Two-Person Game Theory

Anatol Rapoport

• Publisher: Dover Publications
• Year: 2013
• eBook ISBN: 9780486281094
• Topic: Mathematics
• Subtopic: Game Theory
• Index: Mathematics

"Game theory is an intellectual X-ray. It reveals the skeletal structure of those systems where decisions interact, and it reveals, therefore, the essential structure of both conflict and cooperation." — Kenneth Boulding
This fascinating and provocative book presents the fundamentals of two-person game theory, a mathematical approach to understanding human behavior and decision-making, Developed from analysis of games of strategy such as chess, checkers, and Go, game theory has dramatic applications to the entire realm of human events, from politics, economics, and war, to environmental issues, business, social relationships, and even "the game of love." Typically, game theory deals with decisions in conflict situations.
Written by a noted expert in the field, this clear, non-technical volume introduces the theory of games in a way which brings the essentials into focus and keeps them there. In addition to lucid discussions of such standard topics as utilities, strategy, the game tree, and the game matrix, dominating strategy and minimax, negotiated and nonnegotiable games, and solving the two-person zero-sum game, the author includes a discussion of gaming theory, an important link between abstract game theory and an experimentally oriented behavioral science. Specific applications to social science have not been stressed, but the methodological relations between game theory, decision theory, and social science are emphasized throughout.
Although game theory employs a mathematical approach to conflict resolution, the present volume avoids all but the minimum of mathematical notation. Moreover, the reader will find only the mathematics of high school algebra and of very elementary analytic geometry, except for an occasional derivative. The result is an accessible, easy-to-follow treatment that will be welcomed by mathematicians and non-mathematicians alike.

11. An Example: Inspector vs. Evader

We have now discussed the essential ideas which form the basis of the theory of the two-person game. We have used different games to illustrate the different principles: to define the concept of strategy, to reduce the game to normal form, to solve games with and without saddle points, to illustrate the way different bargaining principles are applied to a negotiated game, to exhibit paradoxes immanent in some nonnegotiated games, and to resolve these paradoxes by the use of an inductive theory.

We shall now discuss a single situation in which all of these principles can be illustrated. In thus relating the main problems of the theory to a model of a concrete situation, we hope to give the reader a “feel” for what is involved when one attempts to “apply” the theory, specifically an appreciation of both its conceptual power and its serious limitations.

Imagine an agreement concluded between two sovereign states, Urania and Plutonia, to the effect that each shall refrain from certain acts or activities. The treaty provides for an inspection procedure presumably designed to give assurance that the agreement is not violated.

Urania, for reasons which are irrelevant to our problem, insists on more inspection privileges, while Plutonia is reluctant to grant them. Obviously the entire problem, involving as it does a vast network of strategic, political, psychological, and possibly psychopathological components, is too complex to be formulated as a well-defined game. We can, however, isolate a salient feature of the problem and concentrate the entire conceptual apparatus of two-person game theory upon it. The result will be not a unique prescriptive solution of “the game”; for, as we shall see, even the drastically simplified version offers a large array of “solutions.” The constructive feature of our result will be an unfolding of the problem into its various constituent parts and so a clarification of the issues, which ordinarily remain obscure.

Let us suppose that the issues have for a while revolved about the number of inspections to be allowed in a specified interval of time. As we have said, Urania has been pressing for more, while Plutonia has been holding out for fewer. At long last, however, the two sovereign states have agreed on the frequency of inspections. The only disagreement is on whether this frequency is to be realized by a fixed number of inspections per year or as an average number of inspections whose actual number can fluctuate statistically.

Specifically, suppose the year is divided into two six-month periods. Urania offers the following scheme. At the end of each period a coin is to be tossed. If it falls heads, an inspection is to be allowed; if it falls tails, not. In this way, although there may be one, two, or no inspections in any particular year, on the average there will be one inspection per year. Plutonia opposes this plan. She proposes a scheme whereby the question of whether there will or will not be an inspection in a six-month period shall be decided toward the end of that period by the inspecting party. If it is decided to inspect at the end of the first six months, then no inspection can take place during the last six months of the year. If it is decided to omit the inspection at the end of the first six months, then an inspection will be allowed during the last six months. (We shall also assume that the nature of prohibited activities is such that violations can be detected within six months but not after a longer period.)

From the positions of Urania and Plutonia, it appears that Urania is more interested in preventing violations by Plutonia than in having an opportunity for clandestine evasion. One might also suspect from Plutonia’s position that she is less interested in inspecting Urania than in having an opportunity to evade. Let us therefore refer to the two high contracting parties as Inspector and Evader respectively.

The problem before us is which inspection schedule should Inspector prefer, the single allowed inspection per year or the two potential inspections per year, each with probability one half?

Urania’s strategic experts argue that the probabilistic schedule is preferable to the fixed one. In support of their contention, they cite the so-called “end effect.” For consider what is likely to happen if the fixed schedule is in effect. Clearly Inspector cannot always defer the inspection to the second period. For if he does, Evader can evade with impunity during the first. On the other hand, if Inspector randomizes his choice of period to inspect, Evader can simply wait until Inspector chooses to inspect the first period and then evade with impunity in the second. With a probabilistic inspection schedule, this is not possible, for whatever happened in the first period has no bearing on the second. Therefore Evader always stands the risk of being found out and is thereby deterred from evading.

The argument looks conclusive. Let us see, however, how it looks in the light of game theory. Consider the following game consisting of four moves.³³

Move 1. Evader chooses to evade or not in the first period.

Move 2. Inspector, not knowing how Evader has chosen (naturally, otherwise there is altogether no need for inspections!) chooses to inspect or not to inspect in the first period.

Move 3. Evade...

Table of contents

• DOVER BOOKS ON MATHEMATICS
• Title Page
• Copyright Page
• Preface
• Table of Contents
• 1Games
• 2. Utilities
• 3. Strategy
• 4. The Game Tree and the Game Matrix
• 5. Dominating Strategy and Minimax
• 6. Mixed Strategy
• 7. Solving the Two-Person Zero-sum Game
• 8. The Negotiated Game
• 9. Nonnegotiable Games
• 10. An Inductive Theory of Games: Dynamic Models
• 11. An Example: Inspector vs. Evader
• 12. Opportunities and Limitations
• Notes
• References
• Index