Decision Synthesis: The Principles and Practice of Decision Analysis

DECISION SYNTHESIS The principles and practice of decision analysis

DECISION SYNTHESIS The principles and practice of decision analysis

STEPHEN R. WATSON University of Cambridge, Department of Engineering

DENNIS M. BUEDE Decision Logistics, Reston, Virginia

CAMBRIDGE UNIVERSITY PRESS

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1987 First published 1987 Reprinted 1989, 1994 British Library cataloguing in publication data Watson, Stephen R. Decision synthesis: the principles and practice of decision analysis i. Decision-making. I. Title. II. Buede, Dennis M. 8 ' HD30.23 Library of Congress cataloguing in publication data Watson, Stephen R. Decision synthesis. Bibliography. Includes index. 1. Decision-making. I. Buede, Dennis M. II. Title. HD30.23.w38 1987 6$8.4'c>3 87-10270 ISBN 0 521 31078 4 paperback Transferred to digital printing 2003

CE

CONTENTS

List of List of tables Foreword Preface

figures

I INTRODUCTION AND READER'S GUIDE

page x xiii xv xvii I

PART I THEORY 2 RATIONALDECISION-MAKING

2.1 2.2 2.3 2.4

The difficulties of decision-making The meaning of rationality The rise of operational research Requirements for a satisfactory rationality for decisionmaking

3 DECISIONTHEORY

3.1 Conflicting objectives 3.1.1 Utility and value 3.1.2 Value functions 3.1.2.1 Problem formulation 3.1.2.2 Value functions: definition and alternatives 3.1.2.3 Existence of the value function 3.1.2.4 Additive value functions 3.1.3 Summary Exercises for 3.1 3.2 Uncertainty 3.2.1 Descriptions of uncertainty 3.2.2 Theoretical background to subjective probability 3.2.3 Using probabilities 3.3 Risk preference 3.3.1 Utility theory

9

9 11 13 16 l8

18 19 21 21 22 24 25 27 28 28 29 32 34 35 35

vi

Contents 3.3.2 Comparing risky options 3.3.3 Implications of utility theory 3.3.4 Criticisms of utility theory 3.3.5 The value of information Exercises for 3.3 3.4 Risk preference and conflicting objectives 3.4.1 Introduction 3.4.2 Keeney's approach 3.4.3 The Stanford approach 3.4.4 Comments Exercises for 3.4 3.5 Other normative approaches 3.5.1 Introduction 3.5.2 Cost-benefit analysis 3.5.3 The multi-criterion decision-making movement 3.5.4 Social judgment theory 3.5.5 Outranking 3.5.6 The analytic hierarchy process 3.5.7 Conclusions

4 THEPSYCHOLOGY OF DECISION-MAKING

4.1 Introduction 4.2 Uncertainty 4.2.1 Is man an intuitive statistician? 4.2.2 Heuristics and biases in judgments of probability 4.2.3 Calibration 4.3 Expected utility 4.4 Judgment 4.5 Implications for decision synthesis Exercise for 4 5 DECISION ANALYSIS IN ORGANIZATIONS

5.1 Introduction 5.2 Organizational theory and decision-making 5.3 Normative theories for group decision-making 5.3.1 Collective choice 5.3.2 Game theory 5.4 Using decision analysis in organizations 5.4.1 The individual in the organization 5.4.2 Designing programmed decisions 5.4.3 Management decisions in a participative setting 5.4.4 Public policy analysis 5.4.5 Negotiations 5.4.6 Conclusions

38 45 50 53 59 62 62 63 65 67 67 68 68 69 70 72 73 76 78 8l

81 82 82 84 87 90 95 98 100 IOI

101 102 106 106 109 111 112 112 113 115 117 118

Contents

vii

PART II PRACTICE 6 ANALYTIC STRATEGY

6.1 Introduction 6.2 The MODELING strategy 6.2.1 Background and definition 6.2.2 MODELING application to the Mexican electrical system 6.2.3 MODELING application to Xerox consumables 6.2.4 Requirements for success in the MODELING strategy 6.3 The INTROSPECTING strategy 6.3.1 Background and definition 6.3.2 INTROSPECTING strategy for electrical energy production in Wisconsin 6.3.3 INTROSPECTING strategy for transmission conductor selection 6.3.4 Requirements for success with the INTROSPECTING strategy 6.4 The RATING strategy 6.4.1 Background and definition 6.4.2 RATING application to a research program 6.4.3 Requirements for success in the RATING strategy 6.5 The CONFERENCING strategy 6.5.1 Background and definition 6.5.2 CONFERENCING strategy for tanker safety 6.5.3 CONFERENCING strategy for factory location 6.5.4 Requirements for success with the CONFERENCING strategy 6.6 The DEVELOPING strategy 6.6.1 Background and definition 6.6.2 DEVELOPING strategy for resource allocation 6.6.3 Requirements for success with the DEVELOPING strategy 6.j ALLOCATE and NEGOTIATE decision modes 6.8 Summary 7 ANALYTIC TACTICS

7.1 Introduction 7.2 Problem-and model-structuring 7.2.1 Problem-structuring aids 7.2.1.1 The rational manager - Kepner and Tregoe (1981) 7.2.1.2 Van Gundy's shopping list

123

123 127 127 132 133 134 135 135 137 143 144 145 145 146 148 148 148 150 151 154 155 155 157 159 159 160 163

163 163 164 165 165

viii

Contents

7.2.1.3 Report Card 7.2.2 Model-structuring aids 7.2.2.1 Influence diagrams 7.2.2.2 Cognitive mapping 7.2.3 How to get started 7.2.4 Requisite decision-modeling Exercises for 7.2 7.3 Iterative analysis 7.3.1 Four types of iterative analysis 7.3.2 How much analysis is enough? 7.4 Probability elicitation 7.4.1 Research findings 7.4.2 A recommended elicitation process 7.4.3 Discretizing continuous distributions Exercise for 7.4 7.5 Value and utility elicitation 7.5.1 Research 7.5.2 Definitions 7.5.3 Assessing additive value functions 7.5.4 Assessing additive value weights 7.5.5 Constructing hierarchical additive value weights 7.5.6 Assessing utility functions Exercises for 7.5 7.6 Sensitivity analysis Exercise for 7.6 8 APPLICATIONS

8.1 General introduction 8.2 Problem-structuring 8.2.1 Introduction 8.2.2 Hertz and Thomas (1983) — new product and facilities planning 8.2.3 Dyer and Miles (1976) - selection of trajectories 8.3 Model-structuring 8.3.1 Introduction 8.3.2 North and Merkhofer (1976) - analyzing emission control strategies 8.3.3 Flyback and Keeney (1983)- constructing an index of trauma severity 8.4 Iterative analysis 8.4.1 Introduction 8.4.2 Spetzler and Zamora (1983) - facilities investment and expansion

167 168 168 176 179 180 180 181 181 183 184 184 186 189 189 190 190 191 193 200 203 204 206 209 213 215

215 216 216 216 217 226 226 226 232 236 236 237

Contents 8.4.3 Buede and Choisser (1984) — system design 8.5 Probability assessment 8.5.1 Introduction 8.5.2 Howard, Matheson and North (1972) - hurricaneseeding 8.5.3 North and Stengel (1982) - program choices in magnetic fusion 8.6 Value assessment 8.6.1 Introduction 8.6.2 Dyer and Lund (1982) — merchandizing package 8.6.3 Kuskey, Waslov and Buede (1981) - Marine Corps's POM development 8.7 Sensitivity analysis 8.7.1 Introduction 8.7.2 Macartney, Douglas and Spiegelhalter (1984) cardiac catheterization 8.7.3 Kirkwood (1982) - nuclear power plant site selection 9 DECISION SYNTHESIS APPRAISED

9.1 The validation of decision analysis 9.2 Research directions 9.2.1 Research on how decisions are made 9.2.2 Research on the foundations of decision theory 9.2.3 Research on problems of applying decision theory 9.2.4 Research into methods for communicating analyses 9.2.5 Implications for decision synthesis 9.3 The ethics of decision analysis References Index

ix 243 247 247 247 252 256 256 256 260 263 263 263 266 271

271 274 275 276 276 279 279 279 282 295

FIGURES

I.I 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 4.1 4.2 4.3 5.1 5.2 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

Decision synthesis Dominance Decision tree for a simple decision problem The concept of equivalence Equivalence produced by the substitution principle Decision tree for the decision-maker's extended problem Decision-maker's utility function for final asset position Decision tree and utilities for the decision-maker's extended problem Insuree's decision problem Insurer's decision problem A job choice problem Allais's paradox Decision tree to compute the value of perfect information When the value of perfect information is zero Value of imperfect information A set of empirical graphs describing calibration Ellsberg's paradox Choice situations for Tversky and Kahneman's framing experiment The prisoners'dilemma Decision analysis in negotiations Howard's decision analysis cycle Deterministic model Deterministic sensitivity Stochastic analysis Clairvoyance The revised decision analysis cycle A decision analysis model of the Mexican electrical system A decision analysis of Xerox copier consumables Preferential independence Multi-attributed lottery questions

page 4 23 35 38 39 42 43 43 47 48 49 51 54 56 57 88 91 94 no 118 128 128 129 130 131 131 132 134 138 139

Figures 6.11 6.12 6.13 6.14 6.15 6.16 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19 7.20 7.21 7.22 7.23 7.24 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11

The individual utility functions Transmission utility functions Initial value structure Final value structure DSS architecture DSS evolution Problem-solving funnels Definitions used in influence diagrams New information and influence diagrams Uncoalesced decision tree Coalesced decision tree Manipulation forming cycle not permitted Symbols used in an influence diagram Initial beach investment influence diagram Final beach investment influence diagram An exemplary cognitive map Enjoyment attributes Bisection method Decreasing returns to scale Other shapes for value functions Value function for "walking time" Differences method Value function for "time spent" Relative and cumulative weights Risk preference lotteries Risk preference function Stochastic dominance One-way policy region analysis (proportional) One-way policy region analysis (NPV-constrained) Two-way policy region analysis Initial Egg 'N Foam flow chart Egg 'N Foam investment strategies Graph of cumulative probability density functions for each strategy Illustration of total cost comparison Overview of pollution model Total social cost versus pollution cost, existing coal-fired plant (non-urban) Results for existing coal-fired, non-urban plant Results for new coal-fired, non-urban plant Results for old reconverted urban plant Hierarchy of trauma severity concerns Severity adjustment factor due to age

xi 140 143 153 155 156 158 166 169 170 171 171 172 173 175 176 177 192 195 196 197 197 198 200 204 205 205 209 211 211 212 217 218 219 226 227 228 230 231 231 233 235

xii 8.12 8.13(3) 8.i3(b) 8.14 8.15 8.16 8.17 8.18 8.19 8.20 8.21 8.22 8.23 8.24 8.25 8.26 8.27 8.28 8.29 8.30 8.31 8.32

Figures Facilities investment deterministic model Distributions of individual judgments Probability distribution for raw material costs Facilities investment decision tree WWDSA value tree Plot of initial WWDS A evaluation results Plot of intermediate WWDS A results WWDS A iterative evaluation of alternatives Hurricane-seeding decision tree Maximum sustained winds over time Results for nominal hurricane Decision tree structure for magnetic fusion R & D Influence diagram for assessing probability distributions on fusion reactor outcomes The Christmas tree of merchandizing alternatives Structural model of merchandizing alternatives Allocation of sales margins Evaluation of merchandizing strategies (relative to current) Catheterization decision Two-way policy region analysis of catheterization decision Three-way sensitivity analysis of catheterization Decision tree for nuclear plant siting Three-way sensitivity analysis for nuclear plant siting

237 239 239 240 24 3 244 24 5 246 24 8 249 251 253 254 257 258 259 260 264 266 267 268 270

TABLES

6. i 6.2 6.3 6.4 6.5 6.6 7.1 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13

page Attributes for evaluating energy policy 137 Final utility function 141 Expected utilities 144 Initial alternatives 152 Final alternatives 154 Strategy summaries 162 Beach cottage results 210 The MJS77 science investigations 220 Ordinal rankings and cardinal utility function values 222 Collective choice rankings and values 224 Ordinal rankings for preferred trajectory pairs 225 Expected value of resolving uncertainty 232 Trauma concerns and measures 234 Severity adjustment factors due to other diseases 235 Deterministic results for facilities investment 238 Combined effect of discount rate and risk aversion on the certain equivalent 239 Value of perfect information 241 Initial WWDSA alternatives 24 2 Definition of catheterization variables 265 Attributes for nuclear plant siting 269

