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More About This Textbook
Overview
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 yieldedgame theoryhas since been widely used to analyze a host of realworld 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.
Editorial Reviews
History of Political Economics
Praise for Princeton's previous edition: A rich and multifaceted work. . . . [S]ixty years later, the Theory of Games may indeed be viewed as one of the landmarks of twentiethcentury social science.— Robert J. Leonard
New Scientist
Praise for Princeton's previous edition: Opinions still vary on the success of the project to put economics on a sound mathematical footing, but game theory was eventually hugely influential, especially on mathematics and the study of automata. Every selfrespecting library must have one.— Mike Holderness
SIAM News
While the jury is still out on the success or failure of game theory as an attempted palace coup within the economics community, few would deny that interest in the subject—as measured in numbers of journal page—is at or near an alltime high. For that reason alone, this handsome new edition of von Neumann and Morgenstern's still controversial classic should be welcomed by the entire research community.— James Case
The Journal of Political Economy
The main achievement of the book lies, more than in its concrete results, in its having introduced into economics the tools of modern logic and in using them with an astounding power of generalization.The American Economic Review
One cannot but admire the audacity of vision, the perseverance in details, and the depth of thought displayed in almost every page of the book. . . . The appearance of a book of [this] calibre . . . is indeed a rare event.The Bulletin of the American Mathematical Society
Posterity may regard this book as one of the major scientific achievements of the first half of the twentieth century. This will undoubtedly be the case if the authors have succeeded in establishing a new exact science—the science of economics. The foundation which they have laid is extremely promising.SIAM News  James Case
While the jury is still out on the success or failure of game theory as an attempted palace coup within the economics community, few would deny that interest in the subject—as measured in numbers of journal page—is at or near an alltime high. For that reason alone, this handsome new edition of von Neumann and Morgenstern's still controversial classic should be welcomed by the entire research community.History of Political Economics  Robert J. Leonard
Praise for Princeton's previous edition: A rich and multifaceted work. . . . [S]ixty years later, the Theory of Games may indeed be viewed as one of the landmarks of twentiethcentury social science.New Scientist  Mike Holderness
Praise for Princeton's previous edition: Opinions still vary on the success of the project to put economics on a sound mathematical footing, but game theory was eventually hugely influential, especially on mathematics and the study of automata. Every selfrespecting library must have one.From the Publisher
Praise for Princeton's previous edition: "A rich and multifaceted work. . . . [S]ixty years later, the Theory of Games may indeed be viewed as one of the landmarks of twentiethcentury social science."Robert J. Leonard, History of Political Economics
Praise for Princeton's previous edition: "Opinions still vary on the success of the project to put economics on a sound mathematical footing, but game theory was eventually hugely influential, especially on mathematics and the study of automata. Every selfrespecting library must have one."Mike Holderness, New Scientist
"While the jury is still out on the success or failure of game theory as an attempted palace coup within the economics community, few would deny that interest in the subjectas measured in numbers of journal pageis at or near an alltime high. For that reason alone, this handsome new edition of von Neumann and Morgenstern's still controversial classic should be welcomed by the entire research community."James Case, SIAM News
"The main achievement of the book lies, more than in its concrete results, in its having introduced into economics the tools of modern logic and in using them with an astounding power of generalization."The Journal of Political Economy
"One cannot but admire the audacity of vision, the perseverance in details, and the depth of thought displayed in almost every page of the book. . . . The appearance of a book of [this] calibre . . . is indeed a rare event."The American Economic Review
"Posterity may regard this book as one of the major scientific achievements of the first half of the twentieth century. This will undoubtedly be the case if the authors have succeeded in establishing a new exact sciencethe science of economics. The foundation which they have laid is extremely promising."The Bulletin of the American Mathematical Society
