eBook
Available on Compatible NOOK devices, the free NOOK App and in My Digital Library.
Related collections and offers
Overview
N-person game theory provides a logical framework for analyzing contests in which there are more than two players or sets of conflicting interests-anything from a hand of poker to the tangled web of international relations. In this sequel to his Two-Person Game Theory, Dr. Rapoport provides a fascinating and lucid introduction to the theory, geared towards readers with little mathematical background but with an appetite for rigorous analysis.
Following an introduction to the necessary mathematical notation (mainly set theory), in Part I the author presents basic concepts and models, including levels of game-theoretic analysis, individual and group rationality, the Von Neumann-Morgenstern solution, the Shapley value, the bargaining set, the kernel, restrictions on realignments, games in partition function form, and Harsanyi's bargaining model. In Part II he delves into the theory's social applications, including small markets, large markets, simple games and legislatures, symmetric and quota games, coalitions and power, and more.
This affordable new edition will be welcomed by economists, political scientists, historians, and anyone interested in multilateral negotiations or conflicts, as well as by general readers with an interest in mathematics, logic, or games.
Product Details
| ISBN-13: | 9780486143675 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 05/20/2013 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 332 |
| File size: | 8 MB |
Read an Excerpt
N-Person Game Theory
Concepts and Applications
By Anatol Rapoport
Dover Publications, Inc.
Copyright © 1970 University of MichiganAll rights reserved.
ISBN: 978-0-486-14367-5
CHAPTER 1
Levels of Game-theoretic Analysis
The origins of game theory stem from concerns related to rational decisions in situations involving conflicts of interest. The term game theory itself derives from the analysis of so-called games of strategy such as Chess, Bridge, etc. Serious research in this area was doubtlessly stimulated by a need to bring to bear the power of rigorous analysis on problems faced by persons in the culturally dominant roles of "decision makers." That connections between games of strategy and strategic conflict already exist in the minds of men of affairs appears in the metaphors linking the languages of business, international relations, and war; in short, spheres of activity where strategic shrewdness is assumed to be an important component of competence.
Thus it is easy to portray game theory as an extension of a theory of rational decisions involving calculated risks to one involving calculations of strategies to be used against rational opponents, competitors, or enemies; that is, actors who are also performing strategic calculations with the aim of pursuing their goals and, typically, attempting to frustrate ours.
In short, the game metaphor (business is a game, life is a game, politics is a game) is already firmly established among people whose careers depend on the choice of right decisions and among those who have an appreciation of this process. Consequently, the above mentioned definition of game theory is easily related to what people already know, understand, and appreciate.
Unfortunately, inferences likely to be made from the definition easily lead to a misconception about the scope and the uses of game theory. The widespread appreciation of the decision maker's role makes it easy to put oneself into his shoes. If I were a decision maker (one might, perhaps, ask oneself), what would I expect from a theory which purports to deal with rational decisions in conflict situations? Clearly, I would expect from such a theory some indications of how rational decisions are to be singled out from all the available ones. Certain features of rational decisions follow from common sense considerations. One must know the range of the possible outcomes which can result as consequences of one's own choices and also of choices made by others, as well as, perhaps, certain chance events. Having made a list of these outcomes, one must know one's own order of preference among them and, as appears after a moment of reflection, also the orders of preference of other decision makers, who also exercise partial control over the outcomes by their own choices. Next, one must distinguish between immediate, intermediate, and final outcomes of decisions. Often it is difficult to decide what the preferences are (even one's own) with regard to the immediate and intermediate outcomes. For these can be evaluated only with reference to the final outcomes, to which they eventually lead, via additional choices of all concerned; and the relation between the former and latter is often not clear.
Anyone who has reflected on problems of decision making in conflict situations will usually quickly grasp the significance of these issues. Namely, these issues must be faced if game theory is to be a useful tool in the search for rational decisions. At this point, the game theorist who has undertaken to explain what he is about faces the difficult task of shifting attention away from these issues, which are not the central ones in game theory, toward other much more fundamental issues. He will have to explain the difference between the theory of some specific conflict situation and a general theory of such situations.
Let us first see what might be the shape of a theory of a specific, strictly formalized conflict situation, say a game of Chess. In Chess, the immediate outcomes of choices are "positions," i.e., the dispositions of the pieces on the board. These positions are controlled partially by one player, partially by the other, specifically, by the choices which the players make alternately. The positions must be evaluated only with reference to the bearing they have on the final outcome of the game; that is, on whether the outcome is a win for White, a draw, or a win for Black. The preference order of the players among these outcomes is clear. White prefers them in the order named; Black prefers them in the opposite order. Thus the only real problem the chess player faces is that of estimating the bearing which the immediate and the intermediate situations (the "positions") have on the final outcome.