Xlll

FOREWORD

Decision Synthesis makes up and lives up to its title. The authors have done an exemplary job of piecing together the diverse strands of the emerging, multidisciplinary field of behaviorally oriented decision theory in a way that serves their stated goal - improving decision-making by synthesizing techniques and perspectives with a sensitivity to the peculiarities of individual decision problems and decision-makers. The text draws upon diverse literatures whose creators generally know about one anothers' existence, but never have quite found the time to work out the interrelationships between their respective approaches. Although there are too many of these pairwise relationships for many to be fleshed out in any great detail, seminal ideas can be found at many junctures. As such, it is a book that gives some of the future of decision-making, as well as well-chosen pieces of its past and present. In creating this synthesis, Watson and Buede have adopted a decidedly disorderly approach to the often tidy world of decision theory. They present rudiments of the important normative axiomatizations of formal prescriptive approaches. However, they also caution the reader against taking it all too literally. Following much current thinking, they thoughtfully discuss the limits to imputing descriptive validity to normative theories — that is, assuming that people make decisions the way they are supposed to. Rooted in common sense and psychological research, this consideration of human foibles could be taken as grounds for a blanket endorsement of decision aids (and aides), helping to make people more like the ideal. Indeed, a variety of approaches to aiding are presented. However, the same tale of foibles is also used to show the limits to these helpers. Most pointedly, the authors challenge the common and, by same accounts, essential assumption that people have articulated values and beliefs. If this were true, the aiding procedures could be conceived as inert servants, eliciting from people the pieces of their decision problems and then showing how to reassemble them in order to identify the best course of action. Abandoning this assumption makes decision-making a more creative process in which the decision aide plays a more active role. Rather than reporting their perceptions to an aide entrusted with synthesizing xv

xvi

Foreword

them, decision-makers are seen as synthesizing those perceptions from pieces of their experience. In this conception, the aide may prompt them regarding where to look, suggest alternative perspectives (drawn from the aide's own experience), and even force them to work harder. This is a stance of unabashed activism which conflicts with images endorsed by many people in the question asking business, whether as decision aides or social scientists. Questions are not neutral stimuli eliciting ready responses. Rather, they are opportunities for reflection, requiring an inferential process (as to "just what do I think?"). Such an activist role imposes a particular responsibility on decision aides. They need somehow to engage their clients without taking over the decision-making process. To this end, Watson and Buede provide a consumers' (and producers') guide to decision-aiding approaches, showing their strengths, weaknesses, and manipulative potential. They articulate the decision-makers' right to ignore advice, because a method does not fit their problems, because it places too much power in the hands of those who know the method (as opposed to those who know the problem), or because its complexities lead them to lose cognitive control of the problem (and, with it, their intuitive feel for what is happening). These issues are discussed both in general terms and in the context of a rich set of applied contexts. Here, as elsewhere, the writing allows readers to learn easily enough to know whether they want to learn more. Ample references are applied to that "more." This book, like most written with a feeling of mission, represents a landmark in its authors' eyes. The story whose time has come to be told here seems to be that of thirty odd years of chaotic development in decisionmaking research and application. In it, enough has happened to allow the authors to identify the flowers that have bloomed among the starts that have been tried. To their credit, the authors resist the temptation to begin a period of codification just as they resist the temptation to fill pages with routine technical expositions. For each approach, they force themselves to ask and answer the hard question of "what good is it?" For the field as a whole, they force us to ask ourselves the hard question of, "What is it all about, this decision-making business?" The text provides many hints on more detailed issues such as how we know what we want, how we know the limits to our own knowledge, how we tell people what we are thinking with enough precision to enlist their aid or win their support, and how much credit do people deserve for managing their own affairs and how much help can aides and aids give them? Cambridge and Reston are quiet, thoughtful, verdant communities - albeit of quite different vintages - many of whose inhabitants are dedicated to providing decision-makers on the outside with defensible, practical tools. Decision Synthesis is a worthy contribution to that clumsy enterprise of producing wisdom. Baruch Fischhoff, Eugene, Oregon

PREFACE

During the four years that have elapsed from the initial concept to the publication of this book, we have received helpful advice from many people. We want to record our particular thanks to a number of our colleagues who have read part or all of the manuscript, and given constructive criticism. These are, in alphabetical order, Val Belton, Terry Bresnick, Rex Brown, Sue Brownlow, Nick Butterworth, Elizabeth Collins, Baruch Fischhoff, Vivien Fleck, Anthony Freeling, Gary Frisvold, Elizabeth Garnsey, Colin Gill, Lisa Griesen, Kenneth Kuskey, Warner North, James Onken, and Jake Ulvila. In recording our thanks, however, we also want to state that any inaccuracies in our text are our responsibility alone, and, if readers find any, we would be grateful to learn of them. Many authors and publishers have kindly granted us permission to quote from their works, and we would also like to record our thanks to all these. In addition we wish to thank Richard Lund, who provided the photograph used to reproduce Figure 8.25. Readers willfinda guide to the book - what it contains, and how to use it - in Chapter 1. We do not mention there the use of the exercises in chapters 3, 4, and 7, however. These are designed to help readers to apply the ideas in the text. We have not given any answers to the examples, believing that the availability of answers can stand in the way of learning. We would be happy to provide any reader with answers, however, and readers seeking these should make application to either author. Finally we would like to thank our families, and particularly our wives, Rosemary and Anne, for their forbearance and support during the writing. Without such support we doubt that this book would ever have been finished!

INTRODUCTION AND READER'S GUIDE

This is a book about decision-making — about how people do make decisions, and about how they may improve their decision-making. To say this makes some assumptions, and, before we go on to show the reader how decision-making will be tackled in this book, as we will do in the rest of this introductory chapter, we will examine these assumptions. First, can decision-making be studied? One of the authors, when discussing his research work with a senior (and traditionally minded) colleague, was asked what subject he studied. On replying that he studied decision-making, the response was "Is that a subject?" Many readers will not need our prompting to answer that question affirmatively, but it is as well to spell out the nature of our study. It is principally concerned with the determinants of decision - that is to say, what it is that makes us take one course of action rather than another. In some cultural and intellectual traditions there is an easy answer to this question. Extreme forms of Calvinism, for example, itself a particularly severe kind of Protestantism, deny that free will exists; nobody may make decisions, therefore, since everything is fore-ordained by God. Then, as Howard (1980) points out, in an excellent article on the philosophy of analyzing decisions, many Eastern philosophies will not see the study of decision-making as a fruitful endeavor, since in these philosophies one accepts the unfolding of life as it presents itself. The idea of a "decision" is a quintessentially Western idea, an act of hubris to a believer in Eastern philosophy and a joke to the enlightened. (Can you imagine Buddha or Lao-tzu making a decision?) Howard 1980, p.5

We imagine, however, that most readers of this book will sit within the humanist tradition, the dominant world-view in the West since the Enlightenment. To such readers it will seem self-evident that we do make choices, and that we do have the freedom to decide to take one or another of several different avenues open to us. We can then properly ask what it is that makes us take one path rather than another. Psychologists who study human cognition have now been investigating this issue for many decades,

2

Introduction and reader's guide

and we describe some of their findings in Chapter 4 of this book. The principal question addressed has been whether people make decisions in accordance with plausible models of what might be called coherent decision-making; the answers have been mainly negative! Nonetheless, these studies have been fruitful in exploring what drives decision-making; they indicate that decision-making is indeed a proper subject of study. The second implicit supposition of our opening sentence is whether decision-making can be improved. At the intuitive level the reader's response is likely to be that it clearly can, at least in some instances. We are ready to point the finger at many bad decisions, such as Napoleon's decision to march on Moscow, or the investor who places all his savings in a company that goes bankrupt. If bad decisions can be made, so too, presumably, can good ones; and thus decision-making can be improved. But, if we investigate more closely what is meant by "good" and "bad" decision-making, wefindthe idea difficult to define. Is a decision good if it turns out to have led to the ideal outcome given the eventual circumstances? Or should we call a decision good only if the process which leads to it is satisfactory? We will discuss these issues in Chapter 2, where we will define a good decision as a rational one, and a rational one as one that conforms to freely chosen rules of behavior. Assuming that the quality of decision-making can be improved, the next question to ask is how this may be done. Over the last half century an increasingly sophisticated theory has developed, idealizing how decisions might be made. This decision theory has its roots in economics, psychology, and mathematical logic. We give an account of it in Chapter 3. A general point that we may make here is that throughout this book our object will be to give readers an understanding of the key ideas, without dwelling on them at length. We do aim, however, to provide sufficient references to the literature for an interested reader to be able to pursue points of interest elsewhere. For example, in Chapter 3 we describe value theory, subjective probability theory, and utility theory in sufficient detail (we hope) for readers to grasp the essentials, but we give references to the original literature and to other more specialized textbooks where these topics are described in greater detail. There are other subjects that a book such as this might have covered but which we have left out in order not to make the text too unwieldy; for example, we considered including a presentation of the mathematics of probability, a working knowledge of which is necessary to get to grips with much of Chapter 3. Chapter 3, then, presents a normative, but idealized, theory of decisionmaking. An idealized theory of what constitutes a good decision is very different, however, from a guide to making decisions better in practice. The art of applying decision theory to practical situations has had a more recent history, but since about i960 considerable experience has been

Introduction and reader's guide

3

developed in this applied art; the art has come to be called decision analysis. Here we should digress to explain the title of this book. So far as we are aware, the term decision synthesis has not been used previously to describe the craft of taking a decision-maker's problem and providing help in reaching a decision in conformity with the principles of decision theory. We have used it rather than the more traditional term (decision analysis) not only to provide a distinctive title for the book, but also because the craft involvesfirstthe identification of the component parts of the decision, and then the construction, or synthesis, of recommendations for action. Despite our title we will talk mostly of decision analysis (rather than synthesis) in this book, since this is the term which is now imbedded in the literature. We still feel, however, that decision synthesis is a better term for describing what the craft involves. The next question the reader will ask is how we should do decision synthesis; how may an idealized normative theory be used to help real people with real problems? We give our account of the application of decision theory to real problems in Part II of this text. First, in Chapter 6 we describe different approaches to decision synthesis that have been successfully practiced. There are now many different people in the Western world with experience of applying decision theory. As one might expect in an applied subject which involves as much art as science, different schools of practice have emerged. We call them analytic strategies, and we hope that, by studying Chapter 6, readers will be able to gain an impression of the different flavors of decision analysis that are currently practiced. Secondly, in Chapter 7, we discuss some of the techniques that are available for carrying out an analysis effectively. These include: how to structure a problem, and to model that problem once structured; how to develop an analysis iteratively; and how to elicit numerical representations of subjective judgments from decision-makers. We refer to these as analytic tactics. Then we go on, in Chapter 8, to give examples of a number of successful applications of decision theory; it is here that readers will be able to judge for themselves whether decision synthesis can improve decision-making. We hope that they will agree with us that, if good decision-making is judged by its conformity to principles of consistency, its clarity of reasoning, and its ability satisfactorily to articulate goals, then decision synthesis, properly carried out, can improve decision-making. A more detailed representation of the structure of this book is given in Figure 1.1. Notice that certain topics, such as the modeling of uncertainty, of risk preference, and of conflicting objectives, are discussed in most of the chapters. One way of using this book could be to concentrate on just one of these issues at a time, looking at first the normative theory (in Chapter 3), then the experimental evidence of whether that theory

Rationality

Psychology

2.1 The difficulties of decision-making

4.1 Introduction

2.2 The meaning of rationality

Strategy

Organizations

7

8

Tactics

Applications

5.1 Introduction

7.1 Introduction

5.2 Organization theories

7.2 Structurings:

k

5.3 Normative group theories

2.3 The rise of operational research

5.4 Decision analysis in organizations

8.1 Introduction

»-6.1 Introduction "Why do DA?."

• 7.3 Iterative analysis "How much analysis?"

8.3 Model structuring

» 8.4 Iterative analysis

(6.2-6.7) 2.4 A satisfactory rationality

7.4 Probability elicitation

6.4 RATING

Figure I.I

8.5 Probability assessment

6.2 MODELING •6.3INTROSPECTIN

3.4 Risk preference and conflicting objectives

6.5 CONFERENCING

3.5 Other normative approaches

6.7 ALLOCATE and NEGOTIATE

Decision synthesis.

6.6 DEVELOPING

9.1 The validation of decision analysis

"• 8.2 Problem structuring

7.5 Value/utility elicitation

» 8.6 Value assessment

7.6 Sensitivity analysis

• 8.7 Sensitivity analysis

9.2 Research directions

9.3 The ethics of decision analysis

Introduction and reader's guide

5

describes behavior (Chapter 4), then the way that issue is handled by different schools of practice (Chapter 6), next the practical approaches available for coping with the concept (Chapter 7), and finally how other analysts have put these ideas together (Chapter 8). We recommend this mode of study as a possibly more creative learning procedure than the usual one of reading through chapter by chapter. We have not yet mentioned Chapters 5 and 9. Chapter 9 is a concluding statement, where we discuss, in the light of the rest of the book, why, and in what context, decision synthesis is a valid and useful procedure to adopt; we also give our view of the research topics that need to be addressed to render the approach of yet more value. Chapter 5, on the other hand, addresses a more substantial issue which affects the validity of many applications of decision analysis. Decision theory is a normative theory for how an individual decision-maker might think through his or her decisions and determine sensible actions; it does not set out to do the same for groups of people, or for corporations, or for public bodies. Yet decision analysis has been widely used in such contexts. What we do in Chapter 5 is first to give a brief review of what is known about decision-making in organizations, and an introduction to the literature here. Next we describe some of the normative approaches that have been developed for group decision-making, and discuss their limitations. Finally, however, we show why, and in what contexts, it is valid to apply decision theory to organizational decision problems. We describe the bounds for organizational decision synthesis. The final topic we want to discuss in this introductory chapter is our motive for writing this book. So much has been written on decision theory and its applications that it behooves the authors of a new text to explain why they wish to add to that literature. Many excellent texts give a satisfactory account of decision theory; amongst these are Luce and Raiffa (1957), Raiffa (1968), Schlaifer (1969), Lindley (1973), Brown et al. (1974), Keeney and Raiffa (1976), Moore and Thomas (1976), Morris (1977), Behn and Vaupel (1982), Howard and Matheson (1984), Bunn (1984), and Bodily (1985). Other texts have put the theory into its psychological context (Lee 1971, Wright 1984, or von Winterfeldt and Edwards 1986) or its organizational setting (Clough 1984). Despite this existing coverage, we have not found a text that quite shares the emphasis that we have sought here; indeed the idea of writing this text came from one of us who had found no suitable single text available to support his teaching of decision analysis to final-year engineering undergraduates at Cambridge. What we feel is needed in a single book is an account of decision analysis that: (i) describes the nature of decision theory in a way that students with a quantitative background could find appealing; (ii) shows interested readers gateways into the literature, allowing them to

6

Introduction and reader's guide

pursue topics more intensively if they so wish; (iii) gives to the readers interested in the practical utility of these ideas a handbook for ways to do decision analysis in practice; and (iv) gives an up-to-date account of the practice of decision synthesis. Readers must decide for themselves whether we have achieved these goals.

