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Game theory has brought into eco nomics an uncertainty principle similar to the one brought into physics by the quantum theory. Professor Borch's main purpose in this book is to show how one can construct realistic eco nomic theories by taking uncertainty into account instead of assuming it away. He brings together a number of recent developments in different fields of economics and other social sciences in which uncertainty and chance play a central role and shows that they really constitute an entity. These de velopments include the revisions of utility, the introduction of game theory as a basic tool, and the view of the foundations of probability theory adopted by the modem Bayesian statis ticians. Woven together these elements form the fabric of an extremely reada ble book, enlivened and illuminated by many simple and original examples. It does not presuppose more than two or three years of college mathematics. KARL HENRIK BORCH is Pro fessor of Insurance, The Norwegian School of Economics and Business Administration, Bergen, Norway. Princeton Studies in Mathematical Economics, No. 2 THE ECONOMICS OF UNCERTAINTY PRINCETON STUDIES IN MATHEMATICAL ECONOMICS EDITED BY OSKAR MORGENSTERN, HAROLD W. KUHN, AND DAVID GALE 1. Spectral Analysis of Economic Time Series, by C.W.J. Granger and M. Hatanaka 2. The Economics of Unce~inty, by Karl Henrik Borch 3. Production Theory and Indivisible Commodities, by C.R. Frank, Jr. 4. Theory of Cost and Production Functions, by R. W. Shephard THE ECONOMICS OF Uncertainty - BY KARL HENRIK BORCH Princeton, New Jersey Princeton University Press Copyright© 1968 by Princeton University Press L.C. Card: 68-10503 ISBN 0-691-04124-S Publication of this book has been aided by the Whitney Darrow Publication Reserve Fund of Princeton University Press Printed in the United States of America by Princeton University Press Second Printing, 1969 Third Printing, 1972 All rights reserved. No part of this book may be repro duced in any form or by any electronic or mechanical means including information storage and retrieval sys tems without permission in writing from the publisher, except by a reviewer who may quote brief passages in a review. Preface At the end of World War II, Field-Marshal Wavell published an anthology of poems, which he called Other Men's Flowers. This would have been a suitable title for this book if it did not create the impression that economists do not take themselves seriously. To prepare this book, I have freely picked the results of other people when they seemed beautiful to me, and when I could fit them into an arrangement which I hoped would be attractive. Not being Japanese, I will not claim that flower arrangement is an art, and I find it difficult to argue that it is useful. I hope, nevertheless, that this book will be useful to stu dents of economics. I hope the book will enable them to see that a number of recent results from very different fields of economics-and from other social sciences-really constitute an entity. These results are impressive and often of a startling nature. They seem to fascinate many young economists and, to use another botanical metaphor, make them see the trees instead of the wood. The purpose of this book is to try to convince the potential reader that, although the trees may be extremely interesting, it is the beautiful wood which is really important. We live in it. It is impossible to prepare a book of this kind, without incurring a heavy debt to a great number of colleagues. The debt is obvious where I quote the work of others. The debt is real, but less obvious, in many other places. There is an oral tradition in the teaching of the subject of this book. This tradition consists of a number of examples, counter-examples, and para doxes which have been created by ingenious teachers, but which too often are taken as part of the common knowledge. I have drawn heavily on this tradition, and I find it impossible to mention by name all those who should be mentioned and to acknowledge my particular debt to them. I must, however, express my particular thanks to Professor Oskar Morgenstern. Without his constant prodding and encouragement over a number of years, this book would never have been written. It is also his work which led to the breakthrough which opened the way to the results which I have tried to survey and bring together in this volume. The others who had deserved a special expression of thanks, will, I be lieve, acknowledge their own debt to Professor Morgenstern, and I hope they will not be offended when I single him out in this way. The book has found its present form after a number of years, and is the outcome of lectures I have given to graduate students at The Norwegian School of Economics and Business Administration in Bergen, The Institute c J v Preface of Advanced Studies in Vienna, and The Graduate School of Business Administration of the University of California in Los Angeles. I probably owe a greater debt than I realize to these students. Their questions and com ments have certainly helped to clear up a number of points which I fear must have been obscure when I first presented them. Karl Borch Bergen, August 1966 [ Vl ] Contents Preface v I The Economics of Uncertainty 3 II Economic Decisions under Uncertainty 11 III The Bernoulli Principle 23 IV Applications of the Bernoulli Principle 34 v Portfolio Selection 47 VI The Bernoulli Principle- Observations and Experiments 62 VII Decisions with Unknown Probabilities 77 VIII Market Equilibrium under Uncertainty 88 IX The Two-person Zero-sum Game 109 x The General Two-person Game 129 XI Elements of the General Game Theory 150 XII The Objectives of the Firm 166 XIII Survival and the Objectives of the Firm 181 XIV Credibility and Subjective Probabilities 202 xv Group Decisions 214 Index 225 J [ vii