New Scientist
Praise for Princeton's previous edition: "Opinions still vary on the success of the project to put economics on a sound mathematical footing, but game theory was eventually hugely influential, especially on mathematics and the study of automata. Every selfrespecting library must have one.— Mike Holderness
The Bulletin of the American Mathematical Society
Posterity may regard this book as one of the major scientific achievements of the first half of the twentieth century. This will undoubtedly be the case if the authors have succeeded in establishing a new exact sciencethe science of economics. The foundation which they have laid is extremely promising.History of Political Economics
Praise for Princeton's previous edition: "A rich and multifaceted work. . . . [S]ixty years later, the Theory of Games may indeed be viewed as one of the landmarks of twentiethcentury social science.— Robert J. Leonard
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Meet the Author
John von Neumann (19031957) was one of the greatest mathematicians of the twentieth century and a pioneering figure in computer science. A native of Hungary who held professorships in Germany, he was appointed Professor of Mathematics at the Institute for Advanced Study (IAS) in 1933. Later he worked on the Manhattan Project, helped develop the IAS computer, and was a consultant to IBM. An important influence on many fields of mathematics, he is the author of "Functional Operators, Mathematical Foundations of Quantum Mechanics", and "Continuous Geometry" (all Princeton). Oskar Morgenstern (19021977) taught at the University of Vienna and directed the Austrian Institute of Business Cycle Research before settling in the United States in 1938. There he joined the faculty of Princeton University, eventually becoming a professor and from 1948 directing its econometric research program. He advised the United States government on a wide variety of subjects. Though most famous for the book he coauthored with von Neumann, Morgenstern was also widely known for his skepticism about economic measurement, as reflected in one of his many other books, "On the Accuracy of Economic Observations" (Princeton). Harold Kuhn is Professor Emeritus of Mathematical Economics at Princeton University. Ariel Rubinstein is Professor of Economics at Tel Aviv University and at New York University.
Read an Excerpt
Theory of Games and Economic Behavior
By John von Neumann Oskar Morgenstern
Princeton University Press
Copyright © 2004 Princeton University PressAll right reserved.
Introduction
HAROLD W. KUHNAlthough John von Neumann was without doubt "the father of game theory," the birth took place after a number of miscarriages. From an isolated and amazing minimax solution of a zerosum twoperson game in 1713 to sporadic considerations by E. Zermelo, E. Borel, and H. Steinhaus, nothing matches the pathbreaking paper of von Neumann, published in 1928.
This paper, elegant though it is, might have remained a footnote to the history of mathematics were it not for collaboration of von Neumann with Oskar Morgenstern in the early '40s. Their joint efforts led to the publication by the Princeton University Press (with a $4,000 subvention from a source that has been variously identified as being the Carnegie Foundation or the Institute for Advanced Study) of the 616page Theory of Games and Economic Behavior (TGEB).
I will not discuss here the relative contributions of the two authors of this work. Oskar Morgenstern has written his own account of their collaboration, which is reprinted in this volume; I would recommend to the reader the scholarly piece by Robert J. Leonard, who has noted that Morgenstern's "reminiscence sacrifices some of the historical complexity of the runup to 1944" and has given a superb and historically complete account of the two authors' activities in the relevant period. Onbalance, I agree with Leonard that "had von Neumann and Morgenstern never met, it seems unlikely that game theory would have been developed." If von Neumann played both father and mother to the theory in an extraordinary act of parthenogenesis, then Morgenstern was the midwife.
In writing this introduction, I have several goals in mind. First, I would like to give the reader a sense of the initial reaction to the publication of this radically new approach to economic theory. Then, we shall survey the subsequent development of the theory of games, attempting to explain the apparent dissonance between the tenor of the book reviews and the response by the communities of economists and mathematicians. As a participant in this response (from the summer of 1948), my account is necessarily colored by subjective and selective recollections; this is a fair warning to the reader.
