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General introduction to the theory of games
1.1 CONFLICT OF INTEREST
In all of man's written record there has been a preoccupation with conflict of interest; possibly only the topics of God, love, and inner struggle have received comparable attention. The scientific study of interest conflict, in contrast to its description or its use as a dramatic vehicle, comprises a small, but growing, portion of this literature. As a reflection of this trend we find today that conflict of interest, both among individuals and among institutions, is one of the more dominant concerns of at least several of our academic departments: economics, sociology, political science, and other areas to a lesser degree.
It is not difficult to characterize imprecisely the major aspects of the problem of interest conflict: An individual is in a situation from which one of several possible outcomes will result and with respect to which he has certain personal preferences. However, though he may have some control over the variables which determine the outcome, he does not have full control. Sometimes this is in the hands of several individuals who, like him, have preferences among the possible outcomes, but who in general do not agree in their preferences. In other cases, chance events (which are sometimes known in law as "acts of God") as well as other individuals (who may or may not be affected by the outcome of the situation) may influence the final outcome. The types of behavior which result from such situations have long been observed and recorded, and it is a challenge to devise theories to explain the observations and to formulate principles to guide intelligent action.
The literature on such problems is so vast, so specialized, and so rich in detail that it is utterly hopeless to attempt even a sketch of it; however, the attempt to abstract a certain large class of these problems into a mathematical system forms only a small portion of the total literature. In fact, aside from sporadic forays in economics, where for the most part attempts have been made to reduce it to a simple optimization problem which can be dealt with by the calculus, or in more sophisticated formulations by the calculus of variations, the only mathematical theory so far put forth is the theory of games, our topic. In some ways the name "game theory" is unfortunate, for it suggests that the theory deals with only the socially unimportant conflicts found in parlor games, whereas it is far more general than that. Indeed, von Neumann and Morgenstern entitled their now classical book Theory of Games and Economic Behavior, presumably to forestall that interpretation, although this does not emphasize the even wider applicability of the theory.
1.2 HISTORICAL BACKGROUNDS
The modern mathematical approach to interest conflict—game theory—is generally attributed to von Neumann in his papers of 1928 and 1937 [1928; 1937]; although recently Frechet has raised a question of priority by suggesting that several papers by Borel  in the early '20's really laid the foundations of game theory. These papers have been translated into English and republished with comments by Frechet and von Neumann . Although Borel gave a clear statement of an important class of game theoretic problems and introduced the concepts of pure and mixed strategies, von Neumann points out that he did not obtain one crucial result—the minimax theorem—without which no theory of games can be said to exist. In fact, Borel conjectured that the minimax theorem is false in general, although he did prove it true in certain special cases. Von Neumann proved it true under general conditions, and in addition he created the conceptually rich theory of games with more than two players.
Of more interest than a debate on priority is the fact that neither group of papers—the one in France and the other in Germany—attracted much attention on publication. There are almost no other papers than those mentioned before the publication in 1944 of von Neumann and Morgenstern's book, and those were confined to the mathematical journals. Apparently little interest was stimulated in the empirical sciences most concerned with conflict of interest, but this is not surprising since the original papers were written for mathematicians, not for social scientists. Fortunately, von Neumann and Morgenstern attempted to write their book so that a patient scientist with limited mathematical training could absorb the motivation, the reasoning, and the conclusions of the theory; judging by the attention given it in non-mathematical journals, as well as in the mathematical ones, they were not without success in this aim. Only a very few scientific volumes as mathematical as this one have aroused as much interest and general admiration. Yet we know that much of the material had lain dormant in the literature for two decades. Presumably the recent war was an important contributing factor to the later rapid development of the theory. During that period considerable activity developed in scientific, or at least systematic, approaches to problems that had been previously considered the exclusive province of men of "experience." These include such topics as logistics, submarine search, air defense, etc. Game theory certainly fits into this trend, and it is one of the more sophisticated theoretical structures so far resulting from it.