I THEORY

After defining what we mean by rationality in decision-making in Chapter 2, we give an account of decision theory in Chapter 3. Chapter 4 consists of a review of psychological research in decision-making, and we discuss organizational decision-making in Chapter 5.

RATIONAL DECISION-MAKING

2.1 THE DIFFICULTIES OF DECISION-MAKING

In some ways we can describe decision-making as the most common human activity. Almost everything we do involves decisions. Whether we are driving a car, preparing a menu for a meal, working out what to say next in a piece of writing, or handling a difficult social situation, to give just a few examples, we have to make choices and decide to adopt one course of action rather than another. Since we are involved in so much decision-making, we might expect that the human species has evolved to be good at it. Viewed in one way this is indeed the case, in that we have developed many innate mechanisms for effective intuitive decisionmaking in our best interests. As babies, our unconscious decision to cry when hungry, or in pain, surely is the wisest way of improving our lot! As mature adults, our instinct to dodge a stone thrown at us is an intuitive decision that maximizes our welfare. These are at the unconscious level. At the conscious level, however, we have also developed ways of making decisions without much thought. Habits of driving, acting according to conventions of social behavior, following norms of hygiene or diet, all lead to good decision-making. Are we to conclude, then, that human decision-making is as good as it can be, and that nothing is to be gained by studying how it is, or should be, done? Unfortunately, this is not so. In the rest of this section we will discuss why it is that some kinds of decision-making are difficult and give rise to the need to think through the process of decision-making. We usually think of decision-making as being good if it is rational. We discuss the key concept of rationality in Section 2.2. One expression of rationality that has emerged in the last forty years is in the methods of operational research or systems analysis. We give a brief history of these ideas in Section 2.3, together with an appreciation of the difficulties tha't some versions of this interpretation of rationality lead to. Then in Section 2.4 we argue for a different emphasis on the meaning of rationality — more subjective, and more related to perceptions. We hope that by reading

io

Theory

through this chapter the reader will perceive the need for an expression of rationality that will provide a guide for decision-making; decision theory provides such an expression, and we turn to study it in Chapter 3. Although, as we have observed above, there are many examples of decision-making behavior which are clearly sensible, and are in the best interests of the decider, we still speak of "bad" decision-making, as well as "good." The decision-making that led to the Bay of Pigs incident has been described as bad; the decision of the British and French governments to support the manufacture of Concorde was widely felt to be a bad decision; the reader can probably add to this list from his own experience. In some ways criticisms of this kind can be unfair. It could be that, in the light of the information available to the decision-maker at the time, he made a wise decision, and only after the event did it become clear that some other action would have been better. We could argue, as indeed we will at a later stage in this book, that one should distinguish between the decisionmaking process and the outcomes that may result from the decision. It is quite possible for the outcomes resulting from a chosen alternative to be good, even though the decision-making process was bad, and vice versa. Clearly decisions can be made which are bad in both process and outcome. What are the circumstances that lead to this phenomenon? Perhaps the most obvious is the complexity of a decision. Man has always faced decisions with far-reaching consequences, but it is only in recent times that the complex web of interactions in our world has reached the state of development that it has today. In deciding whether to launch a new product, the managers of a company are faced with a complex set of issues. What effect will the new product have on the rest of their product line? How will their competitors react? Will the reaction of some foreign competitors harm their market position in some overseas markets? Will it affect trade sanctions by foreign governments? How will it be made using foreign labor, or local? Will there be adequate suppliers of components, or will parts have to be made by the company, and if so where? The list of questions can be extended almost indefinitely. As we shall see in Chapter 4, there appear to be rather important limitations on man's ability to take all such factors into account at once. Bad decision-making may result from the decision-maker's inability to incorporate all the important factors into his thinking, thus leaving himself open to a disaster caused by a factor that he did not consider. Many of the questions above involve uncertainty, and decision-making is often difficult just because of the many uncertainties involved. A manufacturing company's director might say: "If only I knew what my competitor intended, then I would see more clearly what to do; but because I am uncertain I find it difficult to decide." Some authors have

Rational decision-making

11

seen uncertainty as the chief problem with decision-making, and it is surely a major aspect of nearly all important decision problems. It may be, moreoever, that not only are we unsure what will happen, or what the nature of the world is, but we are also unclear about how important the outcomes would be to us, even if we knew for sure what would happen. For example, the manufacturing director, in deciding where a new manufacturing plant should go, will want to trade off the low labor costs in some countries with the possibly poorer quality of goods and the stability of the economy in that country. The articulation of trade-off judgments is an important problem in decision-making, and when we find it difficult to make such trade-off judgments we may make the decision badly. It is particularly difficult to come to any agreed solution on trade-offs if, as is usually the case, there are many different actors involved in a decision, with different views on the relative importance of the objectives. The increased size of organizations in the last fifty years has exacerbated this problem. Decision-making in such contexts is hard because of the differences of opinion that will emerge from different parts of the organization, all of which must be involved in reaching a conclusion. In the face of complexity, uncertainty, conflicting objectives, and multiple decision-makers, the typical response, at least of Western man, is to attempt to be rational. We seek a rational framework to help us think through our decisions. Indeed, our intuition is that some decisions are good and some bad, so that it is worthwhile seeking to improve decisionmaking. The intuitive approach is to define a good decision as a rational one. To improve decision-making then is to make each decision conform to a norm of rationality. Rationality is a concept that we intuitively look for; it is, however, more difficult to define than might appear at first sight. 2 . 2 THE MEANING OF RATIONALITY

There are some concepts which we use in common speech whose meaning is rarely questioned, but which are in fact rather elusive. Rationality is one such concept (probability is another). Many philosophers have explored what it means to be rational, and indeed Simon (1983, p.vii) speaks of the study of human reason as being his "central preoccupation for nearly fifty years." We do not have space here to explore this question in depth; what we will do, however, is to make passing reference to some entry points in the literature on rationality, before going on to a statement of what we take it to mean. Apart from the major contributions of H. A. Simon to this subject (see, for example, Simon (1957b, 1969)), the reader is recommended to consult March (1978) for a discussion of theories of rationality. Both Diesing

12

Theory

(1962) and Steinbruner (1974) define several different forms of rationality; these are summarized in Sage (1981). For a stimulating and very wide-ranging account of the problems of what it means to be rational, we recommend Jon Elster's Ulysses and the Sirens (1979); one reviewer described this as "the most exhilarating thing to hit the philosophy of the social sciences since Popper" - a view with which we concur. We argue that rationality has a lot to do with consistency. If I assert that A implies B, and also that B implies C, then I will presumably assert that A implies C. We say that it is irrational to do otherwise. What we are doing is making an inference according to a principle of logic, namely that implication should be transitive. We are being inconsistent if we assert that A does not imply C in this example. Similarly, we might invoke as a principle that our preferences should display transitivity. If I assert that I prefer X to Y, and Y to Z, but I do not prefer X to Z, I might be accused of irrationality. This accusation is probably based on the assumption that I wish my preferences to display transitivity, and on the observation that the expressed preferences are inconsistent with this. Notice that, in both these examples, the demonstration of irrationality depended on the presence of a rule and showing that statements were inconsistent with that rule. This provides the model for our concept of rationality. We will say that we are rational when, having adopted rules which our statements or actions should conform to, we act in a way that is consistent with them. Note that this definition allows many different kinds of rules to be adopted, and that therefore whether a behavior is rational will depend on what rules have been adopted. For example, is it rational to say that I am indifferent between A and B, and between B and C, but that I prefer A to C? If we require our indifference judgments to be transitive, then this is irrational. If we only require clear preference judgments to be transitive, then this is not irrational. To be rational in decision-making, therefore, we will need to construct a set of rules that we will wish to adopt in determining what to do in complex decision-making situations; then we will satisfy our need for rationality by conforming to them. Chapter 3 is devoted to describing the rules of decision theory, and we will argue that they constitute a sensible set of rules to follow. By our definition of rationality, however, it does not necessarily follow that people who do not abide by the precepts of decision theory are irrational; they may well have perfectly sensible rules of their own which they are following most rationally. Before developing decision theory, however, we shall digress- to discuss the development of operational research and systems analysis, approaches to decision-making which are often characterized as rational. We do so in order to put the rationality of decision analysis in a wider context. The

Rational decision-making

13

reader who wants to get on with the main theme of this book is recommended to skip the next two sections, or to come back to them at a later stage. 2.3 THE RISE OF OPERATIONAL RESEARCH

To be rational is one thing; to be scientific is another. To apply the scientific method, a hypothesis must be constructed about the world, and then data must be collected to attempt to falsify the hypothesis; in the absence of such falsification, the hypothesis may be used to explain phenomena. Engineers take the hypotheses of science, and use them to construct artifacts to perform useful functions. The procedures of science and engineering are sometimes described as "scientific rationalism," and this nomenclature fits in with our definition of rationality, by taking these procedures as the rules to which we wish to conform. The application of scientific rationality to the organization and management of human activity has come to be referred to as operational research, or systems analysis. In this section we shall give a brief account of the history of this subject and the difficulties it now faces, before going on to describe, in Section 2.4, what is needed to meet those difficulties. The meaning of the term operational (or operations) research has always been a matter of contention. It is a great pity that it is one of the few academic subjects whose title does not convey to the layman some idea of its area of study. The term grew up in the Second World War to describe the activities of units researching into how to carry out military operations more effectively. It was recognized that considerable improvements in performance might be obtained if engineering methods were applied to logistics. For example, in determining whether large or small Atlantic convoys stood the best chance of moving material from the US to Europe against the threat of German attack (small convoys stood a better chance of passing undetected, large convoys a better chance of resisting an attack, at least in part, because of the greater number of support ships), a mathematical model was built of this situation, and used to explore the probabilities. On the strength of this analysis the policy on convoy size was changed, leading to an improvement in the Allied position. In many other contexts the operational research groups proved very valuable to the war effort (see Waddington (1973) for an account of the wartime contribution of OR). The novelty of the approach was to take the methods which had been successfully used in engineering contexts to design effective machinery, and apply them to the design of effective logistical operations. That is to say, once the purpose of the activity had been defined, a mathematical model of the system was set up, and used to evaluate different design options. One could ask questions of the model

14

Theory

without having to try out the many different practicable possibilities to see which was best; this was the approach that had been used by engineering designers for a long time. OR proved to be so successful during the war that it was widely recognized to have potential value as an approach to the management of civilian operations, particularly in manufacturing industry, but also wherever managers had to decide which of several options to pursue in organizing their activities. During the fifties and sixties there was considerable enthusiasm for OR, particularly in the US and the UK, but also in the rest of the Western world. Considerable successes were achieved. For example, the recognition that the problem of how to schedule the operations of an oil refinery could be formulated as a linear program led to considerable savings in costs. Now no refinery operates without extensive use of linear programming. Similarly, the design of road systems, manufacturing systems, multi-user computer systems, and even queuing systems in banks and public offices, to avoid congestion, has benefited from the same kind of analysis; by investigating mathematical models of such systems with the mathematical tools of queuing theory, it is possible to construct systems which do not lead to unacceptable congestion. Because of the obvious successes of the OR approach to some problems there was great interest in applying it to others. In the United States, perhaps the most famous institution set up to apply OR methodology was the RAND corporation created by the US Army Air Forces in the late forties. (For a history of this influential organization, see Smith (1966).) The term used by this organization for the work they performed was "systems analysis"; although there were differences of emphasis, it is fair to think of the RAND approach as being essentially that of OR. RAND was not the only organization practicing OR, however. The approach gained much in popularity, and by 1965 it had become so entrenched in the US government that the Department of Defense appointed an Assistant Secretary (a very senior appointment) specifically for systems analysis. Other governments, and private industry, followed the fashion of the times and set up large departments to carry out OR studies. At about the same time, however, criticisms of OR began to emerge, and during the 1970s this led to what some commentators have described as an intellectual crisis. Hoos (1972) gave a most significant criticism of systems analysis; later the OR literature began to contain many critical contributions as well (see, for example, Ackoff (i979a,b), Majone and Quade (1980), Geoffrion (1983), Tinker and Lowe (1984), Tomlinson and Kiss (1984)). The problem was that OR methods were, no longer providing the same noticeable improvements to management problems that had been the case in the early years of the subject. To some extent this was a consequence of OR being oversold, as often happens with inno-

Rational decision-making

15

vations in management methods. A more fundamental problem was that the mind-set of the OR approach, while ideal for problems of logistics, for which the approach was developed, was inadequate for the higher-level management problems that were beginning to be faced. There is now a sizeable literature on this issue. For example, Linstone (1984) in a stimulating recent book gives a personal view of three different perspectives to analyzing decisions: the technological, by which he means the OR approach; the organizational; andthepersonal. We have not sufficient space to repeat Linstone's arguments here, but his last two perspectives stress the importance of organizational forces and personal perceptions in the making of decisions, aspects often ignored in an OR approach. H. Boothroyd, who at the time had worked for ten years in operational research at the National Coal Board in the UK, states the inadequacies well: By 1966 . . . I had begun to be aware of two particular problems: (a) clients were inclined to treat the findings of investigations as complete and final, rather than as a stage in a joint process of exploration and experiment — this was related to their picture of science as providing definitive final answers on both facts and desirable choice of action; (b) in the literature of operational research and systems analysis the implicit picture of decision-making was quite unlike the untidy, episodic nature of the experience of making decisions about the conduct of investigations. Boothroyd 1984, p.35

