This book contains an exposition and various applications of a mathematical theory of games. The theory has been developed by one of us since 1928 and is now published for the first time in its entirety. The applications are of two kinds: On the one hand to games in the proper sense, on the other hand to economic and sociological problems which, as we hope to show, are best approached from this direction.
The applications which we shall make to games serve at least as much to corroborate the theory as to investigate these games. The nature of this reciprocal relationship will become clear as the investigation proceeds. Our major interest is, of course, in the economic and sociological direction. Here we can approach only the simplest questions. However, these questions are of a fundamental character. Furthermore, our aim is primarily to show that there is a rigorous approach to these subjects, involving, as they do, questions of parallel or opposite interest, perfect or imperfect information, free rational decision or chance influences.
JOHN VON NEUMANN
OSKAR MORGENSTERN.
PRINCETON, N. J.
January, 1943.
PREFACE TO SECOND EDITION
The second edition differs from the first in some minor respects only. We have carried out as complete an elimination of misprints as possible, and wish to thank several readers who have helped us in that respect. We have added an Appendix containing an axiomatic derivation of numerical utility. This subject was discussed in considerable detail, but in the main qualitatively, in Section 3. A publication of this proof in a periodical was promised in the first edition, but we found it more convenient to add it as an Appendix. Various Appendices on applications to the theory of location of industries and on questions of the four and five person games were also planned, but had to be abandoned because of the pressure of other work.
Since publication of the first edition several papers dealing with the subject matter of this book have appeared.
The attention of the mathematically interested reader may be drawn to the following: A. Wald developed a new theory of the foundations of statistical estimation which is closely related to, and draws on, the theory of the zero-sum two-person game (“Statistical Decision Functions Which Minimize the Maximum Risk,” Annals of Mathematics, Vol. 46 (1945) pp. 265–280). He also extended the main theorem of the zero-sum two-person games (cf. 17.6.) to certain continuous-infinite-cases, (“Generalization of a Theorem by von Neumann Concerning Zero-Sum Two-Person Games,” Annals of Mathematics, Vol. 46 (1945), pp. 281–286.) A new, very simple and elementary proof of this theorem (which covers also the more general theorem referred to in footnote 1 on page 154) was given by L. H. Loomis, (“On a Theorem of von Neumann,” Proc. Nat. Acad., Vol. 32 (1946) pp. 213—215). Further, interesting results concerning the role of pure and of mixed strategies in the zero-sum two-person game were obtained by I. Kaplanski, (“A Contribution to von Neumann’s Theory of Games,” Annals of Mathematics, Vol. 46 (1945), pp. 474–479). We also intend to come back to various mathematical aspects of this problem. The group theoretical problem mentioned in footnote 1 on page 258 was solved by C. Chevalley.
The economically interested reader may find an easier approach to the problems of this book in the expositions of L. Hurwicz, (“The Theory of Economic Behavior,” American Economic Review, Vol. 35 (1945), pp. 909–925) and of J. Marschak (“Neumann’s and Morgenstern’s New Approach to Static Economics,” Journal of Political Economy, Vol. 54, (1946), pp. 97–115).
JOHN VON NEUMANN
OSKAR MORGENSTERN
PRINCETON, N. J.
September, 1946.
The Third Edition differs from the Second Edition only in the elimination of such further misprints as have come to our attention in the meantime, and we wish to thank several readers who have helped us in that respect.
Since the publication of the Second Edition, the literature on this subject has increased very considerably. A complete bibliography at this writing includes several hundred titles. We are therefore not attempting to give one here. We will only list the following books on this subject:
(1) H. W. Kuhn and A. W. Tucker (eds.), “Contributions to the Theory of Games, I,” Annals of Mathematics Studies, No. 24, Princeton (1950), containing fifteen articles by thirteen authors.
(2) H. W. Kuhn and A. W. Tucker (eds.), “Contributions to the Theory of Games, II,” Annals of Mathematics Studies, No. 28, Princeton (1953), containing twenty-one articles by twenty-two authors.
(3) J. McDonald, Strategy in Poker, Business and War, New York (1950).
(4) J. C. C. McKinsey, Introduction to the Theory of Games, New York (1952).
(5) A. Wald, Statistical Decision Functions, New York (1950).
(6) J. Williams, The Compleat Strategyst, Being a Primer on the Theory of Games of Strategy, New York (1953).
Bibliographies on the subject are found in all of the above books except (6). Extensive work in this field has been done during the last years by the staff of the RAND Corporation, Santa Monica, California. A bibliography of this work can be found in the RAND publication RM-950.
In the theory of n-person games, there have been some further developments in the direction of “non-cooperative” games. In this respect, particularly the work of J. F. Nash, “Non-cooperative Games,” Annals of Mathematics, Vol. 54, (1951), pp. 286–295, must be mentioned. Further references to this work are found in (1), (2), and (4).
Of various developments in economics we mention in particular “linear programming” and the “assignment problem” which also appear to be increasingly connected with the theory of games. The reader will find indications of this again in (1), (2), and (4).
The theory of utility suggested in Section 1.3., and in the Appendix to the Second Edition has undergone considerable development theoretically, as well as experimentally, and in various discussions. In this connection, the reader may consult in particular the following:
M. Friedman and L. J. Savage, “The Utility Analysis of Choices Involving Risk,” Journal of Political Economy, Vol. 56, (1948), pp. 279–304.
J. Marschak, “Rational Behavior, Uncertain Prospects, and Measurable Utility,” Econometrica, Vol. 18, (1950), pp. 111–141.
F. Mosteller and P. Nogee, “An Experimental Measurement of Utility,” Journal of Political Economy, Vol. 59, (1951), pp. 371–404.
M. Friedman and L. J. Savage, “The Expected-Utility Hypothesis and the Measurability of Utility,” Journal of Political Economy, Vol. 60, (1952), pp. 463–474.
See also the Symposium on Cardinal Utilities in Econometrica, Vol. 20, (1952):
H. Wold, “Ordinal Preferences or Cardinal Utility?”
A. S. Manne, “The Strong Independence Assumption—Gasoline Blends and Probability Mixtures.”
P. A. Samuelson, “Probability, Utility, and the Independence Axiom.”
E. Malinvaud, “Note on von Neumann-Morgenstern’s Strong Independence Axiom.”
In connection with the methodological critique exercised by some of the contributors to the last-mentioned symposium, we would like to mention that we applied the axiomatic method in the customary way with the customary precautions. Thus the strict, axiomatic treatment of the concept of utility (in Section 3.6. and in the Appendix) is complemented by an heuristic preparation (in Sections 3.1.–3.5.). The latter’s function is to convey to the reader the viewpoints to evaluate and to circumscribe the validity of the subsequent axiomatic procedure. In particular our discussion and selection of “natural operations” in those sections covers what seems to us the relevant substrate of the Samuelson-Malinvaud “independence axiom.”
JOHN VON NEUMANN
OSKAR MORGENSTERN
PRINCETON, N. J.
January, 1953.