The book reviews that greeted the publication of TGEB were extraordinary, both in quantity and quality; any author would kill for such reviews. Consider the following partial list of the reviews, paying special attention to the length of these reviews, the quality of the journals, and the prominence of the reviewers:
H. A. Simon, American Journal of Sociology (1945) 3 pages* A. H. Copeland, Bulletin of the American Mathematical Society (1945) 17 pages* L. Hurwicz, The American Economic Review (1945) 17 pages* J. Marschak, Journal of Political Economy (1946) 18 pages T. Barna, Economica (1946) 3 pages* C. Kaysen, Review of Economic Studies (1946) 15 pages D. Hawkins, Philosophy of Science (1946) 7 pages J.R.N. Stone, Economic Journal (1948) 16 pages E. Ruist, Economisk Tidskrift (1948) 5 pages G. Th. Guilbaud, Economie Appliquée (1949) 45 pages E. Justman, Revue d'Economie Politique (1949) 18 pages K.G. Chacko, Indian Journal of Economics (1950) 17 pages
The quotes from these reviews are a publisher's dream. Thus: Simon encouraged "every social scientist who is convinced of the necessity for mathematizing social theoryas well as those unconverted souls who are still open to persuasion on this pointto undertake the task of mastering the Theory of Games."
Copeland asserted: "Posterity may regard this book as one of the major scientific achievements of the first half of the twentieth century."
Hurwicz signaled that "the techniques applied by the authors in tackling economic problems are of sufficient generality to be valid in political science, sociology, or even military strategy" and concluded "the appearance of a book of the caliber of the Theory of Games is indeed a rare event."
After praising the "careful and rigorous spirit of the book," Jacob Marschak concludes: "Ten more such books and the progress of economics is assured."
If the quantity of reviews and the quality of the journals in which they were published are impressive, the choice of reviewers and their positions in the social sciences are equally impressive. Two of the reviewers, H. A. Simon and J.R.N. Stone, were awarded Nobel Memorial Prizes in Economics.
The first review to appear was that of Herbert Simon. By his own account, he "spent most of [his] 1944 Christmas vacation (days and some nights) reading [the TGEB]." Simon knew of von Neumann's earlier work and was concerned that the TGEB might anticipate results in a book that he was preparing for publication.
The first review that was directed at mathematicians was that of A. H. Copeland, a specialist in probability theory and professor at the University of Michigan. Copeland's only significant work in social science is the socalled "Copeland method" for resolving voting problems: simply, it scores 1 for each pairwise win and 31 for each pairwise loss, and declares the alternative with the highest score the winner. His review gave the mathematical community an extremely complete account of the contents of the TGEB. As is typical of almost all of the reviewers, although Copeland pointed to the research challenges opened by the TGEB, he never engaged in research in game theory as such. The only paper in his prolific output that is marginally related to game theory is a joint paper on a oneplayer game which must be categorized as a game of chance. Copeland's principal contribution to game theory consists in the fact that he was Howard Raiffa's thesis adviser; the book Games and Decisions, written by Raiffa with R. Duncan Luce (published by Wiley in 1957 and reprinted by Dover Publications in 1989) was the first nonmathematical exposition that made the theory of games accessible to the broad community of social scientists.
Another reviewer, David Hawkins, is permanently linked to H. A. Simon for their joint discovery of the "HawkinsSimon conditions," a result that every graduate student in economics must study. Hawkins was a young instructor at the University of California at Berkeley when his friend, J. Robert Oppenheimer, picked him as the "official historian" and "liaison to the military" at Los Alamos, where the first atomic bomb was produced. Hawkins later had a distinguished career at the University of Colorado, where he was chosen in the first class of MacArthur "genius" scholars in 1986. Hawkins did no research in game theory.
The pattern of extravagant praise and no subsequent research is repeated with more significance in the cases of Jacob Marschak and Leonid Hurwicz. Marschak was head of the Cowles Commission at the University of Chicago when he reviewed the TGEB. He had survived a tumultuous early life that took him from Russia, where he was raised, to Berlin, where he trained as an economist, to the United States, where he ran an influential econometric seminar at the New School for Social Research. Leonid Hurwicz preceded Marschak on the staff of the Cowles Commission and continued as a consultant after Tjalling C. Koopmans became director and the commission moved from the University of Chicago to Yale University. Both Marschak and Hurwicz were in a position to influence the research done at the Cowles Commission, but it is an astounding fact that the extensive research output of the commission did not encompass game theory until Martin Shubik joined the Yale faculty in 1963. Eight years after reviewing the TGEB, Hurwicz posed the question: What has happened to the theory of games? His answer, published in The American Economic Review, contains conclusions that are echoed in this introduction.
