theoretical
183
Description of Strategies Heading
theoretical
184
Strategies in the simplified poker game: pure strategies, mixed strategies, and conditional action probabilities
theoretical
185
Parameter reduction through reduced probability vectors
theoretical
186
Problem formulation and minimum conditions for good strategies
theoretical
187
Transition from discrete hands to a continuous hand-strength scale
theoretical
188
Integral reformulation of the reduced strategy conditions
theoretical
189
Closing Good-Strategy Criterion and Exclusion of Revealing Low Bids
theoretical
190
Intermediate Values and the Average Probability Condition
theoretical
191
Existence and Plateau of Intermediate Values
theoretical
192
Upper-Interval Strategy from Gamma Comparisons
theoretical
193
Lower Endpoint of the Mixed Region
theoretical
194
Determination of the Threshold and Final Strategy
theoretical
195
Graphical Representation and Uniqueness of the Good Strategy
theoretical
196
Payoff Coefficient Curves
theoretical
197
Verification of Optimality and Game Value
theoretical
198
Interpretation of Bluffing in the Poker Model
theoretical
199
Deviations, Permanent Optimality, and Loss Conditions
theoretical
200
Defensive significance of correct bluffing
theoretical
201
Offensive exploitation of incorrect bluffing
theoretical
202
Toward more general forms of poker
theoretical
203
Discrete hands and the discrete poker solution
theoretical
204
Discrete and Multiple-Bid Bluffing; Alternating Bidding Rules
theoretical
205
Pure Strategies in the Alternating-Bidding Poker Variant
theoretical
206
Pure-strategy payoff matrix for the asymmetric poker variant
theoretical
207
Mixed strategies and expected payoff in the poker model
theoretical
208
Conditional probability reduction of mixed strategies
theoretical
209
Good-strategy criteria for the asymmetric poker model
theoretical
210
Continuous-card limit and integral criteria for good strategies
theoretical
211
Support Conditions for Good Strategies
theoretical
212
Mathematical Description of All Solutions
theoretical
213
Coefficient Differences and Graphical Optimality Criteria
theoretical
214
Verification of Good Strategies and Value of the Game
theoretical
215
Interpretation of the Solution: Zones and Bluffing Motives
theoretical
216
Consequences: Aggressive and Defensive Bluffing
theoretical
217
End of Borel-Style Max-Min Comparison
theoretical
218
Chapter V: Three-Person Zero-Sum Games
chapter
219
Preliminary Overview: General Viewpoints and Coalitions
theoretical
220
The Simple Majority Game for Three Persons: Description of the Game
theoretical
221
Necessity of Agreements in the Simple Majority Game
theoretical
222
Coalitions and Symmetry in the Three-Person Majority Game
theoretical
223
Further Examples: Compensation and Coalitions of Different Strength
theoretical
224
Inequality, Symmetric Coalition Incentive, and Equivalent Payoff Formulas
theoretical
225
Exhaustive Analysis of Inessential and Essential Three-Person Zero-Sum Games
theoretical
226
Complete Formulas for Coalition Values and Basis Values
theoretical
227
Discussion of an Objection: Complete Information and Its Significance
theoretical
228
Detailed Critique: Necessity of Compensations Among Three or More Players
theoretical
229
Chapter VI: Formulation of the General Theory of n-Person Zero-Sum Games
chapter
230
The Characteristic Function in n-Person Zero-Sum Games
theoretical
231
Constructing a Game with a Given Characteristic Function
theoretical
232
Proof of the Zero-Sum Property and Characteristic Function Equality
theoretical
233
Summary: Characterization of Characteristic Functions
theoretical
234
Strategic Equivalence and the Reduced Form
theoretical
235
Inequalities and the Gamma Parameter
theoretical
236
Inessential and Essential Games
theoretical
237
Criteria for Inessential Games, Non-Additive Utility, and Essential-Case Inequalities
theoretical
238
Vector Operations on Characteristic Functions
theoretical
239
Groups, Symmetry, and Fairness: Permutations and Invariant Groups
theoretical
240
Set-Transitive Symmetry and the Definition of Fairness
theoretical
241
Reduced Games, Extra Payments, and Fundamental Strategic Asymmetry
theoretical
242
Qualitative Reconsideration of the Three-Person Zero-Sum Game
theoretical
243
Heuristic Stability and Quantitative Blocking in the Three-Person Game
theoretical
244