The analyses failed to involve the problem-owners; and the methods used to decide how to do the analysis were very different from those recommended by the analysis. Checkland has long made similar criticisms of OR, or systems analysis. In Checkland (1981), he gives a careful account of what he calls "hard" systems thinking (in which category he places OR), and points out its inadequacies when applied to certain kinds of problem. One of his chief criticisms is that the "hard" systems thinker is concerned with how to achieve a given end most efficiently, rather than with what that end should be in the first place. The exploration of goals, their creation, and their expression are not important parts of the analysis. But, as Checkland points out, in many practical problems, such investigation of goals is probably the most essential need in coping with the problem. He reiterates his criticisms in a recent paper (Checkland 1983). In summary then, we conclude that inadequacies have appeared in some (though not all) attempts to apply the scientific rationality of OR to management. These include: - failure to involve the decision-maker in the experimental and iterative nature of the analysis; -failure to appreciate the organizational and personal context of decision-making;

16

Theory

- failure to explore ends as well as means. We now go on to see what is required to meet these inadequacies. 2.4 REQUIREMENTS FOR A SATISFACTORY RATIONALITY FOR DECISION-MAKING

If the OR approach is an inadequate methodology for the synthesis of decisions (at least as articulated in textbooks and taught in many university courses), we must ask what may be done to improve matters. Checkland's answer is to use his "soft" systems approach. We have not the space here to describe this approach in detail, and any summary may well fail to capture the essence of what is a very sophisticated approach to thinking about problems. The reader is referred to Checkland (1981) for an account of "soft" systems thinking. We will note, however, that its essence involves a more open-minded, iterative, and flexible way of thinking about a problem than is typical of the traditional kinds of OR and the harder kinds of systems thinking. Although Checkland's approach emphasizes the importance of including problem-owners (a slightly more general term than decision-makers) in the analysis and analyzing the problem as seen by them, it does not provide a specific methodology for eliciting the problem-owners' perceptions of the real world they face. An interesting and novel approach which does just that has been developed by Colin Eden and his colleagues at Bath University in the UK. They stress the need to understand the problem-owners' perceptions of their problems if an effective way is to be found for creating a rationality that will satisfy them and help them in decision-making (see, for example, Eden et al. (1983)). They use the notion of a "cognitive map," a representation of an individual's thinking, to help him to structure his perceptions of the decision problem that he faces. They provide a formal procedure for constructing cognitive maps, and have prepared an interactive computer program to assist with this procedure. In contrast to the "hard" systems approach, to use Checkland's characterization, Eden's approach unashamedly attempts to model, not the real world in some objective sense, but that world as perceived by the problem-owners. We believe that, as a way to structure problems, cognitive mapping has great potential. It stops short, however, of providing a formal calculus for fleshing out the problem structure, once it has been identified by mapping. We make further reference to this idea in Section 7.2.2. These two ways of thinking provide very useful insights, but, in our view, they give only part of the answer. We shall nowfinishthis chapter by drawing together the various threads that we have explored, and argue the need for a calculus of decision-making with particular properties. We first argued that decision-making was difficult because of the

Rational decision-making

17

complexity of the decision being faced, the uncertainty that generally surrounds the problem, the existence of conflicting objectives, and the presence of multiple actors in the decision situation. We then suggested that, if we wished to handle these problems in some rational way, it was necessary for a set of rules of decision-making to be established; rationality would then be achieved by conforming to these rules. We digressed to describe the procedures of OR, often identified with rationality, and highlighted some of the inadequacies that have been seen in applications of the OR method. What then is needed? The properties listed below follow from our analysis above. (i) A set of rules for decision-making must be defined in order that we may know what it is to be rational. (ii) The rules must address the values of the problem-owners. It must be possible to articulate their preferences and perceptions. (iii) The rules must state what it is to be rational in the face of perceptions of uncertainty. (iv) The rules must provide a calculus for steering the thinking of the decision-makers through complex problems. The next chapter gives an account of decision theory, a body of ideas with a long intellectual history and one which meets these requirements.

DECISION THEORY

In the previous chapter we argued that, to assist people in thinking through complex decision problems, a quantitative framework is needed to encourage consistency, the essence of rationality. Decision theory provides such a framework, and in this chapter we give an account of this theory. We start, in Section 3.1, with value function theory, which is concerned with conflicting objectives. In Section 3.2 we discuss uncertainty, and how the calculus of probability may be used to describe it. Section 3.3 contains a description of utility theory, which is concerned with decision-making in the face of uncertainty, and in Section 3.4 we show how uncertainty and conflicting objectives may be handled at the same time. Finally, in Section 3.5, we discuss other approaches to creating a calculus for decision-making. 3.1 CONFLICTING OBJECTIVES

We all make decisions all the time. What clothes shall I wear this morning? What shall I have for lunch? Shall I watch television or read a book? Most of us have no difficulty in deciding matters like these, but on occasion we find ourselves spending a lot of mental effort in making up our minds. Choosing a career and then a particular job, or where to live, often falls into this category. The stakes at risk are more significant on such occasions, and we find that we need to spend more effort than normal insuring that we choose correctly. In our working lives, when we are part of a decision-making group, or when we have responsibility for decisions that affect many other people, we are highly motivated to get the decision right. Whether it is the choice between several different locations for a new plant, or the decision to institute a new payments system in a factory, or the selection of a new computer system, we seek to satisfy ourselves that the choice we make is consistent with our own beliefs and values, and with the values of the organization that we serve. The practical role of decision theory is to provide a framework that assists people in achieving such consistency. In 18

Decision theory

19

this chapter we will set out the theory, and leave until Part II descriptions of how the theory is applied to practical problems. Notice how, in many of the problems mentioned above, the essence of the difficulty is that some of the objectives conflict with each other. Do I take the job with the higher salary, the better prospects or the nicer working conditions? Do I choose a computer system which is compatible with an older system currently in use, or the one which provides new and useful features? Of course, while conflict between objectives may seem the most significant factor in such decisions, we are generally also worried by uncertainty. In fact most decisions that cause us to pause and think involve both uncertainty and conflicting objectives. In Section 3.4 we shall see how decision theory can provide a framework for thinking about these two aspects at once; but we shall devote the current section to the handling of conflicting objectives when we have no uncertainty at all about the outcomes of our decision-making. We do this for three reasons: first, there are a small number of problems where no uncertainty exists (although most practical problems of interest are not in this class); secondly, many real problems are dominated by conflicting objectives, so that a satisfactory framework for decision-aiding can be obtained by ignoring uncertainty; and, thirdly, the theory of how to handle both problems at once is founded on a theory for how to handle each problem separately. 3.1.1 Utility and value Choice is simple if we have only one thing to worry about. If we can buy the same product in several different shops, we usually use cost as a single criterion and choose the shop where the price is lowest. In awarding a university prize, examiners could concentrate on aggregate marks alone and give the prize to the student with the highest score. In both these examples, though, other things could well be taken into account. The closer shop, or the one with the more pleasant salespeople, might be worth going to even if we have to spend a few pennies more. Perhaps the prize should go to the student with a slightly lower aggregate score but a better all-round performance. If we could establish a single numerical measure which took these other factors into account, then our choice procedure would again be straightforward; the difficulty is how to construct such a measure in the first place. The concept of a numerical measure to describe the value of alternative choices has come to be referred to as utility theory, with the 'Utility function being the numerical measure itself. Utility appears to have been first discussed by Daniel Bernoulli (1738). His concern was to explain the St Petersburg paradox, which involves the proper evaluation of a par-

20

Theory

ticular uncertain gamble (see Section 3.3.1 for a discussion of this problem). Bernoulli's explanation was that money was not an adequate measure of value. Instead he suggested that the worth, or utility, of money for each individual was non-linear and had a decreasing slope; marginal utility decreased as wealth increased. At this stage, however, the concept of utility had not been refined beyond this simple idea. In the following century, economists continued to use and develop the concept of utility to explain economic behavior, even if no uncertainty was present; indeed, in such a standard work as Marshall's Principles of Economics (1890), the idea of utility is a central notion in explaining economic phenomena. A thorough review of the development of different concepts of utility in economic enquiry is given by Stigler (1950). The utilitarian philosophers, such as Bentham and Mill, also talked of utility. To them, however, it was not a device for understanding economic behavior; rather, it was a tool for constructing a theory of ethics. Bentham's dictum that "the greatest good for the greatest number of people", should be pursued carries the implication that utility, a measure of the benefit of the options available to a person, is to be maximized in sum. While both theories were attractive, they failed in practice, for it was not possible at that time actually to measure a person's utility function in a satisfactory way. In the 1920s, several economists argued that economic theory could be founded on a less demanding assumption about measuring utility (see, in particular, Pareto (1927)). Instead of making the assumption that a number was available representing the worth of any bundle of goods {cardinal utility), one could simply assume that it was possible to say, for any two bundles of goods, which was preferred (ordinal utility). This latter notion almost entirely replaced the former in economic thinking in the inter-war years. It was not until the 1940s and 1950s that the concept of cardinal utility was resurrected, and this happened in two rather different guises. In the study of human interaction, the theory of games was developed both to describe how people behave when engaging with others with conflicting goals and to prescribe how rational people ought to behave in such situations. In their seminal book on this subject, the great mathematician John von Neumann and the no less eminent economist Oskar Morgenstern found the need to develop a cardinal theory of utility to prescribe how people should evaluate options about which they were uncertain (von Neumann and Morgenstern 1944). While their utility theory is expounded almost as an aside in their book, it has probably been the most influential contribution to the development of modern decision analysis. When the concept wasfirstsuggested, it sparked considerable controversy (see, for example, the excellent paper by Ellsberg (1954), who gives a very

Decision theory

21

clear account of the distinctions between different utility theories). We shall study von Neumann and Morgenstern's utility theory in Section 3.3 below. A separate development of ideas about utility was based on new procedures for measuring the utility of the older kind; most notably the theories of difference measurement and conjoint measurement (see Krantz et al. (1971)). Although ordinal utility is all that is needed to direct choice between multi-attributed alternatives, as the economic theorists noted, it is a difficult task to attempt to elicit such a function by plotting indifference curves. Luce and Tukey's theory of conjoint measurement (Luce and Tukey 1964) and other developments in measurement theory have spawned a revival of interest in cardinal utility, distinct from that of von Neumann and Morgenstern but applicable in the absence of uncertainty. This cardinal utility turns out to be more easily measured, and therefore more useful, than ordinal utility. We will describe these recent developments in the remainder of Section 3.1. Decision analysts have made this distinction between two types of utility, which can be characterized as the distinction between value and risk preferences, ever since the subject first developed. Value preferences are made between competing objectives or attributes when no uncertainty is present. Risk preference (as measured by the utility functions discussed above) addresses the decision-maker's aversion to, indifference to, or desire for risk-taking. We will discuss the former in Section 3.2.1 and the latter in Section 3.3. There have been many reviews of different concepts of utility in the last forty years. As well as Stigler (1950) and Ellsberg (1954) mentioned above, the interested reader is referred to Edwards (1954) and the many writings of P.C. Fishburn, and in particular Fishburn (1976). A useful discussion of the theoretical and practical distinctions between these concepts is given in Barron et al. (1984). 3.1.2 Value functions 3.1.2.1 Problem formulation As with much of this text, our account of value functions must necessarily be succinct, because of the breadth of issues we discuss. We will state the key elements of value function theory, and direct the reader elsewhere for a fuller account. For much of this chapter the reader will find the presentation of Keeney and Raiffa (1976, pp. 66-129) an excellent expansion of ours. This section will be concerned with the properties a value function should have; first, however, we will review why we want to construct one in the first place. We introduced the idea of a value function to assist people faced with decision problems; we can characterize this problem as follows. A set of

22

Theory

options has been well defined. Note that the terms alternatives, choices, decisions, or actions are used by some writers in place of the word options-, the literature sometimes makes careful distinctions between them and sometimes, as here, uses them interchangeably. We assume that we know what the options are and, perhaps more importantly, that we know what the attributes (or factors) are which discriminate amongst them. We suppose that there are n such attributes, named Xx, X 2 ,..., Xn, and that it is possible unequivocally to provide a numerical score for each option with respect to each attribute. The set of scores with respect to the attributes for a particular option will be represented by a vector x = {xx, x2, . . ., xn). The problem is now reduced to deciding which vector of scores is most attractive. For example, consider the problem of deciding which car to buy — a problem often addressed in discussions of multi-attribute decision-making. The options are the different cars we might choose to buy; the attributes are things like purchase price, mileage, reliability, and so on. Although at first sight the idealization above may seem to be a reasonable abstraction, real problems are rarely like this, at least to begin with. First, it is often the case that the option set is not completely known, or at the very least we may not know whether all the possible options have been identified. There may, to follow our example, be cars available which we have overlooked, or models which differ from what we had imagined them to be. We could decide not to bother about this point, arguing that decision theory is only concerned with what to do once the options are specified. It is obviously of practical importance, however, for people wanting guidance with real problems. The generation and delineation of decision options have been active research interests in recent years, and we shall discuss this point again in sections 7.2 and 7.3. Secondly, it may not be clear how to characterize the attributes that distinguish between the options, nor how to measure each option with respect to some of the attributes. We know, for example, that reliability is one attribute that distinguishes among cars; but is reliability to do with the frequency of breakdown, or the severity of breakdown, or both? Even if we can be precise, how should it be measured? We will return to these issues later; for the present we will assume that they can be overcome sufficiently for the analysis of our idealized problem to be useful in tackling real-choice problems. 3.1.2.2 Value functions: definition and alternatives Suppose that we can construct a real-valued function v{ . ), with the property that ^(x^) > v{xB) if and only if we prefer the option whose vector of scores is xA to another with score vector xB; and with the further property that v{xA) = v(xB) if and only if we are indifferent between xA and xB. We