Among the reviews and reviewers, the review of G. Th. Guilbaud is surely unique. Occupying 45 pages in the journal, Economie Appliquée, it contained not only an account of the main themes of the TGEB, but also went further into consideration of the difficulties that the theory then faced. Guilbaud himself was unique in that he was the only reviewer who has contributed to the theory; his book Eléments de la Theorie des Jeux was published by Dunod in Paris in 1968. However, he failed to convince the economic community in France to join him. Guilbaud's seminar in Paris in 195051 was attended by such mathematical economists as Allais, Malinvaud, Boiteux, and myself, but none of the French engaged in research in game theory. I am pleased to report that Guilbaud, a very private person, is still with us at 91 years of age, living in St. GermaineenLaye. It was he who discovered the minimax solution of 1713, when he purchased the treatise on probability written by Montmort from one of the booksellers whose stalls line the river Seine in Paris.
Given the extravagant praise of these reviewers, one might have expected a flood of research. If nowhere else, surely the Princeton economics department should have been a hotbed of activity. When Martin Shubik arrived in Princeton to do graduate work in economics in the fall of 1949, he expected to find just that. Instead, he found Professor Morgenstern in splendid isolation from the rest of the department, teaching a seminar with four students in attendance. Morgenstern's research project consisted of himself assisted by Maurice Peston, Tom Whitin, and Ed Zabel, who concentrated on areas of operations research such as inventory theory, but did not work on game theory as such. If Shubik had come two years earlier, he would have found the situation in the mathematics department somewhat similar. Samuel Karlin (who received his Ph.D. at Princeton in mathematics in the spring of 1947 then took a faculty position at Cal Tech, and almost immediately started to consult at the RAND Corporation under the tutelage of Frederic Bohnenblust) has written that he never heard game theory mentioned during his graduate studies.
Nevertheless, many observers agree that in the following decade Princeton was one of the two centers in which game theory flourished, the other being the RAND Corporation in Santa Monica. The story of the RAND Corporation and its research sponsored by the Air Force has been told on several occasions. We shall concentrate on the activity in the mathematics department at Princeton, a story that illustrates the strong element of chance in human affairs.
The story starts with two visits by George Dantzig to visit John von Neumann in the fall of 1947 and the spring of 1948. In the first visit Dantzig described his new theory of "linear programming" only to be told dismissively by von Neumann that he had encountered similar problems in his study of zerosum twoperson games. In his second visit, Dantzig proposed an academic project to study the relationship between these two fields and asked von Neumann's advice about universities in which such a project might be pursued. Dantzig was driven to the train station for his trip back to Washington by A. W. Tucker (a topologist who was associate chairman of the mathematics department at that time). On the ride, Dantzig gave a quick exposition of his new discoveries, using the Transportation Problem as a lively example. This recalled to Tucker his earlier work on electrical networks and Kirkhoff's Law and planted the idea that the project to study the relationship between linear programming and the theory of games might be established in the mathematics department at Princeton University.
In those halcyon days of no red tape, before a month had elapsed Tucker hired two graduate students, David Gale and myself, and the project was set up through Solomon Lefshetz's project on nonlinear differential equations until a formal structure could be established through the Office of Naval Research's Logistics Branch. And so, in the summer of 1948, Gale, Kuhn, and Tucker taught each other the elements of game theory.
How did we do this? We divided up the chapters of the Bible, the TGEB, as handed down by von Neumann and Morgenstern, and lectured to each other in one of the seminar rooms of the old Fine Hall, then the home of the mathematics department at Princeton. By the end of the summer, we had established that, mathematically, linear programming and the theory of zerosum twoperson games are equivalent.