Exact Definitions: Imputations, Effective Coalitions, Domination, and Solutions
theoretical
245
Discussion of Solutions as Stable Behavioral Standards
theoretical
246
The Concept of Saturation and Symmetrized Relations
theoretical
247
Definitions of R-Conformity, R-Compatibility, and R-Saturation
theoretical
248
Examples and General Properties of R-Saturation
theoretical
249
Why Maximal Conformity Does Not Yet Prove Existence of a Solution
theoretical
250
Limits of Symmetric Saturation and Extending Consistent Standards
theoretical
251
Three Immediate Questions About Solutions
theoretical
252
First Consequences: Convexity, Flatness, and Domination Criteria
chapter
253
Transition: further results and questions answered
theoretical
254
Terminology for certainly necessary and certainly unnecessary coalitions
theoretical
255
Elementary criteria for omitting coalitions
theoretical
256
Coalition convexity, flatness, and inessential games
theoretical
257
Flat Coalitions and Unessential Games: Closing Consequences
theoretical
258
Necessary and Unnecessary Coalitions from Flatness
theoretical
259
The System of Imputations and Asymmetric Domination
theoretical
260
Undominated Imputations and Unessential Games
theoretical
261
Inessential Games and Singleton Solutions
theoretical
262
The Isomorphism Corresponding to Strategic Equivalence
theoretical
263
Formulation of the Mathematical Problem and the Graphical Method
chapter
264
Determination of All Solutions
chapter
265
Conclusions: Diversity of Solutions, Discrimination, and Its Significance
theoretical
266
Statics and Dynamics
theoretical
267
Chapter VII: Four-Person Zero-Sum Games
chapter
268
Preliminary Overview: General Considerations
theoretical
269
Formalism and Cube Representation of Essential Four-Person Zero-Sum Games
theoretical
270
Player Permutations and Geometric Symmetries of the Cube
theoretical
271
Special points in cube Q: advantaged players, straw men, symmetry, and interior variation
theoretical
272
Heuristic Investigation of the Main Diagonal Near Corner VIII
theoretical
273
Exact Investigation and the Solution Range Near Corner VIII
theoretical
274
Exact proof of theorem (36:A), necessary coalitions, and initial no-domination checks
theoretical
275
Completion of Internal Domination Check for Criterion (30:5:a)
theoretical
276
External Stability Proof and Remaining Main-Diagonal Solution Intervals
theoretical
277
Orientation Toward Solutions Around the Center of the Cube
theoretical
278
The Symmetric Midpoint and Its Coalition Alternatives
theoretical
279
Symmetry, Alternatives, and the Scope of the Heuristic Search
theoretical
280
Discriminatory Entry Alternative and the Candidate Symmetric Solution
theoretical
281
Equal-Treatment Alternative for the Three-Person Coalition
theoretical
282
Unique Symmetric Extension of the Equal-Share Midpoint Coalition
theoretical
283
Comparison of the Two Midpoint Solutions and Opening of Asymmetric Solutions
theoretical
284
Asymmetric Midpoint Solution with Player 4 Distinguished
theoretical
285
Continuation of the Four-Person Solution: Privileged Group and Replacement Imputation
theoretical
286
Social Interpretation of the Discriminatory Four-Person Solution
theoretical
287
A Family of Solutions for a Neighborhood of the Midpoint
theoretical
288
Number of Parameters in Classes of n-Person Zero-Sum Games
theoretical
289
Formalism of the Symmetric Five-Person Zero-Sum Game
theoretical
290
The Two Extreme Cases of the Symmetric Five-Person Game
theoretical
291
Mapping the Symmetric Five-Person Game to a 1,2,3-Symmetric Four-Person Game
theoretical
292
Solution Zones for the Symmetric Five-Person Game
theoretical
293
Heuristic Meaning of Player Fusion and Opening of Chapter IX
theoretical
294
Search for Solvable n-Person Games
theoretical
295
The First Type: Composition and Decomposition
theoretical
296
Exact Definitions of Composition and Decomposability
theoretical
297
Criteria for Decomposability and Component Games
theoretical
298
Need to Modify the Decomposition Concept
theoretical
299
Partial Retention of the Zero-Sum Condition
theoretical
300
Strategic Equivalence and Constant-Sum Games
theoretical
301
Characteristic Functions, Imputations, and Essentiality in the Constant-Sum Theory
theoretical
302
Decomposability in the Constant-Sum Theory
theoretical
303
Splitting Sets and Components
theoretical
304
Closure Properties of Splitting Sets and Components
