What Is the Theory of Games?
The most casual observer of the social behavior of mathematicians will remark that the solitary mathematician is a puzzle-solver while mathematicians together consume an extraordinary amount of time playing games. Coupling this observation with the obvious economic interest of many games of skill and chance, it does not seem strange that the basis for a mathematical theory of games of strategy was created by the mathematician John von Neumann ∗ and elaborated with the economist Oskar Morgenstern . In this series of lectures we shall attempt to present the salient results of this theory.
Although our primary concern will be with the mathematical aspects, a complete understanding of the theory can only come from the recognition that it is an interpreted system. This means that, like any satisfactory physical theory, it consists of (1) an axiom system and (2) a set of semantical rules for its interpretation, that is, a set of prescriptions that connect the mathematical symbols of the axioms with the objects of our experience. In the immediate interpretation, these objects are ordinary parlor games such as chess, poker, or bridge, and the success of the axiom system that we choose is measured by the number of interesting facts that we can derive about these games. There are, of course, other interpretations of the axioms, and one of the more ambitious hopes of the theory is that eventually these will include a substantial portion of the science of economics . In these lectures we will restrict ourselves, with few exceptions, to the principal interpretation in terms of games.
Ideally, such an interpreted system could be presented by giving its axioms and the rules for their interpretation and then deriving all the important theorems of the theory by the application of mathematical techniques. But practically the situation is such that most fields of science seem at the present time to be not yet developed to a degree which would suggest this strict form of presentation; only certain branches of physics and geometry qualify by their apparent completeness. Fortunately for the researcher, the theory of games is by no means complete. Indeed, in some portions, such as games played by more than two players, it seems likely that the final formulation will be far removed from the present theory. Accordingly, the ideal order of presentation will be reversed in these lectures, and it will only be after a number of specific games have been examined in detail that the axioms of the theory will appear.
*Numbers in square brackets refer to explanatory material appended to each chapter. In some cases
this material leads to sources and advanced topics in the theory.