Enthused by the research potential of the subject we had just learned, we wanted to spread the gospel. We initiated a weekly seminar in the department centered on the subjects of game theory and linear programming. To understand the importance of this development, one must contrast the situations today and then. Today, the seminar lists of the university and the Institute of Advanced Studies contain over twenty weekly seminars in subjects such as number theory, topology, analysis, and statistical mechanics. In 1948, there was a weekly colloquium that met alternate weeks at the university and the institute. The topologists and statisticians had weekly seminars and my thesis advisor, Ralph Fox, ran a weekly seminar on knot theory; but that was that. So the addition of a new seminar was an event that raised the visibility of game theory considerably among the graduate students in the department and among the visitors to the institute.
The speakers included von Neumann and Morgenstern, visitors to the institute such as Irving Kaplansky, Ky Fan, and David Bourgin, as well as outside visitors such as Abraham Wald, the Columbia statistician who had made significant connections between game theory and statistical inference. (Wald had done the review of the TGEB for Mathematical Reviews and had tutored Morgenstern in mathematics in Vienna.)
More importantly it provided a forum for graduate students in mathematics who were working in this area to present new ideas. As Shubik has reminisced: "The general attitude around Fine Hall was that no one cared who you were or what part of mathematics you worked on as long as you could find some senior member of the faculty and make a case to him that it was interesting and that you did it well ... To me the striking thing at that time was not that the mathematics department welcomed game theory with open armsbut that it was open to new ideas and new talent from any source, and that it could convey a sense of challenge and a belief that much new and worthwhile was happening." He did not find that attitude in the economics department.
A crucial fact was that von Neumann's theory was too mathematical for the economists. To illustrate the attitude of a typical economics department of the period and later, more than fifteen years after the publication of TGEB the economists at Princeton voted against instituting a mathematics requirement for undergraduate majors, choosing to run two tracks for students, one which used the calculus and one which avoided it. Richard Lester, who alternated with Lester Chandler as chairman of the department, had carried on a running debate with Fritz Machlup over the validity of marginal product (a calculus notion) as a determinant of wages. Courses that used mathematical terms and which covered mathematical topics such as linear programming were concealed by titles such as "Managerial theory of the firm." Given such prevailing views, there was no incentive or opportunity for graduate students and junior faculty to study the theory of games.
As a consequence, the theory of games was developed almost exclusively by mathematicians in this period. To describe the spirit of the time as seen by another outside observer, we shall paraphrase a section of Robert J. Aumann's magnificent article on game theory from The New Palgrave Dictionary of Economics.
The period of the late '40s and early '50s was a period of excitement in game theory. The discipline had broken out of its cocoon and was testing its wings. Giants walked the earth. At Princeton, John Nash laid the groundwork for the general noncooperative theory and for cooperative bargaining theory. Lloyd Shapley defined a value for coalitional games, initiated the theory of stochastic games, coinvented the core with D. B. Gillies, and together with John Milnor developed the first game models with an infinite number of players. Harold Kuhn reformulated the extensive form and introduced the concepts of behavior strategies and perfect recall. A.W. Tucker invented the story of the Prisoner's Dilemma, which has entered popular culture as a crucial example of the interplay between competition and cooperation.
(Continues...)