theoretical
305
Minimal Splitting Sets and Reconstruction of All Splitting Sets
theoretical
306
The Decomposition Partition and Its Extreme Cases
theoretical
307
Decomposable Games and the Question of Solutions
theoretical
308
Composition and Decomposition of Imputations
theoretical
309
Composition and Decomposition of Solutions
theoretical
310
Extended Imputations and External Sources
theoretical
311
The Excess of an Extended Imputation
theoretical
312
Bounds on Excess and the Non-isolated Character of Games
theoretical
313
The New Framework E(e0) and F(e0)
theoretical
314
Lower Bound for Excess and Definition of |Gamma|1
theoretical
315
Upper Bound for Excess: Separated and Completely Separated Extended Imputations
theoretical
316
Separated Imputations and Domination
theoretical
317
Relationship Between |Gamma|1 and |Gamma|2
theoretical
318
Opening of the E(e0) and F(e0) Solution Comparison
theoretical
319
Footnote on the n=4 Bound Example
footnotes
320
D*(e0) and the Bijection Between E(e0) and F(e0) Solutions
theoretical
321
Auxiliary Lemma on Coordinatewise Order and Domination
theoretical
322
Lemmas K, L, and M for the E/F Solution Correspondence
theoretical
323
Lemma N and Completion of the E/F Solution Bijection Proof
theoretical
324
Summary of Excess Regimes and Solution Relations
theoretical
325
Maximum, Minimum, and Range of Excess in F(e0) Solutions
theoretical
326
Determination of All Solutions of a Decomposable Game: Elementary Properties of Decompositions
theoretical
327
Lemma on Choosing a Dominating Coalition within One Component
theoretical
328
Decomposition and Solutions: Initial Results for F(e0)
theoretical
329
Lemma 46:C and Compactness Properties of Solutions
theoretical
330
Openness of dominated complements in solution sets
theoretical
331
Component recombination criterion and excess formula
theoretical
332
Nondecomposability condition and proof that phi plus psi equals e0
theoretical
333
Proof that the J-component set is a solution
theoretical
334
Beginning of inverse construction from component solutions
theoretical
335
Constraints on component solution extrema
theoretical
336
Converse construction of solutions for a decomposable game
theoretical
337
Explicit conditions for the complete result in F(e0)
theoretical
338
Summary theorem 46:H on explicit alternatives
theoretical
339
Completion of the Proof for Case (b)
theoretical
340
Complete Result for E(e0) and F(e0) Solutions
theoretical
341
Graphical Representation: Introduction and Excess Distribution
theoretical
342
Footnote on the Transformation Used in Section 46.6
footnotes
343
Graphical Representation of the φ, ψ Plane and the e0 Line
theoretical
344
Interpretation: The Normal Zone and Propagation of Properties
theoretical
345
Propagation of Normal Behavior and Need for the Extended Theory
theoretical
346
Fifth and Sixth Remarks: Old-Theory Failure, Zero Excess, and Straw-Man Setup
theoretical
347
Blowing Up Solutions by Adding Inessential Straw Men
theoretical
348
Old Theory Loses Propagation Outside the Straw-Man Case
theoretical
349
Defining the Embedding of a Game
theoretical
350
The Question of Solutions Across All Embeddings
theoretical
351
Answer: Embedded Components Equal Normal-Zone Solutions
theoretical
352
Proof of the Embedding Result Using an Auxiliary Three-Person Game
theoretical
353
Remarks on Propagation, Straw Men, and the External Source
theoretical
354
Methodological Conclusion on the Passage from the Old to the New Theory
theoretical
355
Meaning of the Normal Zone: Transfers Between Components of a Composite Game
theoretical
356
Interpretation of Tribute and Its Bounds in Decomposable Games
theoretical
357
Interpretation of Bounds for a Component Game
theoretical
358
First Appearance of the Transfer Phenomenon: n = 6
theoretical
359
Essential Three-Person Game: Motivation, Preliminaries, and Cases I–III
theoretical
360
Cases IV and V: Geometric Solution Structure, Curves, and Surface Regions
theoretical
361
Case VI and Interpretation of Curve Components in Solutions
theoretical
362
Interpretation of Two-Dimensional Solution Regions and Transition to Simple Games
theoretical
363
Winning and Losing Coalitions and Their Axiomatic Characterization
theoretical
364
General Concepts of Winning and Losing Coalitions
theoretical
365
The Special Role of Singleton Coalitions
theoretical
366