Decision theory

2.3

Figure 3.1 Dominance. In the figure £, F, G are dominated, E by B, F by C, and G by B and C. A, B, C, D are not dominated, and form the efficient frontier. shall refer to such a function as a ftf/«e function. Then the choice of option is simple. We merely choose the option with the score vector x for which v(x) is largest. The difficulty will be in constructing this function v( . ). If the only way to do so were to elicit from the decision-maker his preference between all possible score vectors, then we would not have achieved anything by trying to construct a value function. The choice problem would have been answered by direct intuition. It is the fact that we find it so difficult to compare sets of scores directly that leads us to choose some other route to decision-aiding. Fortunately, as we shall see, there are ways of computing v{ . ) that rely on general properties of our preference structure and just a few simple comparative judgments; the power of the method is that if our preferences obey certain conditions (and they often do) we can synthesize v{ . ) from an analysis of only parts of our set of preferences. Before we discuss this theory, we note that many authors have investigated what may be done without a complete specification of v( . ). The simplest observation is that the notion of dominance might lead to a unique choice. An option A is said to dominate another option B if, for all / in 1,. . . , «, xf ^ xf and, for at least one /, x£ > xf, using an obvious notation. The notion of dominance is illustrated in Figure 3.1 in the case of two attributes. Clearly, if one option dominates all the others, it is

24

Theory

rational to choose it; that is to say, the choice of any other option is inconsistent with our statement of preferences. (Note that we have assumed that scores have been arranged so that an increase in the score on any attribute is a good thing; it is always possible to do this.) Needless to say, dominance rarely occurs with problems that are taxing enough to require analysis, and we need to search further for appropriate methods for most problems we might face. Note, in passing, that there are more sophisticated methods which use the dominance concept (see Keeney and Raiffa, 1976, p. 128). Another device that is often very valuable in seeking the best from a set of options is related to dominance. If we remove from the option set all options dominated by some other option, the remaining set is called the efficient frontier, or the Fareto-optimal set (see Figure 3.1). If our preference structure implies a best choice, it is certainly on the efficient frontier; but to find it we need more information. One idea, which does not involve the complete construction of v( . ) to complete the search, is to conduct a myopic search along the efficient frontier. We will mention some of this work in Section 3.5; here we just note that the input information for procedures of this kind is preferences between neighboring points on the efficient frontier. Within the community of decision analysts, there are some who approach problems with multiple attributes by pricing out all but one attribute (see, for example, Howard et al. (1972), or the more extended discussion in Keeney and Raiffa (1976, Section 3.8)). This approach finds a point on the same indifference surface as the point in question, but where all attributes but one have a standard set of values. Comparison of score vectors is then done by the simple comparison of values on the last attribute, which is usually taken to be money. The main difficulty with this approach is the problem of tracing along indifference surfaces. We will now restrict attention to methods that construct an explicit value function, since these are the most tractable and useful in practice. 3.1.2.3 Existence of the value function The notion of effecting an explicit construction of v{ . ) as a way of solving the idealized decision problem makes intuitive sense. There are many practical situations where this approach has been widely accepted as a sensible device for routine decision-making. Credit-scoring, examination marks, and rules for the UK Rate Support Grant are obvious examples. What are of interest to us here are the procedures that ought to be employed in constructing such evaluation indices. Note that the existence of a value function to represent preferences is equivalent to there being a complete preference order over the options. Clearly if a value function exists it induces a complete ordering on the

Decision theory

25

vectors x. Conversely, it is easy to see that if we do have a complete preference order, then we can construct a value function by merely placing the vectors in increasing preference order and attaching any increasing sequence of numbers to the vectors. If then we feel confident that, given enough introspection, we could produce a complete ordering on the vectors x, there must be some value function v( . ) that can represent our preferences. Some authors have criticized utility and value theory on the grounds that, if we could articulate all such preferences, then we would not need to construct v( . ); we could decide by simple introspection. The fact that we find it difficult to make such comparisons must imply, they argue, that either no value function exists or, if it does, it cannot be found. Such criticism misses the point. So long as we assert that there must be some preference ordering of all vectors x but that we are unable to articulate all of the constituents of this ordering, then we can use a value function to express that ordering. This is a most important point in the interpretation and use of value function theory. Since a partial ordering of our preferences can be inferred from any set of decisions we make, the point in creating a value function is to help us construct preferences between score vectors which we find difficult to compare directly. The problem, then, is to construct our preferences within and across the attributes Xj of x in such a way that we have a complete ordering of all vectors x. This theory is normative, showing us how we should behave if we want to be rational, rather than descriptive of how we do behave. We may use value function theory to construct preferences which we find difficult to articulate directly, using only those preferences that we can express easily and adopting the principle of complete transitivity for all our preferences. Having assumed the existence of v( . ) for a particular problem (and note that there is no reason why v( . ) should remain constant in time, or from problem to problem), we now have to measure it. Clearly any monotonic function of v( . ) will now also be a value function, since it will satisfy the definition at the beginning of Section 3.1.2.2. This means that the critical aspect of the value function is the set of indifference curves it implies, and, as we have already noted, it is cumbersome and difficult to elicit this, particularly if the number of variables is large. Fortunately, however, there are properties that our preference structure might (and often does) have which lead us to special forms of the value function which can be more easily elicited. 3.1.2.4 Additive value functions The most often used form for a'value function is the additive one: n

~

.•(*,•).

(3-i)

z6

Theory

The simplicity of this form and the ability to work on each attribute separately make it an attractive candidate for our consideration, although it is obviously quite possible for a particular preference structure to lead to a non-additive value function. For example, suppose we wish to evaluate different possible designs for a microwave relay station as part of a communications network. The attributes of importance to us might be the ability to receive signals clearly, the ability to transmit them, and the cost. Suppose that these are measured by the variables R, T, and C respectively. Then it would be plausible that someone might have the following preferences: and

(RH, TH, CH) preferred to (RH, TL, CL) (RL, TL, CL) preferred to (RL, TH, CH)

where the suffixes L and H indicate specific high or low values of the variables. The first preference expresses the view that it is worth paying the higher price, CH, for the improved transmission ability {TH instead of TL) if the receiving ability is good. The second preference states that the high price is not worthwhile if receiving ability is poor. In this case no additive value function may be found consistent with both these preferences, as may be readily demonstrated. Despite examples such as these, it turns out that we can get quite a long way with additive value functions. We will now, therefore, explore what the conditions on our preferences have to be to justify additivity. The essential condition is that of preference independence. A pair of attributes, Xi, X2, is said to be preference independent of all the other attributes {X,, / = 3, . . . , « } if preferences between different combinations of levels of Xi, X2, with the level of all other attributes being held at constant values, do not depend on what these constant values are. Thus, in the car-buying example, if we say that cost and mileage are preference independent of reliability, it means that if we prefer a car with cost cu mileage m^ and reliability r (measured on some suitable scale) to one with cost c2, mileage m2, and reliability r, then we also prefer a car with cost cu mileage mi, and reliability r' to one with cost c2, mileage w2, and reliability r', no matter what r' is. Moreover, this holds for all possible cu c2, mi, m2. The microwave relay station example above shows a situation where preference independence does not hold. Now we can quote what Keeney and Raiffa (1976) refer to as a "truly remarkable" theorem. First we define mutual preference independence. If our preference structure is such that any two attributes are preference independent of all the others, we say that the attributes display mutual preference independence. It can now be shown that, if this condition is satisfied by our preference structure, then there must be a value function with the additive form given by Equation (3.1) above. Moreover, the

Decision theory

27

function v( . ) is unique up to a positive linear transformation; that is to say, if there is another function v'( . ) with the same properties, there must be some numbers a and b, with b > o, such that v'(x) = a + bv{x). This type of function obeys the properties of an interval (or cardinal) scale; that is, an interval of some given number of units on the scale represents the same increment in value no matter what the starting point on the scale is. The analyst can arbitrarily assign numbers to any two points on the scale; or, equivalently, he may choose a and b as he wishes. An example of such a scale is temperature; the Fahrenheit and Celsius scales are but two possible ways to measure temperature. Fishburn (1970, chapters 4 and 5) and Krantz et al. (1971, Section 6.2.4) give detailed proofs of this result, and Keeney and Raiffa (1976, Sections 3.5 and 3.6) give a rather more readable discussion of the same body of theory. We will not reproduce these accounts here, and merely refer the interested reader to these other texts. Mutual preference independence is often a plausible condition and, as we shall see, it is possible to test whether our preference structure does satisfy this condition in many practical contexts. While being the most common form of value function, both in theoretical accounts and in practice, the additive form is not the only possibility. Several authors have discussed multiplicative or other forms. An interesting possibility is a part additive and part multiplicative form. Keeney and Raiffa (1976) give a fuller account of this point in their Section 3.6. 3.1.3 Summary In this section we have described the purpose and theory of value functions. We first discussed the problem of decision-making in the absence of uncertainty, and the appropriateness of the standard idealization of the problem. Then after a brief history of ideas in this area, we described the axiomatic approach to value functions, and in particular the conditions which lead to an additive form. We will show how additive functions may be elicited and used in Section 7.5. There is much more that needs to be said, and will be said, about this topic in this book. The problem of structuring a decision prior to applying the theory raises some important philosophical and practical issues, and these will be discussed in Section 7.2. Eliciting values is also a problem area, as will be seen in Section 7.5. The use of this theory, both for individual decision-making and for decision-making within organizations, has implications which go far beyond its mathematical'formalism; these are discussed in Chapter 5 and in Part II. Other formalisms have been suggested for addressing the same idealized problem that we have discussed here. These differ from the theory we

28

Theory

have presented here, sometimes in significant and subtle ways. We review some of them in Section 3.5. EXERCISES FOR 3.1

Sue has a job decision to make. She has been offered four jobs, each having different starting salaries, promotion opportunities, locations, and required levels of creativity. Sue has had a course in decision analysis and has structured her decision problem with the following attributes, values, and weights:

Jobs A B C D

Weights

Starting salary

Promotion opportunity

Location

Required level of creativity 60 40

0

20

100

0

30

65

100

0

0

2-5

75

100

100

0.4

0.2

O.I

0.3

(a) Assuming that all pairs of attributes are preference independent of the other attributes, which job maximizes Sue's value function? (b) If Sue changes her weights for the attributes so that starting salary has a weight of 0.2 and promotion opportunity a weight of 0.4, does the job with the highest value change? (c) Do either of these results change if job C is removed from the analysis? 2.

Select a decision problem of concern to you, such as the choice of a job, and apply the methods of multi-attribute value analysis to help you think through it.

3.

The committee of a students' union is considering whether to introduce a copying machine for students' use and, if so, which machine to purchase. It is recognized that the available machines differ with respect to quality of copying, reliability, ease of maintenance, and cost. Describe in detail how the methods of multi-attribute value theory may be used to assist with this decision problem. Pay particular attention to the assumptions necessary for the method to work, and their likely validity in this case. 3.2 UNCERTAINTY

In Section 3.1 we analyzed the problem of choosing between options when our objectives conflicted. We assumed that we knew with certainty

Decision theory

29

what the results of choosing an option would be, so that the only problem was how to cope with the conflict between our objectives. All these assumptions are unrealistic for most real decision problems, but the problem of uncertainty is of particular importance, since it is very hard to discover any real decision problems where we are not at least slightly uncertain about the outcomes. In the car choice problem, for example, we may be unsure about how reliable the particular car we will buy will be, or how we will rate the car's comfort once we are using it regularly, and so on. In order to use value function theory to aid in the car choice problem, we had to ignore our uncertainty and hope that this would not affect our conclusion. This is a judgment that can only reasonably be made if we have some idea of what kind of analysis we might do if we did address the uncertainty explicitly, and what its outcome might be; such an analysis will be described in this chapter. We will find that it will generally be reasonable to ignore uncertainty in the car choice problem, but there are many problems where the chief difficulty is uncertainty. In most financial decisions, for example, the key to a satisfactory decision is the proper assessment of uncertainty within an analysis. If I am wondering whether to buy stocks or fixed interest securities with some spare cash, I have to balance the surer return from the latter against the possibly higher return from the former. If a company is deciding whether or not to build a new warehouse, it is usually balancing the uncertain cost streams of doing so against not doing so. Some years ago, most texts on decision analysis presented the subject as being almost entirely concerned with uncertainty, and generally where there was only one attribute of interest, usually money (see, for example, Raiffa (1968) or Lindley (1973)). The subject has advanced, however, and, as we argued above, the analysis of conflicting objectives is now emerging as the more useful area in practice. Nonetheless, dealing with uncertainty is vital in many applications. 3.2.1 Descriptions of uncertainty If we are to incorporate uncertainty into our theories of choice, then a calculus for handling uncertainty must be developed. The theory that is most commonly employed to do this is that of probability. Why should we use this theory to describe uncertainty? Might it be that the only reason for using it is that the notion is so well imbedded in our scientific culture that we do not stop to question if it is the only possible, let alone the correct, calculus to employ? Other calculuses have recently been suggested for handling uncertainty, and so we shall need to argue for the use of probability theory rather than some other. Probability theory is itself not very old, being usually dated back to the study of games of chance by seventeenth- and eighteenth-century mathematicians such as Pascal, the

