Matrix Games, Programming, and Mathematical Economics. Mathematical Methods and Theory in Games, Programming, and Economics

427 Pages·1959·22.31 MB·English

Mathematical
in Games, Programming,
Volume I
MATRIX GAMES, PROGRAMMING,
AND MATHEMATICAL ECONOMICS
Methods and Theory
and Economics
by
SAMUEL KARLIN
Stanford University
ΎΎ
PERGAMON PRESS
LONDON-PARIS
1959
ADDISON-WESLEY PUBLISHING COMPANY, INC.
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Copyright © 1959
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PREFACE
The mathematics involved in decision making is playing an increasingly
important role in the analysis of management problems, in economic
studies, in military tactics, and in operations research. These areas of
application have led to challenging new types of mathematical structures.
The development of game theory, linear and nonlinear programming, and
mathematical economics marks one significant aspect of this kind of
mathematics.
Essentially, these volumes are an attempt at a preliminary synthesis of
the concepts of game theory and programming theory, together with the
concepts and techniques of mathematical economics, into a single sys­
tematic theory. Secondarily, I hope they will prove useful as textbooks
and reference books in these subjects, and as a basis for further study
and research.
The coverage of these volumes is in no sense exhaustive; I have stressed
most what appeals to me most. At the same time, I have tried to supply
the necessary routine and basic information on which a proper under­
standing of game theory and programming depends. The presentation of
each subject is designed to be self-contained and completely rigorous, but
every effort has been made to bring out the conceptual depth and formal
elegance of the over-all theory. In both volumes, the principles of game
theory and programming are applied to a variety of simplified problems
based on economic models, business decision, and military tactics in an
attempt to clarify the key mathematical concepts involved and suggest
their applicability to other similar problems.
Each chapter includes a certain amount of advanced exposition, which
is generally incorporated in starred (*) sections. Elementary background
material is presented in the appendixes, and more advanced background
material is incorporated directly in the text.
Each chapter contains problems of various degrees of difficulty, many
of them leading to extensions of the theory. Solutions to most of the
problems and hints for solving others are given at the end of each Part.
The three Parts (Parts I and II in Volume I, and Volume II, which
comprises Part III) are independent of each other except for the material
in starred sections, and may be studied independently in conjunction
with the relevant appendixes. For the readers convenience, a diagram
showing the interrelationships of the various nonadvanced portions of
Parts I and II of Volume I is presented on the following page.
Each chapter ends with a section of historical remarks, general comments
on technical details, and citations of books and articles that may profitably
be consulted for further information. All references cited are listed in the
v
VI PREFACE
bibliography at the close of each book. In some instances the historical
remarks consist in assigning priorities to notable discoveries; any inac­
curacies or omissions along these lines are wholly unintentional and
deeply regretted.
My indebtedness extends to many, and not least to Stanford University,
California Institute of Technology, the RAND Corporation, and the
Office of Naval Research, all of which provided facilities, intellectual
stimulation, and encouragement without which these books might never
have been written.
Among my academic colleagues, I am grateful to Professors Bowker,
Lieberman, Madow, Parzen, and Scarf of Stanford for their constant en­
couragement; to Professor H. F. Bohnenblust of California Institute of
Technology, who introduced me to this field of study and who taught me
more than I know how to acknowledge; to my students Rodriguez Restrepo
and R. V. Miller, who wrote up my initial lectures on which the material
of these books is based; to my students William Pruitt and Charles Stone,
each of whom made valuable suggestions for organizing and improving
the final manuscript; to Professors Hirofumi Uzawa and K. J. Arrow, who
taught me most of what I know about mathematical economics; to my
friends Melvin Dresher, Ray Fulkerson, and Harvey Wagner for their
helpful comments on the first six chapters; and to Irving Glicksberg for
assistance in developing the appendixes.
Finally, I am indebted to J. G. Bell, Jr., for helping to straighten out
my syntax; to Frieda Gennerich, Sally Morgan, and Babette Doyle for
their superb technical typing; and above all to my wife, who has offered
warm encouragement, unbounded patience and kind consideration during
these trying years of writing.
SAMUEL KARLIN
Stanford, California
August 1959
LOGICAL INTERDEPENDENCE OF THE VARIOUS CHAPTERS
{applies only to unstarred sections of each chapter)
PART I PAR! 1 II
Chap. 1 Chap. 5
/ \
Chap. 2 Chap. 6 Chap. 7
/ \ 1
Chap. 3 Chap. 4 Chap. 8
1
1
Chap. 9
INTRODUCTION
THE NATURE OF THE MATHEMATICAL THEORY
OF DECISION PROCESSES
Section 1. The background. The art of making optimal judgments
according to various criteria is as old as mankind ; it is the essence of every
field of endeavor from volleyball to logistics. The science of making such
judgments, as opposed to the mere art, is a newer development, and the
mathematical theory of decision-making is newer still—only a little over
twenty years old. It has grown fast. Particularly in this last decade, we
have seen a rapid spread of the use of scientific disciplines in economic
planning, military strategy, business management, and systems engineer­
ing. There has been a steady interplay between the more careful formula­
tion of business and logistics problems and the development of the mathe­
matical tools needed for their solution. Along with and partly as a result
of advances in game theory, linear programming, and queueing theory,
there have grown up new practical disciplines whose very names were un­
known forty years ago : operations research, management science, systems
analysis, and a score of others. All these disciplines are concerned with the
same fundamental task, that of scientifically analyzing various possible
courses of action with a view to determining which course is "best" under
the circumstances.