Characterization of W and L Systems Arising from Actual Games
theoretical
367
Exact Definition of Simplicity
theoretical
368
Elementary Properties, W/L Systems, and Minimal Winning Coalitions in Simple Games
theoretical
369
Solutions of Simple Games
theoretical
370
Majority Games: Direct and Weighted Majority Examples
theoretical
371
Homogeneity and the Economic Imputation Concept
theoretical
372
Direct Procedure and Formal Solution Conditions
theoretical
373
Reformulation of the Solution Criterion
theoretical
374
Interpretation and Difficulty of Choosing U
theoretical
375
Relation to Homogeneous Weighted Majority Games
theoretical
376
Preliminary Remarks on Enumerating Simple Games
theoretical
377
The Saturation Method for Describing Winning Coalitions W
theoretical
378
Transition from W to Minimal Winning Coalitions and an Asymmetric Saturation Criterion
theoretical
379
Construction of Simple Games Using Minimal Winning Coalitions W*
theoretical
380
Simplicity and Decomposition of Games
theoretical
381
Composition, Excess, Straw Men, and the Setup for Decomposability
theoretical
382
Distinguished Players and the Split Set I0
theoretical
383
The Decomposition Partition of a Simple Game
theoretical
384
Indecomposability and the Core of a Simple Game
theoretical
385
Classifying Simple Games by Two-Person Minimal Winning Coalitions
theoretical
386
Decomposable Cases and Complete Classifications for Four and Five Players
theoretical
387
Patterns Observed Before Six Players and the First Multiplicity Phenomenon
theoretical
388
Six-Player Counterexamples: Weighted but Nonhomogeneous, and Nonweighted with a Principal Solution
theoretical
389
A Six-Player Game That Is Neither Weighted nor Has a Simple Principal Solution
theoretical
390
A Seven-Player Projective-Geometric Example with Strict Inequality in the Principal-Solution Criterion
theoretical
391
Determining All Solutions: Why Main Solutions Are Insufficient
theoretical
392
Residual Footnotes on Player Properties and Symmetry
footnotes
393
Value of Isolated Results Across Arbitrary n
theoretical
394
Known Cases Where All Solutions Are Determined
theoretical
395
The Simple Game [1,...,1,n-2]_h and Player n’s Privilege
theoretical
396
Introductory Remarks on the Simple Game [1,...,1,n-2]_h
theoretical
397
Domination, the Principal Player, and Cases I and II
theoretical
398
Case I: Fixed Principal-Player Payoff and Stability Conditions
theoretical
399
Case I solution characterization and transition to Case II analysis
theoretical
400
Case II: Completion of the Main Player's Total Defeat Result
theoretical
401
Case II: Preliminary Bounds for the Main Player's Best Outcome
theoretical
402
Case II: Exact Form of the Main Player's Maximum-Imputation Set
theoretical
403
Case II: Verbal Interpretation of the Main Player's Best Outcome
theoretical
404
Case II: Beginning the Analysis of A and S Star
theoretical
405
Completion of Case II General Properties and Interpretation
theoretical
406
Case II Prime: The Unique Finite Non-Discriminatory Solution
theoretical
407
Opening of Case II Double Prime: Defining \mathfrak{G} and V Prime
theoretical
408
Completion of Proof that V' Satisfies the Solution Condition
theoretical
409
Characterization of Domination within Q
theoretical
410
Case II'': Bounds and Dimension of the Region A
theoretical
411
Determining V': Omega*, the Maximal Point, and Uniqueness by the n-th Coordinate
theoretical
412
Proof of Interval Exhaustion and Parametrization of V Prime
theoretical
413
Properties of Parameter Functions and Complete Summary of Case II Double Prime
theoretical
414
Remark on Degenerate Cases in the (55:L') Construction
theoretical
415
Completeness Proof for Case (II'') and Extension to Case (II')
theoretical
416
Exclusion of the Empty S-Star Case and Final Classification
theoretical
417
Final Observations on Case (II') Parameters
theoretical
418
Case II'' Boundary Conditions and Rejection of the Empty-Set Candidate
theoretical
419
Opening Caveats on Interpreting the Result
theoretical
420
Interpretive Remarks on Symmetric, Exclusionary, and Intermediate Solutions
theoretical
421
Conclusion of the Verbal-Proof Discussion and Opening of the Fourth Remark
theoretical
422
Remaining Players, Separation, and Classification by S Star
theoretical
423
Chapter XI: General Non-Zero-Sum Games
chapter
424
Problem of Extending Beyond Zero-Sum Games
theoretical
425
The Fictitious Player and the Zero-Sum Extension