Though it is not directly relevant to the theory itself, it is worth emphasizing again that game theory is primarily a product of mathematicians and not of scientists from the empirical fields. In large part this results from the fact that the theory was originated by a mathematician and was, to all intents and purposes, first presented in book form as a highly formal (though, for the most part, elementary) structure, thus tending to make it accessible as a research vehicle only to mathematicians. Indeed, we believe that so far the impact of game theory has been greater in applied mathematics, especially in mathematical statistics, than in the empirical sciences.
1.3 AN INFORMAL CHARACTERIZATION OF A GAME
Game theory does not, and probably no mathematical theory could, encompass all the diverse problems which are included in our brief characterization of conflict of interest. In this introduction we shall try to cite the main features of the theory and to present some substantive problems included in its framework. The reader will easily fill in examples not now in the domain of the theory, and as we discuss our examples we shall point out some other important cases which are not covered.
First, with respect to the possible outcomes of a given situation, it is assumed that they are well specified and that each individual has a consistent pattern of preferences among them. Thus, if we ignore the fact that the player is in conflict with others and concentrate only on the outcomes, it is supposed that one way or another it can be ascertained what choice he would make if he were offered any particular collection of alternatives from which to choose. This problem of individual decision making is crucial to the whole superstructure we shall discuss, and it is important that there be no confusion about the assumptions that are made. For this reason we have devoted the whole of Chapter 2 to the topic of modern utility theory. In order that those familiar with the classical, and somewhat discredited, uses of the word "utility" not be misled, we shall expend some effort in establishing how the modern work on utility differs from the earlier ideas. In brief, the current theory shows that if one admits the possibility of risky outcomes, i.e., lotteries involving the basic alternatives, and if a person's preferences are consistent in a manner to be prescribed, then his preferences can be represented numerically by what is called a utility function. This utility has the very important property that a person will prefer one lottery to another if and only if the expected utility of the former is larger than the expected utility of the latter. Thus, the assumed individual desire for the preferred outcomes becomes, in game theory, a problem of maximizing expected utility.
Second, the variables which control the possible outcomes are also assumed to be well specified, that is, one can precisely characterize all the variables and all the values which they may assume. Actually, one may best think of them as partitioned into n + 1 classes if there are n individuals in the situation or, in the terminology of the theory, if it is an n-person game. To each person is associated one of the classes, which represents his domain of choice, and the one left over is within the province of chance.
As we said earlier, in this type of conflict situation we are interested in only some of the resulting behavior. Actually, our curiosity may encompass all of it—the tensions resulting, suicide rates or frequency of nervous disorder, aggressive behavior, withdrawal, changes in personal or business strategy, etc.—but of these, any one theory will, presumably, deal with only a small subset. At present, game theory deals with the choices people may make, or, better, the choices they should make (in a sense to be specified), in the resulting equilibrium outcomes, and in some aspects of the communication and collusion which may occur among players in their attempts to improve their outcomes. Although much of what is socially, individually, and scientifically interesting is not a part of the theory, certain important aspects of our social behavior are included.
A theory such as we are discussing cannot come into existence without assumptions about the individuals with which it purports to be concerned. We have already stated one: each individual strives to maximize his utility. Care must be taken in interpreting this assumption, for a person's utility function may not be identical with some numerical measure given in the game. For example, poker, when it is played for money, is a game with numerical payoffs assigned to each of the outcomes, and one way to play the game is to maximize one's expected money outcome. But there are players who enjoy the thrill of bluffing for its own sake, and they bluff with little or no regard to the expected payoff. Their utility functions cannot be identified with the game money payments. Indeed, there are many who feel that the maximization assumption itself is tautological, and that the empirical question is simply whether or not a numerical utility exists in a given case. Assuming that behavior is correctly described as the maximization of utility, it is quite another question how well a person knows the functions, i.e., the numerical utilities, the others are trying to maximize. Game theory assumes he knows them in full. Put another way, each player is assumed to know the preference patterns of the other players.