30

Theory

Bernoullis, and Laplace. A theory describing games of chance is not necessarily appropriate for describing uncertainty in other contexts, however. Even if we adopt probability theory as our tool for describing uncertainty, we must decide which of the many versions of that theory to employ. Readers whose only exposure to probability has been in high school mathematics courses, or even in university statistics courses, may be surprised to learn that there is any dispute about probability. The theory is often presented as a branch of pure mathematics, and as such it is as non-controversial as arithmetic or set theory. Contention arises, however, as to the meaning of probability, and in particular its relation to real-world phenomena. Over the last two centuries, theories of probability have been many and varied. Most prominent philosophers have contributed to the debate, as have scholars in other areas who needed the concept of probability in their own work. Most authors presenting new theories of probability also argue against older theories, for obvious reasons. Very few comparative accounts exist which allow a disinterested reader to balance the arguments in favor of one or another interpretation. Exceptions are the text by Fine (1973), which gives a detailed account of the axiomatic development of the competing theories, and the more recent comparative study of Weatherford (1982). Hacking (1975) provides an excellent historical account of the development of probability ideas. While recognizing that other categorizations of probability theories are possible, Weatherford sees four distinct interpretations for probability. We have not the space here to describe these distinct theories in any detail, and the reader interested in pursuing them is recommended to consult Weatherford (1982). It is important to note, however, that the original ideas of the classical theorists, such as Laplace, were found to be inadequate in dealing with the real world, since the theories did not incorporate a satisfactory method for measuring probabilities. The a priori, or logical, theories of Keynes and Carnap, the relative frequency theories of Venn, von Mises, and Reichenbach, and the subjective theories of Ramsey, de Finetti, and Savage all claimed to improve on these ideas, but in different ways. The most important conflict that we need to mention here is between the last two of these theories. The relative frequency theory holds that the probability of an event is the long-run frequency with which the event occurs in an infinite repetition of an experiment; it is thus seen as an objective property of the real world. Subjective theories present probability as the degree of belief which an individual has in a proposition; this is a property of the individual's subjective perception (or state of knowledge) of the real world. The former theory has dominated scientific thinking for many years, because it has the appearance of conforming to the empirical objectivity which

Decision theory

31

science is sometimes claimed to need and possess. It will not do, however, as a theory for helping an individual to model his perceptions of uncertainty when making decisions. The interpretation of probability which must be adopted for the application of decision theory has to be the subjective one, and we will describe this in detail in the next section. The reader should be aware, however, that not all scholars are prepared to accept this theory (see, for example, the discussion in Weatherford (1982)), but that, if the subjective theory of probability is not adopted, the intellectual justifications for the procedures of decision analysis are much weakened. Alternatives to probability theory are beginning to emerge. The following brief account of some of these alternatives may appear frustratingly inadequate to readers. Our purpose, however, is not to give a detailed comparative account of these theories; we have insufficient space to do this. Rather, we mention them so that readers who wish to consider alternatives to probability may know that such alternatives exist and where to go to find more detailed accounts. First, we mention Dempster's theory of upper and lower probabilities (Dempster 1968) and the related but more general theory of belief functions, due to Shafer (1976). These endeavor to construct a richer description of uncertainty than is provided by probability theory, allowing people to refrain from making judgments on relative likelihoods between some pairs of events. These theories have not yet, to our knowledge, been used satisfactorily with practical problems. Secondly, Zadeh's fuzzy set theory has also been suggested as an alternative to probability theory (see, for example, Zadeh (1983)). Zadeh's theory is intended to describe imprecision, rather than uncertainty, and, although these two concepts are obviously related, they are distinct. The use of fuzzy set theory to describe the imprecision that an individual might perceive about his probabilities, or utilities, has been suggested by Watson et al. (1979) and by Freeling (1980). A good set of readings on fuzzy set theory has been given by Mamdani and Gaines (1981).

Thirdly, we should make brief mention of a calculus for uncertainty developed in the course of constructing the expert system for medical diagnosis called MYCIN. Its constructors needed to be able to represent uncertainty and propagate it through the system. Shortliffe (1976, Chapter 4) gives a detailed account of the "certainty factors" employed in MYCIN, including why they were created and how they are to be manipulated. The theory has presented problems in application, however, despite its attractive properties, and the Stanford University research group responsible for MYCIN is now considering the use of Shafer's theory instead (see Buchanan and Shortliffe 1984, Chapter 13).

32

Theory

Finally, we wish to mention L. J. Cohen's theory of inference (Cohen 1977). Cohen is concerned with evidence and how to present its uncertain implications in terms of the number of different kinds of ways the conclusion could turn out to be false. This theory has not yet been developed into a practical tool for representing uncertainty, but it may well be so developed in the future.

3.2.2 Theoretical background to subjective

probability

Probability theory is an obvious construct for describing subjectively perceived uncertainty. Not only has it been deep in scientific culture for the last fifty years, at least; in chance situations, such as card games, it is the proper calculus to describe relative proportions. To apply this calculus in other situations, however, requires more than an act of faith. What is needed is a set of assumptions about an individual's judgments which, if satisfied, leads us to infer that a set of numbers must exist which describe that individual's perceptions of uncertainty; and, moreoever, that these numbers should be combined using the rules of the probability calculus to infer what numbers should be used to describe other uncertainties. This development was successfully undertaken by Ramsey (1931), Savage (1954), and, most extensively, by De Finetti (1974). (See Kyburg and Smokier (1964) for a collection of the most important papers on this topic.) Most accounts of this work are far from transparent, since considerable mathematical development is required. DeGroot (1970, Chapter 6) and Lindley (1973, Chapter 2) both give a clear presentation. The behavioral suppositions are set out as axioms in Exhibit 3.1. We first suppose that, given two uncertain events, a person can say which is more likely, or whether they are equally likely. Next, we suppose that three highly plausible conditions are satisfied by these judgments of relative likelihood (axioms 2, 3, and 4 in Exhibit 3.1). Finally we suppose the existence of a set of events which acts as a standard in probability judgment. Lindley (1973) takes this to be the angular position adopted by a pointer on a perfectly balanced wheel after it has been spun; but many possible sets of events will do. All that is needed is to be able to find one member of the set whose likelihood is judged to be equal to any uncertain event being considered. With these apparently flimsy assumptions it can be proved (see DeGroot 1970, Chapter 6) that a set of numbers exists which corresponds to the judgments of relative likelihood, in the sense that, if A is judged to be more likely than B, then the number for A is bigger than the number for B. What is more, these numbers must, collectively, satisfy the properties of probabilities; that is:

Decision theory (1) (2) (3)

33

O^p(A)^l; p(A or B) = p{A) + p{B) - p{A and B); p(A and B)=p(A)p(B\ A).

Here A and B are any events, and we represent the probability of an event X by p(X); the probability of B occurring conditional on A occurring is written p(B \ A). Axioms for probability 1.

For any two uncertain events, A is more likely than B, or B is more likely than A, or they are equally likely.

2.

If /A1 and A2 are any two mutually exclusive events, and 81 and B2 are any other mutually exclusive events; and if /*-, is not more likely than B-,, and A2 is not more likely than B2\ then (A| and A2) is not more likely than (B-, and B2). Further, if either A he will not mind, presumably, if we replace the prize S by the right to enter a second gamble {L, H;u(S)}. (This is the substitution principle, a source of considerable controversy.) We can make a similar argument about the prize T. Symbolically, we are asserting that the decision-maker ought to have the preferences: {S,T;q}~{{L,H-MS)},{L,H;u(T>};q}. This is represented in Figure 3.4. Now the two-stage gamble has four possible branches, but only two monetary outcomes, namely L and H. The probability of getting H is q' = qu(T) + (l-q)u(S). The two-stage gamble is clearly equivalent to the one-stage gamble {L, H; qu(T) + (1 — q) u(S)}9 as indicated in Figure 3.4. Observe now that S, T and q were arbitrary, so that we can assert that the decision-maker should be indifferent between any simple gamble with two prizes in the range [L, H] and a simple gamble on the standard prizes (L and H), with a probability of getting H given by the expression above.

Decision theory

39

Figure 3.4 Equivalence produced by the substitution principle. The decisionmaker is always ready to swap the sure thing X for a gamble on H and L with a probability u(X) of getting H.

Presumably the decision-maker will prefer gambles with a bigger chance of winning H. So, faced with a choice between several simple gambles of the form {S/5 T/;^,}, he should choose the gamble for which the expression is largest. This is just the expected utility of the gamble, however; that is, it is the probability-weighted sum of the possible utilities. We have thus concluded that the best course of action is to choose the option with the highest expected utility. This definition of utility has justified Bernoulli's insight that expected utility is a proper measure for the value of a gamble. Several times in this discussion we have made assumptions about the preferences of the decision-maker. The reader may or may not agree with us that they are plausible statements of how a rational decisionmaker might want to think, even if he did not always carry them out in practice. It is possible to take these assumptions, or others equivalent to them, as axioms, and from them deduce quite rigorously what we have just outlined above, namely that the axioms imply that the decisionmaker ought to choose the option which maximizes his expected' utility. One can axiomatize expected utility in several ways, and this has been achieved by several authors; but Savage (1954), in his joint axiomatization of subjective probability and utility, gives what is probably the

40

Theory Axioms for utility

1.

The decision-maker can place all possible lotteries in a preference order. That is, for any two lotteries, A and B, he can state which he prefers, or whether he is indifferent between them; furthermore, if he prefers A to B and B to C, then he prefers A to C.

2.

If the decision-maker prefers P^ to P2 and P2 to P3, where P^, P2, P3 are three prizes, then there is some probability p, between 0 and 1, such that he is indifferent between getting P2 for sure and a lottery giving him P: with probability p and P3 with probability (1 - p).

3.

If the decision-maker is indifferent between two prizes, P^ and P2, then he is also indifferent between two lotteries, the first giving P, with probability p and a prize P3 with probability (1 - p), and the second giving P2 with probability p and P3 with probability (1 - p). This must hold whatever P3 and pare.

4.

If two lotteries each lead only to the two prizes P^ and P2l and the decision-maker prefers P-i to P2, then he prefers the lottery with the higher chance of winning P1#

5.

Suppose lottery A\ gives a prize P^ with probability pit and a prize P2 with probability (1 - pi), for / = 1, 2, 3; and suppose lottery B gives entry to lottery A2 with probability g, and entry to lottery A3 with probability (1 - q); then the decision-maker is indifferent between lottery A^ and lottery B if, and only if; Pi = QP2 + (1 - q)Ps-

Exhibit 3.2 If an individual can subscribe to these properties of his preferences for lotteries, then it can be shown (see Baumol 1972, pp. 548-51) that a utility function must exist and that the rational action is to choose the option with the highest expected utility. most comprehensive account available. Exhibit 3.2 contains a version of these axioms, and the reader is referred to Baumol (1972, pp.548-51) for a proof that they imply that expected utility should be maximized. We shall return later to a discussion of whether the behavioral postulates set out in Exhibit 3.2 are reasonable ones to follow; for the present we point out that this theory merely shows that, if we wish to act in accordance with them, then we should always maximize expected utility. Several observations need to be made about the utility we have defined. First we see that, since L ~~ {L, H; «(L)}, u(L) = o. Similarlyit follows that u(H) = 1. Secondly, any other utility function will differ from u by at most a positive linear transformation; that is, if u' is another utility function, then

Decision theory

41

tt'(X) = a + bu(X) for some positive number b. To see this, note that the only arbitrary things about the definition of u are the reference prizes L and H. So suppose we construct another utility function using some other reference prizes, U and H', and, without loss of generality, we suppose that L' < L 0.1 x 1 + 0.89 x u + 0.01 x 0

52

Theory

or that u > 10/11. On the other hand, the judgment that C is better than D implies that 0.1 x 1 + 0.9 x 0 > 0.11 x « + 0.89 x 0 or that u < 10/11! Raiffa (1968, pp.80-6) provides an interesting discussion of the implications of this paradox. He gives two arguments why people might wish to reverse their preferences, and an argument that brings in another attribute, the presence of which makes the judgments consistent with expected utility theory. Our view of this paradox is slightly different. There is no moral imperative to adopt decision theory. Decision-makers are free to make their decisions in whatever way takes their fancy. The theory exists as a guide for those who wish to abide by its precepts, because they see them as providing a sensible framework for thinking through complex decision problems involving uncertainty. So if, after the inconsistencies of a set of judgments are pointed out to a decision-maker, he refuses to change any of them, the matter must rest. Rationality is defined in terms of adherence to a set of rules; there is no reason why a decision-maker ought to abide by the rules of decision theory, or indeed any rules for that matter! More recently there have been some more trenchant criticisms of the methods of decision analysis, particularly of the ideas of subjective probability and utility theory. Tocher (1977) is a good example of these. He argues that we do not obey the axioms of subjective probability, so that it has to be a normative, rather than a prescriptive, theory. With this we agree; he then goes on to say that proponents of subjective probability accuse others of irrationality if they are not prepared to attempt to make their judgments conform to the principles of subjective probability. Here he is too sensitive, for, if rationality is defined as conforming to a particular set of behavioral axioms, then a failure to do so is irrational by definition. What Tocher should be criticizing is an arrogant assertion by some theorists that their rules of behavior, defining what it is to be rational, are the only ones. As we argued in Chapter 2, we believe that each person is free to choose these rules as he or she thinks fit. If he or she wishes not to behave in conformity with the axioms of subjective probability, so be it. We must point out, however, that other frameworks for supporting decision-making in a consistent manner are quite scarce! Tocher has a further criticism of utility theory which has more practical import. He points out that to apply the theory a utility function has to be

Decision theory

53

elicited, and to do that the decision-maker has to ask himself hypothetical questions about his choice between gambles which he may never face in actuality. As Tocher says: Such experiments in the mind offer no guarantee as to the subject's reaction when faced with a similar real choice; we all know from bitter experience that men do not act as they say that they will. Tocher 1977 We will return in Chapter 4 to discuss what is known about man's ability to make such judgments. It will be sufficient here to reiterate that one is not bound to follow the procedures of decision analysis. We see them rather as a sensible set of ideas to structure, in a consistent and comprehensible way, a complex and uncertain problem. We use a utility function not because we are convinced that it is the "correct" one, but because it represents our risk preferences at the time we are making the decision as well as we can determine; the adoption of a utility function is the adoption of a basic set of attitudes, in conformity to which we wish to act, and from which we can use the precepts of decision analysis to determine a consistent set of judgments. Still more recently, Dreyfus (1984) has argued that the analytical philosophy of decision analysis, which breaks a decision problem down into small judgmental tasks, such as eliciting separate probabilities and utilities, is at variance with the holistic approach to problems taken by most experienced decision-makers. He argues that decision analysis has not caught on as some were predicting twenty or thirty years ago because real decision-makers are typically not interested in the analytical approach (and here he echoes another point made by Tocher). In an interesting reply, Brown (1984) argues that expected utility theory can be, and has been, of use. Brown has also been an author of two interesting reviews of the use of decision analysis (Brown 1970, Ulvila and Brown 1982) which show that, despite these important criticisms of subjective probability and utility theory, there are many people, in many different contexts, who have used the ideas effectively. 3.3.5 The value of information Some decision analysts argue that one of the most valuable insights provided by expected utility theory is its ability to handle the value that some potential piece of information might have; we will discuss this notion in this section. Faced with a difficult decision involving uncertainty, many a decision-maker will seek to reduce the uncertainty before making the decision. Very often, however, getting the information will be a costly and time-consuming business; so before collecting the information it would be well to assess whether the cost is worth the value gained.