Indeed, the theory of Chess deals entirely with this problem. In the theory of Chess, certain frequently occurring situations are analyzed with a view of deciding to which of the three possible final outcomes they are likely to lead. Sometimes analysis yields a definitive answer. It is known, for example, that if all the pieces have been captured except one rook, then the possessor of the rook must win the game provided he guides his choices by certain specified rules. Definitive prognoses can be made also in other more complex "end game" situations (i.e., when only a few pieces and pawns remain). Prognoses on the basis of situations arising in the beginning or the middle of the game are, as a rule, not definitive. But this is because the analysis of such situations is too complex to allow the investigation of all possible lines of play. Nevertheless, some prognoses can be made with considerable confidence on the basis of several centuries of experience. Other prognoses are noncommittal except to the extent of assertions like "White (or Black) has a better position." These too are based on intuitive judgments derived from experience of able players.
Chess theory is, then, essentially concerned with the search for "effective strategies," i.e., with the search for choices which are "likely" to lead to definitely winning positions or to prevent the opponent from achieving them. It is important to note that "likely" in this instance is not to be understood in the sense that Chance intervenes in the development of the situations the way it does in games of chance. Chance has "nothing to say" about what positions will emerge in a game of Chess, since each position results from a deliberate choice by a player. The term "likely" is unavoidable in the formulation of Chess theory, because the situations are too complex to be analyzed in their entirety. It is conceivable that, if a complete analysis were carried out, the outcome of every Chess situation could be predicted with certainty (assuming that such analysis could be carried out by both players). In fact the end game situations mentioned above are precisely such; and so are Chess problems, whose solution depends on complete analysis. Such situations can be analyzed completely, and for this reason, when they arise in games played by experienced players, the game is broken off, the outcome being known to both. Therefore it makes sense to conceive of "progress" of Chess theory in terms of subjecting more and more situations to deeper and deeper analysis. It follows that the Chess player able to pursue such analysis more deeply acquires thereby a greater control over the situations and is in a better position to win the game.
All this makes good sense to a decision maker faced with choices in situations involving a conflict of interest. He may be well aware that the situations with which he is faced are far from being as clear cut as a game of Chess. The range of alternative choices (especially those available to an opponent or opponents) may not be known with certainty, nor the opponents' preference orders for the outcomes. It may be difficult to decide whether situations are to be assigned the status of intermediate or final outcomes and so to decide whether they are to be evaluated as "means to an end" or as "ends," etc. Nevertheless, a decision maker sophisticated in the ways of science (where concrete problems must always be translated into simplified or idealized "models" and where hypothetical assumptions must always be made because knowledge is never complete) can conceive of conflict situations which, under certain conditions, can be formulated as well-defined "games." If so, he may be willing to examine the sense in which game theory can be relevant to the problem of choosing rational decisions.
After all the preliminary conditions have been fulfilled, what is the problem from the decision maker's point of view? It is that of pursuing analysis sufficiently far so as to single out strategic choices which will either bring about the preferred outcomes or are "likely" to bring them about ("likely" in the estimation of persons with experience in similar situations).
In other words, it seems to the decision maker quite natural to see the task of game theory as a generalization of the task posed by the theory of Chess: the search for effective strategic decisions (after the problem has been sufficiently defined). Therein lies a misconception, because the search for effective decisions is not a central problem of game theory.
Game theory (as developed by people who have come to be recognized as game theorists) is properly a branch of mathematics. To the extent that many problems of mathematics are (or, at least, have been) instigated by abstractions from real life situations, game theory, too, can be so viewed. However, the mathematicians' research tools are different from those of the natural scientist (who deals with the world perceived by the senses). Consequently the problems usually posed by the mathematician are typically not at all the problems posed by "life," although they may have been instigated by impressions gathered from life.
The mathematician pursues his science by ascending to ever higher levels of generality, hence of abstraction. It seems natural to suppose that by solving a "more general class of problems" than those originally posed, the mathematician is thereby enabled to solve also the original problems; for does not the general case embody the special case? It does indeed happen that by ascending to a higher order of abstraction the mathematician is enabled to solve the problem originally posed and all the other problems of the same type. Frequently, however, this change of perspective has a different consequence, namely, the abandonment of a class of problems in favor of another class which may never have arisen in the original context.
As an elementary example, consider the problem of solving an equation with one unknown. At first, solutions of special cases of such equations were found; then a general method of solving such problems. Using this method, one could, of course, solve any special case.
However, in the process of establishing a method of solving linear equations in the first degree, certain problems appeared which had nothing to do with the problems that initiated this search. A special problem of solving a first degree equation might have arisen in the context of searching for a specific number among a set of known numbers. For example, a money changer, in converting one currency to another and charging a fixed fee for his service, might want to solve a linear equation. If he sells shekels at the rate of 350 minas per shekel and charges 4 minas for performing the transaction, he may want to know how many shekels he should give for 1000 minas. The equation to solve for x is
350x +4 = 1000. (1.1)
The problem "has an answer" in terms of the operations which the money changer habitually performs, because problems "without an answer" do not as a rule arise in this context.