This, and the kindred assumptions about his ability to perceive the game situation, are often subsumed under the phrase "the theory assumes rational players." Though it is not apparent from some writings, the term "rational" is far from precise, and it certainly means different things in the different theories that have been developed. Loosely, it seems to include any assumption one makes about the players maximizing something, and any about complete knowledge on the part of the player in a very complex situation, where experience indicates that a human being would be far more restricted in his perceptions. The immediate reaction of the empiricist tends to be that, since such assumptions are so at variance with known fact, there is little point to the theory, except possibly as a mathematical exercise. We shall not attempt a refutation so early, though we feel we have given some defense in later chapters. Usually added to this criticism is the patient query: why does the mathematician not use the culled knowledge of human behavior found in psychology and sociology when formulating his assumptions? The answer is simply that, for the most part, this knowledge is not in a sufficiently precise form to be incorporated as assumptions in a mathematical model. Indeed, one hopes that the unrealistic assumptions and the resulting theory will lead to experiments designed in part to improve the descriptive character of the theory.
In summary, then, one formulation of a class of conflicts of interest is this: There are n players each of whom is required to make one choice from a well-defined set of possible choices, and these choices are made without any knowledge as to the choices of the other players. The domain of possible choices for a player may include as elements such things as "playing an ace of spades" or "producing tanks instead of automobiles," or, more important, a strategy covering the actions to be taken in all possible eventualities (see below). Given the choices of each of the players, there is a certain resulting outcome which is appraised by each of the players according to his own peculiar tastes and preferences. The problem for each player is: what choice should he make in order that his partial influence over the outcome benefits him most? He is to assume that each of the other players is similarly motivated. This characterization we shall come to know as the normalized form of an n-person game. Two other forms—the extensive and the characteristic function form—will play important roles in our subsequent discussion; but there is no need to go into them now.
1.4 EXAMPLES OF CONFLICT OF INTEREST
Next, we should consider what significant problems of conflict of interest are included in this formulation. Our brief examination will cite examples in each of three areas: economics, parlor games, and military situations. From these it is easy to generate analogous examples for other substantive disciplines.
One basic economic situation involves several producers, each attempting to maximize his profit, but each having only limited control over the variables that determine it. One producer will not have control over the variables controlled by another producer, and yet these variables may very well influence the outcome for the first producer. One may object to treating this as a game on the grounds that the game model supposes that each producer makes one choice from a domain of possible choices, and that from these single choices the profits are determined. It seems obvious that this cannot be the case, else industry would have little need for boards of directors and elaborate executive apparatus. Rather, there is a series of decisions and modifying decisions which depend upon the choices made by other members of the economy. However, in principle, it is possible to imagine that an executive forsees all possible contingencies and that he describes in detail the action to be taken in each case instead of meeting each problem as it arises. By "describe in detail" we mean that the further operation of the plant can be left in the hands of a clerk or a machine and that no further interference or clarification will be needed from the executive.
For example, in the game ticktacktoe, it is perfectly easy to write down all the different possible situations which may arise and to specify what shall be done in each case (and for this reason adults consider it a dull game). Such a detail specification of actions is called a (pure) strategy. There is, of course, no reason why the domains of action need be minor decisions; they may have as elements the various pure strategies of the players. Looked at this way, a player chooses a strategy that covers all possible specific circumstances which may arise. For practical reasons, it is generally not possible to specify economic strategies in full, and as a result a business strategy is usually only a guide to action with respect to pricing, production, advertising, hiring, etc., which neither states in detail the conditions to be considered nor the actions to be taken. The game theory notion of strategy is an abstraction of this ordinary concept in which it is supposed that no ambiguity remains with respect to either the conditions or the actions. With this concept one apparent difficulty in applying the game theoretic model to economic problems evaporates. The notion of a pure strategy, and some related concepts, will receive considerably more discussion in Chapters 3 and 4.
Excerpted from Games and Decisions by Robert Duncan Luce, Howard Raiffa. Copyright © 1957 John Wiley and Sons, Inc.. Excerpted by permission of Dover Publications, Inc..
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Posted December 5, 2009
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