theoretical
426
Questions about the Character of Gamma
theoretical
427
Scope of the Inquiry into Interpreting Γ as a Zero-Sum Game
theoretical
428
Excluding the Fictitious Player from Real Coalitional Transactions
theoretical
429
Why a Fictitious Player Can Still Influence Coalitions
theoretical
430
Limits of Using Γ: Two-Person Example and the Simple Majority Game
theoretical
431
Fictitious Player Compensation and Non-Equivalence of the Zero-Sum Transformation
theoretical
432
First Proposed Escape: Abandoning the Characteristic Function
theoretical
433
Heuristic Versus Exact Theory and the Choice of Solution
theoretical
434
Why the Fictitious Player Must Be Discriminated Against
theoretical
435
Range of c in Discriminatory Solutions
theoretical
436
Worst-Case Exclusion of the Fictional Player
theoretical
437
Alternative Possibilities and the New Reduced Imputation Framework
theoretical
438
Reduced imputation constraints after eliminating the fictitious player
theoretical
439
Interpreting the constraints and checking consistency with the old zero-sum theory
theoretical
440
Zero-sum original games: the fictitious player as a straw man
theoretical
441
First consistency result: solution systems coincide after adding a straw man
theoretical
442
Irrelevance of Choosing Ω′ or Ω″ in the Zero-Sum Special Case
theoretical
443
Old and New Imputation Conditions for n-Person Zero-Sum Games
theoretical
444
Beginning the Analysis of Domination for the Zero-Sum Extension
theoretical
445
Two Cases for Expressing Domination with Only the Real Players
theoretical
446
Interpretation of the Two Domination Conditions
theoretical
447
Continuation of Section 56.10.3: Blocking by Omitted Players and the Social-Welfare Condition
theoretical
448
Section 56.10.4: Rejection of the Irrational Alternative in Domination
theoretical
449
Section 56.11: Stricter Investigation of the New Domination Concept
theoretical
450
Proof that old-sense solutions satisfying the extra-player value condition are new-sense solutions
theoretical
451
Equivalence between new-sense solutions and the system Omega double-prime
theoretical
452
Interpretation: selecting Omega double-prime as the solution system
theoretical
453
The new definition of a solution for general n-person games
theoretical
454
Characteristic functions: extended and restricted forms, and basic properties
theoretical
455
Necessary Conditions for Restricted Characteristic Functions
theoretical
456
Sufficiency of Restricted Characteristic Functions and Construction of a General Game
theoretical
457
Extended Characteristic Functions and Eliminability of Individual Players
theoretical
458
Total Eliminability and Inessential Games
theoretical
459
Extreme Games and Eliminability in Three- and Four-Person Zero-Sum Games
theoretical
460
Strategic Equivalence: Zero-Sum Games and Constant-Sum Games
chapter
461
Strategic equivalence and imputations in general n-person games
theoretical
462
Zero-sum and constant-sum characteristic functions; the fictitious player as a dummy
theoretical
463
Interpretation of the Characteristic Function: Analysis and Definition
theoretical
464
The Wish to Gain versus the Wish to Inflict Loss on the Opponent
theoretical
465
Discussion: Strategic Threats, Compensation, and Collective Gain
theoretical
466
Continuation on Compensation and the Resolution of Difficulties
theoretical
467
Further Objections: Sketchiness, Threats, and Applications
theoretical
468
General Considerations: Program for Applications
theoretical
469
Reduced Forms and Strategic Equivalence
theoretical
470
Null-Reduced Characteristic Functions and the Opening Gamma Notation
theoretical
471
Reduced General Games: Gamma and Coalition Inequalities
theoretical
472
Essential and Inessential General Games
theoretical
473
Domination, Necessity, Convexity, and Flatness in General Games
theoretical
474
Magnitude Measures, Composition, and Decomposition for General Games
theoretical
475
Solutions of General Games with n ≤ 3: The One-Person Case
chapter
476
Completion of the one-person inessential game case
theoretical
477
The two-person case: essentiality and invariants
theoretical
478
Solutions of general two-person games
theoretical
479
The three-person case: reduced form and parameter region
theoretical
480
Solutions and domination geometry for essential three-person general games
theoretical
481
