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.

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 /. Kaplanski, (“A Contribution to von Neumann’s Theory of Games,” Annals of Mathe- matics, Vol. 46 (1945), pp. 474-479). We also intend to come back to vari- ous 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. Hururicz, (“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).