What a decision problem is must be clearly defined. Specifically, the
possible courses of action, their consequences, the objectives of the various
participants, and the nature of the random effects must be ascertained.
From here on, the theory tries to evaluate and make comparisons of the
various alternatives of action in terms of the objectives. A decision
problem typically has four parts: (1) a model, expressing a set of assumed
empirical relations among the set of variables; (2) a specified subset of
decision variables, whose values are to be chosen by the firm or other
decision-making entity; (3) an objective function of the variables, formu­
lated in such a way that the higher its value, the better the situation is
from the viewpoint of the firm ; and (4) procedures for analyzing the effect
on the objective function of alternative values of the decision variables.
The main component of a decision problem is the model. The following
definition may help explain the significance and scope embodied in this
concept. A model is a suitable abstraction of reality preserving the essen­
tial structure of the problem in such a way that its analysis affords insight
into both the original concrete situation and other situations which have
the same formal structure.
1
2 INTRODUCTION
Model building has become increasingly widespread in these past years,
particularly in the social sciences. Modern psychologists approach the
study of learning processes in terms of a model involving stochastic
variables. Econometricians, drawing on game theory and programming
theory, have developed global equilibrium models in an attempt to under­
stand the relationship of consumption, production, and price units in a
competitive economy. Some political scientists are using the concepts of
n-person game theory in an effort to analyze the power relationships that
govern the decisions of legislative bodies. More and more business firms
are applying rigorous mathematical thinking to management problems,
inventory problems, production problems. Engineers are engaged in in­
tensive studies in the new field of systems analysis, whose main technique
consists of simulating idealized models of a given operation. The basic
mathematics used in these studies is a mixture of the mathematical con­
cepts and thinking processes of statistics, probability, and general de­
cision theory.
The past fifty years have been years of fruitful fermentation in the
disciplines of probability, stochastic processes, and statistics on the one
hand, and have witnessed the development of game theory and linear
programming on the other. These mathematical disciplines are the natural
tools for the analytical investigation of problems arising in the social and
managerial sciences. In this respect it is interesting to compare briefly
the usefulness of probability, statistics, and decision theory with that of
the other types of pure and applied mathematics.
The development of classical mathematics was principally inspired by
problems based on phenomena in the fields of physics and engineering.
In the usual classical physics problem, the variables of the system are seen
to satisfy a system of well-defined deterministic relations. All parameters
of the problem can be assumed known or accurately measurable, and one
seeks to determine the relations between them in order to predict exactly
what will happen in similar or analogous circumstances in the future.
Until recently scientists were primarily concerned with just such determin­
istic problems, problems that could be solved, by and large, by the methods
of partial differential equations and related mathematical techniques.
Clearly, however, these deterministic concepts and techniques are not
altogether suitable for the social sciences, which typically present problems
involving uncertainties and variability. Whenever human beings play a
crucial role in the formulation of a model, we must allow for the unknown
and sometimes random character of human actions, and for the randomness
inherent in the environment in general, if we are to arrive at meaningful
results. It is therefore inevitable that students of human behavior should
be concerned prominently with the theories of probability, statistics, and
decision analysis of the kind embodied in game theory and linear pro­
gramming.
INTRODUCTION 3
The reverse is also true: mathematicians and statisticians are increas­
ingly concerned with the analytical problems originating from the analysis
of human behavior. It is now no longer exclusively the physical sciences
and engineering, but also the social and management sciences, that signal
promising new directions for mathematical research.
There are two aspects to this research : the construction of the model, in
which probability and stochastic processes play a fundamental role, and
the normative essence of the model (optimization), the province of statis­
tics, game theory, decision theory, and operations research. In practice
these two aspects are inseparable. Obviously, to understand how to
optimize the objective function, it is necessary to understand the structure
and mechanism of the model; and to construct a model properly, it is
necessary to understand what use is to be made of it.
In order to properly delimit the scope of these researches, we also men­
tion the descriptive significance in the study of mathematical model
building. This is in contrast to the normative essence of the model which
we have emphasized. From the standpoint of mathematical models in
psychology and sociology, the bulk of current literature is concerned with
the orientation of the model toward empirical adequacy. On the other
hand, the model based on situations in management, economics, and
statistics is more concerned with the kind of normative questions of
rational behavior theory (game and decision theory) to which we devote
our main attention.
Inevitably the techniques of probability, statistics, and decision theory
interact, and developments in each lead to developments in the others.
As a result, the determination of optimum procedures for decision problems
must be considered not in terms of half a dozen, or three, or two more or
less separate sets of techniques, but in terms of a single over-all analytical
methodology.