Table of Contents
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 settheoretical 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 SETTHEORETICAL 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 zerosum restriction 84
CHAPTER III: ZEROSUM TWOPERSON GAMES: THEORY
12. PRELIMINARY SURVEY 85
12.1. General viewpoints 85
12.2. The oneperson 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: ZEROSUM TWOPERSON 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: ZEROSUM THREEPERSON 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: ZEROSUM nPERSON 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
27.3. Inessentiality and essentiality 249
27.4. Various criteria. Non additive utilities 250
27.5. The inequalities in the essential case 252
27.6. Vector operations on characteristic functions 253
28. GROUPS, SYMMETRY AND FAIRNESS 255
28.1. Permutations, their groups and their effect on a game 255
28.2. Symmetry and fairness 258
29. RECONSIDERATION OF THE ZEROSUM THREEPERSON GAME 260
29.1. Qualitative discussion 260
29.2. Quantitative discussion 262
30. THE EXACT FORM OF THE GENERAL DEFINITIONS 263
30.1. The definitions 263
30.2. Discussion and recapitulation 265
*30.3. The concept of saturation 266
30.4. Three immediate objectives 271
31. FIRST CONSEQUENCES 272
31.1. Convexity, flatness, and some criteria for domination 272
31.2. The system of all imputations. One element solutions 277
31.3. The isomorphism which corresponds to strategic equivalence 281
32. DETERMINATION OF ALL SOLUTIONS OF THE ESSENTIAL ZEROSUM THREEPERSON GAME 282
32.1. Formulation of the mathematical problem. The graphical method 282
32.2. Determination of all solutions 285
33. CONCLUSIONS 288
33.1. The multiplicity of solutions. Discrimination and its meaning 288
33.2. Statics and dynamics 290
CHAPTER VII: ZEROSUM FOURPERSON GAMES
34. PRELIMINARY SURVEY 291
34.1. General viewpoints 291
34.2. Formalism of the essential zero sum four person games 291
34.3. Permutations of the players 294
35. DISCUSSION OF SOME SPECIAL POINTS IN THE CUBE Q 295
35.1. The corner I. (and V., VI., VII.) 295
35.2. The corner VIII. (and II., III., IV.,). The three person game and a "Dummy" 299
35.3. Some remarks concerning the interior of Q 302
36. DISCUSSION OF THE MAIN DIAGONALS 304
36.1. The part adjacent to the corner VIII.: Heuristic discussion 304
36.2. The part adjacent to the corner VIII.: Exact discussion 307
*36.3. Other parts of the main diagonals 312
37. THE CENTER AND ITS ENVIRONS 313
37.1. First orientation about the conditions around the center 313
37.2. The two alternatives and the role of symmetry 315
37.3. The first alternative at the center 316
37.4. The second alternative at the center 317
37.5. Comparison of the two central solutions 318
37.6. Unsymmetrical central solutions 319
*38. A FAMILY OF SOLUTIONS FOR A NEIGHBORHOOD OF THE CENTER 321
*38.1. Transformation of the solution belonging to the first alternative at the center 321
*38.2. Exact discussion 322
*38.3. Interpretation of the solutions 327
CHAPTER VIII: SOME REMARKS CONCERNING n [equal to or greater than] 5 PARTICIPANTS
39. THE NUMBER OF PARAMETERS IN VARIOUS CLASSES OF GAMES 330
39.1. The situation for n = 3, 4 330
39.2. The situation for all n [equal to or greater than] 3 330
40. THE SYMMETRIC FIVE PERSON GAME 332
40.1. Formalism of the symmetric five person game 332
40.2. The two extreme cases 332
40.3. Connection between the symmetric five person game and the 1, 2, 3 symmetric four person game 334
CHAPTER IX: COMPOSITION AND DECOMPOSITION OF GAMES
41. COMPOSITION AND DECOMPOSITION 339
41.1. Search for nperson games for which all solutions can be determined 339
41.2. The first type. Composition and decomposition 340
41.3. Exact definitions 341
41.4. Analysis of decomposability 343
41.5. Desirability of a modification 345
42. MODIFICATION OF THE THEORY 345
42.1. No complete abandonment of the zero sum restriction 345
42.2. Strategic equivalence. Constant sum games 346
42.3. The characteristic function in the new theory 348
42.4. Imputations, domination, solutions in the new theory 350
42.5. Essentiality, inessentiality and decomposability in the new theory 351
43. THE DECOMPOSITION PARTITION 353
43.1. Splitting sets. Constituents 353
43.2. Properties of the system of all splitting sets 353
