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Game theory : A Nontechnical Introduction

The theory of games was originally created to provide a new approach to economic problems. John von Neumann and Oskar Morgenstern did not construct a theory simply as an appendage, to take its place on the periphery of economics as an afterthought. Quite the contrary. They felt that "the typical problems of economic behavior become strictly identical with the mathematical notions of suitable games of strategy." The term "game" might suggest that the subject is narrow and frivolous, but this is far from the case. Since the classical work of von Neumann and Morgenstern was published, the theory of games has proven to be of sufficient interest to justify its study as an independent discipline. And the applications are not limited to economics; the ef- fects of the theory have been felt in political science, pure mathematics, psychology, sociology, marketing, finance, and warfare. Actually, the theory of games is not one theory but many. This is not surprising. A game, after all, is only a model of reality and it would be too much to expect that a single model could accurately reflect such diversified types of XV xvi Author's Introduction activity. But there are certain elements that are contained in all these models, and it is these that we will be concerned with when we discuss games. The word "game" has a different meaning for the layman and for the game-theorist, but there are some similarities. In both types of games there are players, and the players must act, or make certain decisions. And as a result of the behavior of the players?and, possibly, chance?there is a certain outcome: a reward or punishment for each of the participating players. The word "player," incidentally, does not have quite the meaning one would expect. A player need not be one person; he may be a team, a corporation, a nation. It is useful to regard any group of individuals who have identical interests with respect to the game as a single player. How to Analyze a Game Two basic questions must be answered about any game: How should the players behave? What should be the ultimate outcome of the game? The answer to one of these questions, or to both, is sometimes called the solution of the game, but the term "solu- tion" does not have a universal meaning in game theory; in different contexts it has different meanings. This is true also of the word "rational" when used to describe behavior. These questions lead to others. What is the "power" of a player? That is, to what extent can an individual determine the outcome of a game? More specifically, what is the minimum a player can assure himself, relying on his own re- sources, if he receives no cooperation from others? And is it reasonable to suppose that the other players will in fact be hostile? In order to answer these questions, one must know certain things about the game. First, one must know the rules of the game. These include: Authors Introduction xvii 1. To what extent the players can communicate with one another 2. Whether the players can or cannot enter into binding agreements 3. Whether rewards obtained in the game may be shared with other players ( that is, whether side payments are permitted) 4. What the formal, causal relation between the actions of the players and the outcome of the game is (that is, the payoff matrix ) 5. What information is available to the players In addition, the personalities of the players, their subjective preferences, the mores of society (that is, what the players believe to be a "fair outcome"), all have an effect on the outcome. Perhaps the single most significant characteristic of a game is the number of players, that is, its size; in fact, the games will be discussed in this book in order of size. Games of the same size will be grouped together and then dis- tinguished from one another on the basis of other, less prominent differences. The smallest games will be discussed first. Generally, the fewer the players, the simpler the game. As one progresses from the simplest games to those of greater complexity, the theories become less satisfying. This might almost be expressed in the fonn of a perverse, quasi-conservation law: the greater the significance of a game?that is, the more applications it has to real problems?the more difficult it is to treat analytically. The most satisfying theory, at least from the point of view of the mathematician, is that of the competitive, two-person* game; but in real life the purely competitive game is rare indeed. For the more common, partly competitive, partly cooperative, two-person game, in fact, no generally accepted theory exists. This perverse law, incidentally, is not restricted to game theory alone. In the area of psychology, it is considerably easier to determine how rats learn than to discover the causes of mental illness. xviii Authors Introduction In the more complex games, a player is faced with forces that he cannot control. The less control a player has upon the final outcome, the more difficult it becomes to define rational behavior. What constitutes a wise decision when any decision will have little influence on what ultimately happens? Earlier we asked what a player should do and what the ultimate outcome of a game should be?"should," not in the moral sense, but in the sense of what will most further a player's interests. The question of what people actually do and what actually happens when a game is played is best left to the behavioral scientist. (This is sometimes ex- pressed by saying that the game-theorist is interested in the normative rather than the descriptive aspects of a game. ) This is all very well in the simpler games, but in complex games the distinction becomes blurred. If a player is not in control, he must be concerned with what the other players want and what they intend to do. If, for example, in a labor-management dispute, one party knows the other party has an aversion to risk, he might act more aggressively than he would otherwise. A short-term speculator on the stock market is almost wholly in the hands of others; what he should do depends on what he thinks others will do. As the games grow more complex, it becomes almost impossible to give convincing answers to our two initial questions; we have to be content with less. Instead of determining a precise outcome, we often have to settle for a set of possible outcomes that seem to be more plausible than the rest. These possible outcomes may be more firm (we say stable) than the others in that no player or group of players has both the power and motivation to replace them with one of the less favored outcomes. Or the out- comes may be fair or enforceable in a sense that we will define later. Or, alternatively, we may take an entirely dif- ferent approach and try to determine the average payoff a player would receive if he played the game many times.

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