Introduction to the Theory of Games

J. C. C. McKinsey

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486428116
• eBook ISBN: 9780486154428
• Topic: Mathematics
• Subtopic: Game Theory
• Index: Mathematics

One of the classic early monographs on game theory, this comprehensive overview illustrates the theory's applications to situations involving conflicts of interest, including economic, social, political, and military contexts. Contents include a survey of rectangular games; a method of approximating the value of a game; games in extensive form and those with infinite strategies; distribution functions; Stieltjes integrals; the fundamental theorem for continuous games; separable games; games with convex payoff functions; applications to statistical inference; and much more. Appropriate for advanced undergraduate and graduate courses; a familiarity with advanced calculus is assumed. 1952 edition. 51 figures. 8 tables.

CHAPTER 1

RECTANGULAR GAMES

1. Introduction. In this book we shall be concerned with the mathematical theory of games of strategy. Examples of parlor games of strategy are such games as chess, bridge, and poker, where the various players can make use of their ingenuity in order to outwit each other. Aside from this application within the sphere of social amusement, the theory of games is gaining importance because of its general applicability to situations which involve conflicting interests, and in which the outcome is controlled partly by one side and partly by the opposing side of the conflict. Many conflict situations which form the subject of economic, social, political, or military discourse are of this kind.

Although many real-life conflicts as well as parlor games involve elements of chance (as in the cards dealt in bridge or the weather encountered in a military operation), we shall ordinarily exclude from our discussion games in which the outcome depends entirely on chance and cannot be affected by the cleverness of the players.

The essential difference between games of strategy and games of (pure) chance lies in the circumstance that intelligence and skill are useful in playing the former but not the latter. Thus an amateur would be extremely unwise to play chess for even money and high stakes against a master: he would face almost certain ruin. But, contrary to the stories occasionally heard (stories which are most likely fabricated by the proprietors of gaming houses), there is no “system” for playing roulette on an unbiased wheel: an idiot has as good a chance of winning at this game as has a man of sense. (This is not to say that there do not remain difficult unsolved mathematical problems in connection with games of chance; but there exist, at least, standard methods for attacking such problems, and we shall not treat of them here.)

Although our attention will be devoted almost entirely to the purely mathematical aspects of the theory of games of strategy, it is perhaps well to begin with some brief remarks about the history of economics. These remarks may serve to convince the student that our theory is not altogether frivolous; for buying and selling are customarily regarded as more serious and respectable occupations than is playing poker—or even chess, for that matter.

For many decades economists tended to take as a standard model for their science the situation of Robinson Crusoe, marooned on an uninhabited island and concerned with behaving in such a way as to maximize the goods he could obtain from nature. It was generally felt that it would be possible to get an insight into the behavior of groups of individuals by starting with a detailed analysis of the behavior proper in this simplest possible case: the case of a single individual all alone and struggling against nature.

This line of attack on economic problems, however, suffers from the defect that in going from a one-man society to even a two-man society, qualitatively different situations arise which could hardly have been foreseen from the one-man case.¹ In a society which contains two members, it may happen that each desires a certain commodity (the supply of which is not sufficient for both) and that each member has control of some, but not all, of the factors which determine how the commodity is to be distributed. The behavior of each, then, if it is to...

Table of contents

• Title Page
• Copyright Page
• Foreword
• Table of Contents
• CHAPTER 1 - RECTANGULAR GAMES
• CHAPTER 2 - THE FUNDAMENTAL THEOREM FOR RECTANGULAR GAMES
• CHAPTER 3 - THE SOLUTIONS OF A RECTANGULAR GAME
• CHAPTER 4 - A METHOD OF APPROXIMATING THE VALUE OF A GAME
• CHAPTER 5 - GAMES IN EXTENSIVE FORM
• CHAPTER 6 - GAMES IN EXTENSIVE FORM—GENERAL THEORY
• CHAPTER 7 - GAMES WITH INFINITELY MANY STRATEGIES
• CHAPTER 8 - DISTRIBUTION FUNCTIONS
• CHAPTER 9 - STIELTJES INTEGRALS
• CHAPTER 10 - THE FUNDAMENTAL THEOREM FOR CONTINUOUS GAMES
• CHAPTER 11 - SEPARABLE GAMES
• CHAPTER 12 - GAMES WITH CONVEX PAYOFF FUNCTIONS
• CHAPTER 13 - APPLICATIONS TO STATISTICAL INFERENCE
• CHAPTER 14 - LINEAR PROGRAMMING
• CHAPTER 15 - ZERO-SUM n-PERSON GAMES
• CHAPTER 16 - SOLUTIONS OF n-PERSON GAMES
• CHAPTER 17 - GAMES WITHOUT ZERO-SUM RESTRICTION: THE VON NEUMANN-MORGENSTERN THEORY
• CHAPTER 18 - SOME OPEN PROBLEMS
• Bibliography
• Index