43.3. Characterization of the system of all splitting sets. The decomposition partition 354
43.4. Properties of the decomposition partition 357
44. DECOMPOSABLE GAMES. FURTHER EXTENSION OF THE THEORY 358
44.1. Solutions of a (decomposable) game and solutions of its constituents 358
44.2. Composition and decomposition of imputations and of sets of imputations 359
44.3. Composition and decomposition of solutions. The main possibilities and surmises 361
44.4. Extension of the theory. Outside sources 363
44.5. The excess 364
44.6. Limitations of the excess. The nonisolated character of a game in the new setup 366
44.7. Discussion of the new setup. E(e_{0}), F(e_{0}) 367
45. LIMITATIONS OF THE EXCESS. STRUCTURE OF THE EXTENDED THEORY 378
45.1. The lower limit of the excess 368
45.2. The upper limit of the excess. Detached and fully detached imputations 369
45.3. Discussion of the two limits, [Gamma]_{1}, [Gamma]_{2}. Their ratio 372
45.4. Detached imputations and various solutions. The theorem connecting E(e_{0}), F(e_{0}) 375
45.5. Proof of the theorem 376
45.6. Summary and conclusions 380
46. DETERMINATION OF ALL SOLUTIONS OF A DECOMPOSABLE GAME 381
46.1. Elementary properties of decompositions 381
46.2. Decomposition and its relation to the solutions: First results concerning F(e_{0}) 384
46.3. Continuation 386
46.4. Continuation 388
46.5. The complete result in F(e_{0}) 390
46.6. The complete result in E(e_{0}) 393
46.7. Graphical representation of a part of the result 394
46.8. Interpretation: The normal zone. Heredity of various properties 396
46.9. Dummies 397
46.10. Imbedding of a game 398
46.11. Significance of the normal zone 401
46.12. First occurrence of the phenomenon of transfer: n = 6 402
47. THE ESSENTIAL THREEPERSON GAME IN THE NEW THEORY 403
47.1. Need for this discussion 403
47.2. Preparatory considerations 403
47.3. The six cases of the discussion. Cases (I)(III) 406
47.4. Case (IV): First part 407
47.5. Case (IV): Second part 409
47.6. Case (V) 413
47.7. Case (VI) 415
47.8. Interpretation of the result: The curves (one dimensional parts) in the solution 416
47.9. Continuation: The areas (two dimensional parts) in the solution 418
CHAPTER X: SIMPLE GAMES
48. WINNING AND LOSING COALITIONS AND GAMES WHERE THEY OCCUR 420
48.1. The second type of 41.1. Decision by coalitions 420
48.2. Winning and Losing Coalitions 421
49. CHARACTERIZATION OF THE SIMPLE GAMES 423
49.1. General concepts of winning and losing coalitions 423
49.2. The special role of one element sets 425
49.3. Characterization of the systems W, L of actual games 426
49.4. Exact definition of simplicity 428
49.5. Some elementary properties of simplicity 428
49.6. Simple games and their W, L. The Minimal winning coalitions: W^{m} 429
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 446
51.3. Reasons for passing from W to W^{m}. Difficulties of using W^{m} 448
51.4. Changed Approach: Enumeration by means of W^{m} 450
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^{m} 455
52. THE SIMPLE GAMES FOR SMALL n 457
52.1. Program. n = 1, 2 play no role. Disposal of n = 3 457
52.2. Procedure for n [equal to or greater than] 4: The two element sets and their role in classify ing the W^{m} 458
52.3. Decomposability of cases C*, C_{n2}, C_{n1} 459
52.4. The simple games other than [1, ^{. . .} , 1, n  2]_{h}, (with dummies): The Cases C_{k}, k = 0, 1, ^{. . .} , n  3 461
52.5. Disposal of n = 4, 5 462
53. THE NEW POSSIBILITIES OF SIMPLE GAMES FOR n [equal to or greater than] 6 463
53.1. The Regularities observed for n [equal to or greater than] 6 463
53.2. The six main counter examples (for n = 6, 7) 464
54. 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 NONZEROSUM 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. Zerosum and constantsum 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] 3 548
60.1. The case n = 1 548
60.2. The case n = 2 549
60.3. The case n = 3 550
60.4. Comparison with the zero sum games 554
61. ECONOMIC INTERPRETATION OF THE RESULTS FOR n = 1, 2 555
61.1. The case n = 1 555
61.2. The case n = 2. The two person market 555
61.3. 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 564
62.1. The case n = 3, special case. The three person market 564
62.2. 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 573
63.1. 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