Asymmetric Inner Triangles and Comparison with Zero-Sum Games
theoretical
482
Economic Interpretation of the One-Person Case
theoretical
483
Economic Interpretation of the Two-Person Market
theoretical
484
Continuation of the Buyer-Seller Market Model and Price Interval
theoretical
485
First Remark on the Two-Person Market and Robustness to Bargaining Rules
theoretical
486
Second Remark: Simplifying the Market by Using Compensation Mechanisms
theoretical
487
Characteristic Function and Imputations for the Two-Person Market
theoretical
488
Justification of the Position of Section 58
theoretical
489
Justification of the Characteristic-Function Standpoint
theoretical
490
Divisible Goods and Böhm-Bawerk’s Marginal Pairs
theoretical
491
Price Determination in the Two-Person Market
theoretical
492
Two-Person Market: Price Bounds, Imputations, and Böhm-Bawerk Comparison
theoretical
493
Three-Person Market Special Case: One Seller and Two Buyers
chapter
494
Introductory Discussion of Three-Person Market Solutions
theoretical
495
Coalition Formation and Divergence from the Ordinary Standpoint
theoretical
496
Solutions: First Subcase—Reduction and Normalization
theoretical
497
Graphical and General Solutions for the Buyer-Seller Three-Person Game
theoretical
498
Algebraic Form of the Solution for v ≤ w
theoretical
499
Degenerate Algebraic Case v = w
theoretical
500
Completion of the Three-Person Market Solution by Buyer Strength Cases
theoretical
501
Conventional Price Analysis for One Seller and Two Buyers
theoretical
502
Coalitional Interpretation and the General Three-Person Market with Multiple Units
theoretical
503
Continuation of Footnote and Straw-Man Buyer Case
footnotes
504
Buyer Weakness and Strict Inequality Assumptions
theoretical
505
Reduction When One Buyer Dominates the Other
theoretical
506
Strict General Case and Diminishing Utility Inequality
theoretical
507
Introductory Discussion of Applying the Solution Method
theoretical
508
Imputations, Reduced Form, and Beginning of Triangle Solution Lines
theoretical
509
Graphical Identification of the General Solution for the Three-Person Case
theoretical
510
Algebraic Form of the Area-and-Curve Solution
theoretical
511
Economic Interpretation of Competition and Coalition in the Three-Person Market
theoretical
512
Separate Buyer Prices and Comparison with Böhm-Bawerk Theory
theoretical
513
Price Differentiation in Monopoly versus Duopoly
theoretical
514
The General Market
theoretical
515
Generalized Domination and Solution Concepts; Complete Orders
theoretical
516
Partial Orders and Incomparable Elements
theoretical
517
Partial Orders and Acyclic Relations
theoretical
518
Solutions for Symmetric Relations and Complete Orders
theoretical
519
Solutions for Partial Orders and Relative Maxima
theoretical
520
Partial Orders and Solutions as the Set of Relative Maxima
theoretical
521
Finite Sets and Condition 65:G
theoretical
522
Acyclicity, Maxima of Subsets, and the Well-Ordering Condition
theoretical
523
Strict Acyclicity and Infinite Descending Chains
theoretical
524
Completion of the Infinite-Chain Consequence of Non-Strict Acyclicity
theoretical
525
Finite Cycles and the Summary of Acyclicity Criteria
theoretical
526
Strict Acyclicity, Maximal Elements, and the Setup for Solutions
theoretical
527
Inductive Construction of the Candidate Solution for a Finite Acyclic Relation
theoretical
528
Completion of the Proof of the Unique Constructed Solution
theoretical
529
Uniqueness of Solutions, Acyclicity, and Strict Acyclicity
theoretical
530
Acyclic Sets and Their Game-Theoretic Applications
theoretical
531
Discreteness, Continuity, and the Paradox of Unique Approximate Solutions
theoretical
532
Generalization of the Concept of Utility
theoretical
533
Description of a Discrete-Utility Bargaining Example
theoretical
534
Solution and Interpretation for Equal Discrete Utility Scales
theoretical
535
Generalization to Unequal Discrete Utility Scales
theoretical
536
Conclusions for Bargaining and Utility Discrimination
theoretical
537
Appendix Heading: Axiomatic Treatment of Utility
chapter
538
Problem Statement for the Axiomatic Treatment of Utility
theoretical
539
Derivation of Numerical Utility from the Axioms
theoretical
540
Closing Remarks on Utility Axioms, Complementarity, and Gambling
theoretical
541
List of Figures
bibliography
542
Name Index
bibliography
543
Subject Index
bibliography