It is this methodology, in the context of game theory, programming,
and mathematical economics, that we seek to delineate in this book. We
may summarize our aims as an attempt to unify the mathematical tech­
niques of the field and to help crystallize the concepts underlying these
kinds of decision problems. It is not our purpose to propose a formal
structure which will encompass all problems in the areas covered. We
do not concentrate on definitions and abstractions per se. Rather, we
attempt to develop in detail the principal mathematical methods that are
applicable, putting emphasis wherever possible on mathematical proto­
types and their analyses.
Section 2. The classification of the mathematics of decision problems.
Our purpose in this section is to describe various useful mathematical
classifications of decision problems that are also applicable to the theory
of games, programming, and mathematical economics.
4 INTRODUCTION
I. Decision problems can be divided into two contrasting types—static
problems and dynamic problems. A static decision problem is one in which
the variables do not involve time explicitly. A dynamic model is one in
which time plays a very decisive role.
The distinction between static and dynamic problems is not always
clear. Often a static situation is a stationary state of a dynamic situation,
i.e., an equilibrium phase achieved by a dynamic process that has been in
operation over a long period of time. Or an ostensibly dynamic process
can be regarded as static, as when the same variables are introduced at the
successive time points as new variables. Nevertheless there is an intrinsic
difference in concept and technique between the two kinds of problems
which will be manifest throughout the detailed analysis.
The principal technical difference between dynamic (multistage) models
and static models is the degree of complexity involved in describing a given
strategy or procedure. In a static situation a strategy is selected once and
for all to be carried out directly, whereas the strategies available in the
dynamic situation are usually complicated functions of information re­
ceived and actions undertaken in the preceding stages. Theoretically, by
defining a strategy suitably, we can reduce any dynamic model to a static
model. In practice, however, this operation so complicates the nature of
the strategy that it is often impossible to analyze the problem in the result­
ing terms. The essence of the analysis of a dynamic problem seems to be
the exploitation of the recursive, time-dependent nature of the problem
and the symmetries made available by the infinite horizon of time.
The significance of dynamic decision problems is obvious, since almost
all human activity takes place in time. Events and learning processes are
sequential. Most business decisions involve a time element. Economists
try to take account of dynamic fluctuations in time. Nonetheless, the
recognition and mathematical analysis of dynamic decision problems are
of recent origin. A notable innovator in this regard was Wald, whose
Sequential Analysis of Statistical Data [244]* contained the essence of the
notion of sequential decision-making for the accumulation of information.
Other more special contributors to the sequential decision problem include
Bellman [19], who saw the generality of the sequential idea and used it in
the formulation of several multistage management decision models, and
Arrow, Harris, and Marschak [8], who cast a type of inventory decision
problem in dynamic form. The theory of multistage games is an extension
of the usual theory of games to a dynamic situation. A recent volume [72]
shows the importance of this development.
The study of static models serves two purposes. First, a static model
may be the best way of describing a static problem whose solution is inter-
* References are to the Bibliography.
INTRODUCTION 5
esting in itself and affords insight into the workings and structure of a
given operation. One common static problem of this sort is that of optimiz­
ing a given criterion among different procedures where the variables range
over a prescribed set in which the time variable is not a special variable.
Second, from a technical point of view, many dynamic models must be
approached through the study of modified static problems. For example,
it is often convenient to reduce the infinite version of the dynamic problem
to the case of a finite number of periods N. The solution of the resulting
problem is obtained by a simple extension of the static model, after which
a passage to the limit on N produces the solution of the full dynamic
problem.
II. Decision problems can also be classified in terms of the number of
antagonistic participants involved. We note three examples.
(1) If there are at least two irreconcilable participants, we are dealing
with problems in the theory of games. Where there are just two such
opponents, their objectives are clear: each strives to achieve as great a
return as possible, knowing that his opponent will strive for the same ob­
jective with all his ability and intelligence. Where there are more than two,
their objectives ate not so clear. The conceptual framework of this model
has not yet been well established, and therefore it is difficult to develop
appropriate mathematical methodology for it.
(2) Where there is only one participant involved and his objectives are
clearly defined, the basic decision problem reduces technically to a strict
maximization of the objective functions subject to the natural constraints
of the model. Generally, this situation leads to variational problems, with
the solution usually occurring on the boundaries of the domain of the
variables. To solve such extreme constrained problems entails developing
new techniques. The fields of linear and dynamic programming abound
with models of this kind.
(3) Where there is one participant and uncertainties are present, a
solution can usually be obtained by combining the methods of statistics
with variational techniques.
III. A third breakdown of the general decision model is into determin­
istic and stochastic cases. Naturally the deterministic models are con­
ceptually and technically much simpler and more frequently amenable to
thorough analytical treatment. However, the majority of practical prob­
lems involve uncertainties, variability, and all kinds of randomness. To
solve problems of this sort requires a thorough acquaintance with the
theory of stochastic processes and the corresponding theory of inference
in stochastic processes.
IV. Another classification can be made in terms of the complexity of
the available strategies (the strategy space). Strategy spaces can be divided
into two categories: finite-dimensional and infinite-dimensional. In many