Theory

54

Final asset position (£) 2,800

Keep investment

1,800

800

Buy information about success

2,800 -p Sell investment 0.9/

Associate pays

1,800-p

Associate ~p

does not pay

0-p Keep investment Associate

Sell

\

investment

0.9/

—pays Associate does not pay

1 800

-

-"

800 -p

Figure 3.12 Decision tree to compute the value of perfect information. refers to the node with this label in Figure 3.5.

0

We once more consider the example of Figure 3.5. There are several uncertainties in this problem, but let us concentrate for the moment on whether the investment will succeed, supposing it gets beyond the first stage. Imagine that it is possible to discover whether or not the-investment will succeed, prior to deciding about selling the option, but at a price. How much is it worth paying for this information? The situation is represented by the decision tree of Figure 3.12.

Decision theory

55

Note that Figure 3.12 differs from Figure 3.5. by moving the chance node for investment success from after the final decision to before it. p is the price at which the information is available to the decision-maker and it is conventional to represent the fact that an extra payment is made if information is bought by a gate across the appropriate path in the decision tree, as in Figure 3.1 z. Presumably the probabilities that the information will imply success or failure are the same as those that were assessed before we started talking about information, as indicated in thefigure.It is easy to see, by folding back the tree, that the expected utility from buying information at a price p is 0.7w(2800 - p) + 0.3(0.9^(1800 - p) + 0.1w(800 - p)) and the decision-maker should buy this information so long as this expression exceeds 0.91, the expected utility of selling the investment, the best course of action without further information. We define the value of perfect information here to be equal to the value of p which insures equality. Computing this value otp is quite difficult, since the function u is defined by the empirical curve in Figure 3.6. Even if we find a suitable analytic approximation for this curve the solution may not be straightforward, since the equation may have to be solved numerically. This is merely a computational problem, however; the reader may check that, when p is about £710, the expression above is about 0.91. (We have, in fact, calculated this value by using an analytical form: the utility curve in Figure 3.6 is very close to an exponential, with coefficient of risk aversion equal to 0.00058. If readers wish to follow the details of the calculations in the rest of this chapter, we suggest that they use this analytic form.) We now argue that it is not worth paying more than £710 to learn in advance about how our uncertainties will be resolved. This concept is sometimes referred to as the value of clairvoyance; but we prefer to call it the value of perfect information, to contrast with the value of imperfect information, which we discuss below. To be more formal, we say that the value of perfect information about some uncertain event is equal to the maximum amount that is worth paying for the uncertainty to be resolved prior to making the relevant decision; it is that sum which, when deducted from all possible outcomes on the decision branch "buy information," reduces the expected utility of that branch to be equal to the next highest expected utility on other branches. There are some circumstances where the mere realization that an upper limit exists to the amount worth paying for information may cause a decision-maker to think very carefully before undertaking an expensive information-gathering exercise. The value of perfect information may be zero, of course. Consider, in Figure 3.12, that information may be available about whether the

Theory

Final asset position (£) Buy information about success

Buy information about associate

Investment will fail

Buy information about associate

800 - p - p '

Figure 3.13 When the value of perfect information is zero. It is not worth buying information about the associate, since this information will not change the decisions. [A\ refers to the node with this label in Figure. 3.5.

Decision theory Success

57 Final assets (£) 2,800

0-7

Keep investment

does not pay

800 -p

Figure 3.14 Value of imperfect information. The consultant advises the decisionmaker on whether the investment will succeed. unreliable associate will pay up, but after information about the other uncertainty is resolved. This situation is represented in Figure 3.13. It is easy to see that in this case the value of information about the associate is zero, no matter what the state of knowledge is about the business venture. On folding back the tree of Figure 3.13, we see that, no matter how small the fee paid for this information, it is not optimal to buy it; and this is true on both branches of the tree. The reason for this is that the information will not change our preferred decision; if the venture will be successful then we will not sell it, no matter if our associate turns out to be reliable, and if it will be unsuccessful, then we will sell, even if he does not-pay in the end. Information value is quite a subtle concept; it is not something which can be permanently attached to an item of information, since value

58

Theory

depends very much on context. If, for example, we were to ask w h a t the value of perfect information about the reliability of the associate would be in the absence of information about the outcome of the business venture, then the answer is that it is positive and equal to about £ 8 9 . The reader might like to check this by constructing a decision tree similar to that of Figure 3.12 and folding it back. In practice, perfect information is very rarely available. When, for example, the managers in a company are involved in market research to estimate h o w well a new product will sell, their uncertainties are, typically, reduced rather than removed altogether. This imperfect kind of information can also be evaluated using expected utility theory, although the considerations are a little more involved. To illustrate these ideas, imagine that instead of perfect information being available, as in Figure 3.12, we have the option of buying advice from a consultant. He will give us his opinion on whether the investment will succeed or not. In the light of that advice we then have to decide whether to sell or not. If the consultant were totally reliable, of course, then we would be in receipt of perfect information, and the analysis above would apply. It is more probable, however, that all that the consultant does is to revise our subjective probability of success. The situation is set out in the decision tree which is partially represented in Figure 3.14. Before folding back the tree to compute the m a x i m u m a m o u n t of money worth paying to the consultant, the decision-maker has to determine w h a t probabilities have to be applied to the branches at each chance node. Perhaps the most obvious probabilities to assess first are the ones that have to be applied at the end nodes, namely the conditional probabilities that the investment will succeed, conditional, that is, on what the consultant says. Let us suppose that the consultant is quite reliable, so that we assess the probability of success, given that he predicts success, at 0.8. Suppose also that the probability of failure, given the consultant's prediction of failure, is 0.7. The only probability missing from the tree is n o w that for w h a t the consultant will say! If we are to be coherent, however, the probability that the investment succeeds, which we have previously assessed at 0.7, must also be calculated as equal to 0.7 when it is computed through the appropriate conditional probabilities. Thus to obey the rules of the probability calculus, that is, to be coherent, we must adjust our probabilities so that: Pr(S) = Pr(S I s) Pr(s) + Pr(S \f) Pr(/) where S stands for the success of the investment, and s, f sta-nd for the prediction of success and failure, respectively, by the consultant. It is easy to see that the only value of Pr(s) consistent with those probabilities that have already been assessed (namely Pr(S) = 0.7, Pr(5 | s) = 0.8,

Decision theory

59

Pr(S I f) — 0.3) is 0.8. This is the probability that we have associated with the consultant's prediction of success in the decision tree. Note that this particular sequence for eliciting probabilities is just one among many. In practice it might be necessary to get at the appropriate probabilities in many different ways, and to overspecify them. We discuss the practical elicitation of probabilities in Section 7.4; all we need to stress here is that it is important for all the probabilities which enter a decision tree to be consistent with each other. Now we can find the value of p in Figure 3.14 which reduces the expected utility of the bottom branch to that of the second branch. It is equal to £82, as the reader may check, by folding back the decision tree with this value of p. This is the value of imperfect information in this case. Not surprisingly, the value of the consultant's information is less than the value of perfect information. This will always be the case, although it is quite tricky to prove it in general. Thus the value of perfect information is an upper bound on the value of imperfect information; and if, as is usually the case, the computations required in the case of imperfect information are complex and difficult, a calculation of the value of perfect information may be sufficient to determine whether a proposed information-gathering exercise is worthwhile. EXERCISES FOR

3.3

1. Clare and Adrian, students on a course in decision analysis, take part in some electronic games. The first game involves flipping an electronic coin which has been set to come up heads with probability 0.7 and tails with probability 0.3. Players win £5 on heads, but lose £5 on tails. The second game uses the same electronic coin, but this time, if it lands heads, the player flips a second electronic coin, with a probability of heads equal to 0.2. If this second coin lands heads, the player wins £15; if tails, he or she wins £5. If the first coin lands tails, a second coin is flipped with a tails probability of 0.3. If this coin lands tails, the player loses £15; if heads, the loss is £5. Clare and Adrian may each have one play at one game or the other, but not both. (a) Draw out the decision tree that faces Clare and Adrian. (b) Clare is risk-neutral for the range of prizes offered, but Adrian is risk-averse, and he estimates his utility function for gains or losses to be: u(x) = - 0.0008 x2 + 0.033 x + 0.68. Which game maximizes expected utility for (i) Clare (ii) Adrian?

60

Theory (c) What is the value of perfect information, (i) for Clare (ii) for Adrian, on the outcome of the flip of the coin in the first game? Suppose that information is available, at a price, on the outcomes of the flips of both possible second coins in the second game. What is the value of this information for Clare? (d) Suppose that Clare can perform a test which yields partial information on the outcome of the coin flip in the first game. Analysis of this test shows that the first coin will come up heads with probability 0.9 when the test says "heads will occur," and that the coin will land tails with probability 0.7 when the test says "tails will occur." How much is this test worth to Clare?

2.

A decision-maker's utility function, w, for reward, x, is given by u(x) = a — be~x/5, where a and b are positive constants. The decision-maker is faced with a choice of doing nothing with reward zero, or undertaking one of two gambles, A or B. Depending on which of three events X, Y, or Z occurs, gamble A will yield rewards of — 5, 10, or 15 respectively, and gamble B will yield rewards of — 10, 20 or 30 respectively. One and only one of the events X, Y, and Z is bound to occur and event Y is thought to be twice as likely as event X. Calculate the ranges of values of p, the probability of event X, for which each decision alternative is optimal.

3.

Show that the utility function for money given by ln(x + 35)/ln(35) — 1 represents decreasing risk aversion (i.e. the coefficient of risk aversion is a decreasing function of x). A decisionmaker with this utility function has assets of £50 and realizes that £25 of it is in a risky situation which has one chance in ten of collapsing and losing him his money. What is a reasonable premium for him to pay for insurance against the loss? What would the premium be if his assets were £100?

4.

An insurance company has assets of £40111 (40 million pounds), and its utility is given by w(x) = ln(x + 35)^(35) — 1, where x is measured in millions of pounds. It is asked to insure against an earthquake disaster of £40111 and assesses the chance of the earthquake at 0.05. What is a reasonable premium for it to request? (Remember you are looking at the situation from the company's side, not from that of the earthquake sufferers.) Administrative expenses can be ignored. The company finds a second company with the same assets and utility function and agrees to share the risk. That is, it insures against a loss of £2om with probability 0.05. What is the premium now? Show that even twice this premium is less than the original premium

Decision theory

61

so that the insurance is improved from everybody's point of view by the companies sharing the risk. An individual's current wealth is £x. He is offered the opportunity to invest part of this wealth, an amount £s, in a venture which may yield an immediate benefit of £r where r > s. There is, however, the possibility that the venture will fail, yielding no benefit. Describe the concepts of utility and subjective probability, and outline in this particular case how they might help the individual to decide whether or not to invest. If the individual's utility function for wealth £y is ln{y) and if he considers that the venture has an equal chance of success or failure, show that he should invest if: 6. By choosing appropriate analytic forms for the utility functions, show that the buying and selling prices for afifty-fiftychance of winning £20 are the same for a decision-maker with a constant risk aversion, but differ if he has decreasing risk aversion. Why is this? 7. An entrepreneur feels that there is an even chance that an event X will occur and that a completely independent event Yhas a probability 0.4 of occurring. He owns a business venture which will yield a profit of £100,000 if X occurs but nothing otherwise. He is offered in exchange for ownership of this venture an alternative business opportunity which will yield £80,000 if Y occurs and £30,000 if it does not. He has linear utility for money. Perfect information may be purchased about X and Y. Show that the value of perfect information about Y, once it has become known whether X has occurred, is independent of this information about X, but that the value of perfect information about X, once it becomes known whether Y has occurred, does depend on whether or not Y has occurred. The entrepreneur can buy perfect information about X for £30,000 and about Y for £11,000. He can buy in any order, or not buy at all. What should he do? 8. An investor can speculate on a stock by buying long or by selling short. Alternatively he can make no investment. The investor makes a good investment if he buys long and the stock price rises, or if he sells short and the stock price falls; by doing this he makes a profit/) f £k. If he makes a bad investment he will lose £k. Initially, he thinks the price is equally likely to rise or fall. His utility for a monetary increment of x is u(x).