The mathematician, however, having posed the problem of solving the general linear equation will very quickly come across problems which have "no solutions" in the hitherto conventional sense. In order to make such problems solvable (so as to "round out" the theory), the mathematician invents new number entities. In the context of making all linear equations solvable these are negative numbers. Once this is done, new questions arise of purely mathematical nature; for instance, how are the usual operations of arithmetic to be extended so as to include negative as well as positive numbers? To be sure, contexts in which negative numbers acquire meaning were not long in appearing, e.g., credit accounting. Nevertheless it is quite possible that the concept of negative numbers arose in the purely mathematical context before this concept was put to work in "real life" applications.
The importance of the purely mathematical context becomes much more important in the theory of the quadratic equation. Here "answers" occur which cannot possibly arise in the context of either counting or physical measurement, namely, irrational and imaginary numbers. Hence a large part of the theory of quadratic equations is concerned with matters other than finding "desired magnitudes" as answers to problems posed by life. For instance, the question of when a quadratic equation has rational roots, real roots, two distinct roots, etc., can be asked quite apart from the problem of finding these roots.
As the restriction on the degree of an algebraic equation is removed, the theory becomes even more general. A part of the theory of equations is concerned with certain mathematical systems (called groups) in which the elements (that is, the entities operated upon) are no longer numbers but are themselves operations (called automorphisms). This branch of mathematics later turned out to have extensive applications, for example in crystallography and in quantum mechanics. However, in the process of its development, the original problems which have given impetus to the theory of equations (from which group theory arose) were completely lost sight of. That is, investigations in group theory have next to nothing to do with the solution of algebraic equations in the sense of finding magnitudes which satisfy it. For this reason, the term "theory of equations" may be misleading to someone whose attention is riveted on the original "practical" problems, in the context of which the beginnings of the theory were rooted. Nevertheless the general theory of equations sheds a brilliant light on the "nature" of algebraic equations and brings into focus certain of their aspects which are fundamental in many different branches of mathematics.
To give another even simpler example, the desk calculator was invented for the purpose of quickly adding long columns of whole numbers. It became the ancestor of the electronic computers. A branch of mathematics eventually was developed which deals with the design of such computers. A specialist in this science has neither a special competence nor the slightest interest in adding long columns of figures.
So it is with game theory. The preoccupations of game theorists have next to nothing to do with the problem of finding "effective strategies" in conflict situations. They have to do with matters which shed light on the "logic" of such situations. This logic turns out to be intricate and often perplexing, at times ridden with paradoxes, which, when resolved, provide us with insight concerning matters which had been either ignored or only vaguely understood.
Like all other branches of mathematics, game theory grew by progressive generalization, hence abstraction. Three levels of abstraction are clearly discernible: the theory of games in extensive form, the theory of games in normal form, and the theory of games in characteristic function form.
The fundamental "mathematical object" in the theory of games in extensive form is the so-called game tree. The game tree is determined by the rules of the game. If the game is to be represented by a game tree, the following must be specified in the statement of the rules.
1. A set of players.
2. A set of alternatives open to each player when it is his turn to make a choice among such alternatives (i.e., when it is his move). This set of alternatives will usually depend on the situation, which, in turn, is determined by the choices already made by all the players on their respective moves.
3. A specification of how much a player can know (when it is his move) about the choices already made by the players on previous moves.
4. A termination rule indicating situations which mean that the game is over.
5. A set of payoffs (one to each player) associated with every outcome of the game, the outcome being the situation in which the game has terminated.
(Continues...)
Excerpted from N-Person Game Theory by Anatol Rapoport. Copyright © 1970 University of Michigan. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
Table of Contents
Introduction: Some Mathematical ToolsPart I. Basic Concepts
1. Levels of Game-theoretic Analysis
2. Three-level Analysis of Elementary Games
3. Individual and Group Rationality
4. The Von Neumann-Morgenstern Solution
5. The Shapely Value
6. The Bargaining Set
7. The Kernel
8. Restrictions on Realignments
9. Games in Partition Function Form
10. N-Person Theory and Two-Person Theory Compared
11. Harsanyi's Bargaining Model
Part II. Applications
Introduction to Part II
12. A Small Market
13. Large Markets
14. Simple Games and Legislatures
15. Symmetric and Quota Games
16. Coalitions and Power
17. Experimetns Suggested by N-Person Game Theory
18. "So Long Sucker" : A Do-it-yourself Experiment"
19. The Behavorial Scientist's View
20. Concluding Remarks
Notes
References
Index