6z

Theory A consultant offers to predict how the stock price will move. The investor believes that the consultant has a probability of p {p > 0.5) of correctly predicting whether the stock price will rise or fall. Find an expression for f, the value of imperfect information provided by the consultant. Show that if the investor has a constant coefficient of risk aversion c, then the value of imperfect information is: f = - (1/c) ln[min{l, pe~ck + (1 - p) ec*}]. Check that this formula gives the correct value of /", both when the consultant is completely informative and when he is completely uninformative.

3.4 RISK PREFERENCE AND CONFLICTING OBJECTIVES

3.4.1

Introduction

So far we have seen how to construct a calculus for evaluating decision options when the decision-maker has conflicting objectives and no uncertainty; and also a single objective but considerable uncertainty. In reality decision problems usually involve both. For example, the car choice problem in Section 3.1, while perhaps being satisfactorily approximated by ignoring uncertainty, may well involve uncertainty about how the cars will in fact perform once they have been purchased. In choosing a job, as we discussed in the last section, a job-seeker will probably want to include more attributes than simply the location of possible employments. Even problems which may at first sight appear to be single-attributed may turn out to be better formulated as a multi-attribute problem. Consider the managers of a company, contemplating a new project. There may well be considerable uncertainty about the financial outcome of the project. What will the costs be? Will the cash flow avoid excessive bank loans? — and so on. It would appear that all the outcomes could be described in terms of a single variable, namely money; so, by using discounting to reduce a potential future cash flow to an equivalent present value, the single-attribute theory developed in the last section could be applied directly. Often this will be a sensible approach to decision-aiding for commercial organizations, but sometimes it will be useful to represent the problem as a multi-attribute one. For example, market share, company image or employee morale may be better expressed as separate variables, rather than attempting to find monetary equivalents for them at the beginning of the analysis. Case studies in sections 6.5.3, 8.3.3, 8.4.3

Decision theory

63

and 8.6.2 of this book, as well as the case reported in Phillips (1982), illustrate such multi-attribute problems. We clearly need to be able to handle uncertainty and conflicting attributes at the same time; this section is devoted to discussing how this might be done. Our first observation is that the theory is straightforward for extending the single-attribute utility theory, developed in the last section, to the multi-attribute case. Suppose that a set of attributes, Xi, . . ., Xw, has been identified, and that all conceivable outcomes of the variable x, are less desirable than xj and more desirable than xf; i.e. [xf, xj] includes all possible outcome values. We shall write the vector (x\,. . ., x*n) as x*, and similarly (JC?, . . ., x%) as x°. Then the utility of a set of outcome values x will be defined as equal to the probability p that makes the decision-maker indifferent between x and

{x°, x*;p}.

With this definition of utility, the theory may be developed exactly as before, leading to the recommendation that expected utility should be maximized. The chief difficulty with this approach is its practicability, however. While there are non-trivial problems in determining satisfactory singleattribute utility functions, as we shall discuss in Section 7.5.2, these are magnified when dealing with multiple attributes. Part of the power of decision analysis lies in the replacement of complex judgmental tasks by simpler ones; determining the value p in the expression above is by no means simple. Several different approaches have been suggested for coping with this difficulty and we will describe some of them in the next sections. 3.4.2 Keeney's approach The assessment of a multi-attribute utility function would be much simplified if it could be written as some simple function of single-attribute utility functions; then we could assess the single-attribute functions without reference to the other attributes, and so reduce the complexity of the utility assessment task to manageable proportions. An additive form is perhaps the most simple, and the search for conditions that would justify using the additive form was an active research area for some time. This work is well described in the seminal text by Keeney and Raiffa (1976), which has now become a standard reference work; the reader is recommended to consult it for a fuller account of the theory we present here. Keeney's theorem, which we will now state, gives sufficient conditions for a multi-attribute utility function to have a simple form which is slightly more general than the additive one but which gives additivity in a special

64

Theory

case. The conditions Keeney needs are preference independence, which we defined in Section 3.1, and utility independence. An attribute X is said to be utility independent of other attributes Y, Z, . . . if the decision-maker's preferences amongst gambles on X do not depend on the actual levels of the other attributes. We consider first the case of three attributes only. Suppose that: (a) X and Y are preference independent of Z; (b) X and Z are preference independent of Y; (c) X is utility independent of Y and Z. Then Keeney's theorem proves that the utility function for the attributes X, Y, Z, namely u{x, y, z), must be such that: (1 + ku(x, y, z)) = (1 + kkxux(x))(l

+ kkyuy(y))(l + kkzuz(z))

where k, kx, ky and kz are constants. The proof of this result is given in Section 6.3 of Keeney and Raiffa (1976). What is more, each of the utility functions is normalized so that

u(x°,yoyz°) = 0; u(x*,y*,z*) = 1; ux(x°) = 0; ux(x*) = 1; and similar expressions hold for the other single-attribute functions. We can now use this result to assess a three-attribute utility function, in certain cases, namely those where the independence conditions hold. First we have to check that the decision-maker's preference structure does indeed display these independence properties. Keeney and Raiffa (1976, Section 6.6.1) give some suggestions of how this might be done, largely involving asking the decision-maker whether his judgments of equivalence between many hypothetical options are indeed independent of the other attributes as required. Examples of these questions are presented in Section 6.3.2. Determining whether the independence conditions really hold is a difficult and time-consuming business, and this is one of the many areas where Keeney's approach has been criticized (we shall discuss others later in this section). In the spirit of interpreting decision theory which we have adopted throughout this chapter, however, we argue that analysis might create a preference structure, rather than simply reveal it, at least in part; thus, if the decision-maker is satisfied that, as far as he can tell, he does no injustice to his preferences by assuming that the independence conditions hold, he will not be led astray by assuming this in his analvsis. On the assumption that they do hold, the utility function must take the multiplicative form given above. Moreover, by setting all three variables to their upper limits in that equation, and recalling the normalization conditions above, we see that: 1 + k = (1 + kkX)(l + kky)(l + kkz)

(3.2)

Decision theory

65

By setting x to its upper value, and y and z to their lowest, we get: u(x\ y°, z°) = xk with similar expressions for the other two variables. It is not easy to avoid at least some of the difficult judgments from which we are trying to escape in setting up this theory. Keeney and Raiffa (1976, Section 6.6.6) suggest that the scaling constants may be determined by direct assessment of the appropriate utility. That is, to determine kx we ask the decision-maker what the value of p is such that he is indifferent between (x\y°,z°)

and

{(x°, y°, z°), (**, **); y\ p).

By the definition of utility and the expression above we conclude that kx = p. On deriving each of the scaling constants in this way, they may be substituted in Equation (3.2) above to give a cubic equation for k. The condition that the utility function must lie between zero and unity will determine which root of this equation to choose. Note that if kx + ky + kz = 1, then k = o, and the utility function reverts to the additive form. This theorem may be generalized to arbitrarily many dimensions, and then it can be used to assess utility functions in these more general cases. Considerable prominence has been given to this approach for the multiple attribute problem over the last ten years, and there have been many reports of its use on practical problems. (See Kirkwood 1982 or Bell 1977 as just two among many examples.) The number of practitioners of decision analysis making these applications has remained small, however, and some significant criticisms can be made of this approach. First, to carry it out properly, considerable effort and time is required, both from the analyst and from the decision-maker, and in many contexts this is not available. Secondly, there are some practitioners who use the method to construct a multi-attribute value (rather than utility) function for assessing options in the situations where there is no uncertainty. (Observe that any multi-attribute utility function will also be a value function, but that the reverse is not true.) This may seem a surprising procedure to adopt, since it involves determining risk preferences when they need not be revealed for the decision problem to be solved. It seems very strange to elicit a value function to aid decision-making in the absence of uncertainty by asking somebody questions about his attitudes towards gambles. Finally, other methods exist for establishing satisfactory utility functions which are simpler to use. We go on to discuss these now. Further discussion of this approach is given in Section 6.3. 3.4.3 The Stanford approach The application of decision analysis, in common with other practical subjects, such as psychoanalysis or architecture, has a markedly different

66

Theory

flavor in different schools of practice. The problem of constructing multi-attribute utility functions which properly reflect risk preference occurs whenever decision theory is applied to a practical problem. Yet it appears that the Stanford school, in which the key figure has been R.A. Howard, has not perceived the need to follow the procedures of the last section, which can be associated with the Harvard school, for which the central figures have been H. Raiffa and R. L. Keeney. The procedures followed by the Stanford school can be best seen in the series of readings published by Howard and Matheson (1984). In modeling multi-attribute problems, the Stanford school recommends first constructing a multi-attribute value function, as we described in Section 3.1. Since value is a single attribute, it must be that U{XU X 2 , . . ., Xn) = U'(V{XU X 2 , • • ., X n))

where u'( . ) is a single-attribute utility function and v( . ) is the value function constructed. Thus the problem offindinga multi-attribute utility function is reduced to two perhaps more tractable problems; first a value function is constructed, and then we find the utility of that value function by normal single-attribute methods. This is consistent with the recommendation of the Stanford school that any decision analysis should pass through a deterministic phase, where a model of the problem is constructed without reference to uncertainty, before the probabilistic phase, when uncertainty and risk aversion are explicitly considered. This approach certainly avoids the criticism we mentioned above, namely, the inappropriateness of asking gamble questions in the absence of uncertainty. It is in fact not necessary to construct a complete value function for this method to work. Suppose that instead of constructing value functions with a particularly simple form, as we did in Section 3.1, we trace the curves of constant preference (the isoquants) to one of the boundaries. That is, for any point x, we find the value of x\9 namely X\{x), such that:

This procedure is termed pricing-out, since the variable chosen is usually money. Now we may construct a single-attribute utility function for Xi, conditional on X2,. . ., Xn all having their lowest values. There are problems with this approach, however. Sometimes it is difficult to carry out the pricing-out properly. As Keeney and Raiffa (1976, Section 3.8) point out, it is only proper to price out each variable separately if certain conditions on preference independence hold. Moreover, in some cases it may not be a very easy set of judgments to make. Since one of the chief purposes of decision analysis is to replace difficult

Decision theory

6j

judgments by easier ones, we should be careful that our procedures do, in fact, satisfy this condition. 3.4.4 Comments We have seen two distinct approaches to constructing a multi-attribute utility function, but this does not exhaust the possibilities. A more detailed procedure, for example, is given in Keelin (1981). Moreover, it is possible to carry out utility elicitation procedures which simplify the assessment task, even if the preference structure does not satisfy the required independence conditions for either of the two methods above. For example, Bell (1979a, b) explores some ways to use conditional utility functions. We have not seen reports of the use of these methods on practical problems yet, and it may be that they are too complicated to be used in any but the most sophisticated analyses. Summarizing then, we see that there is a theoretically sound definition for the utility of multi-attribute consequences which may be used in the maximization of expected utility. It is difficult to assess this directly, however, and various different approaches exist to assist a decision-maker in indirect assessment. The approach adopted may be more or less sophisticated, and the analyst will need to exercise judgment here, as in so many other parts of decision synthesis, on the proper balance between accuracy and the use of resources. EXERCISES FOR

3.4

1. In making decisions about this company's future, a manager is concerned about three conflicting attributes: cash flow, safety, and company image. The manager's value structure is such that cash flow is utility independent of safety and company image, cash flow and safety are preference independent of company image, and cash flow and company image are preference independent of safety. Company image and safety are measured on a o— ioo scale, and the manager's utility for these attributes is linear with respect to these measures. He is risk-averse to cashflow,however, with a constant coefficient of risk aversion of (lnc2)/ioo (K£)~a. He assesses the following combinations of attributes to be equally desirable: Cash flow (K£)

Safety

Company image

IOO

0

0

0

IOO

0

0

0

IOO

He also judges that a gamble giving a one-third chance of (ioo, 100, 100), and a two-thirds chance of (o, o, o) is just as desirable as

68

Theory (ioo, o, o) with certainty. Derive a von Neumann—Morgenstern utility function for the manager with respect to these attributes. The manager has to choose between two options; thefirstproduces an uncertain cash flow, whose probability distribution is negative exponential with mean 80 K£, a safety score of 80, and a company image of 20; the second again produces a negative exponentially distributed cash flow, with mean 70 K£, but in this case the safety score is 45 and the company image is 60. Which option should the manager take, and why? (K£ = £1,000)

2. A decision-maker wishes to construct her multi-attribute utility function for choosing her job. She identifies three attributes which will adequately describe the options available to her: initial salary, measured in K£ per year; salary after ten years, also measured in K£ per year; and challenge of the job, which she measures on a o—10 subjective scale. She believes that her preferences for job challenge are utility independent of the other two attributes. What is now needed to insure that her utility function is additive? She assumes that these conditions are satisfied, and goes on to make the following assessments of indifference judgments: [(9, 15, 10); (5, 9, o); 0.3] ~ (9, 9, o) [(9, 15, 10); (5, 9, o); 0.5] ~ (5, 15, o) [(9, 15, 10); (5, 9, o); 0.4] ~ (5, 9, 10) where {a,fo,c) means the values for the three attributes above, in that order, for a particular outcome, [(a, b, c); (d, e, /); p] means an uncertain outcome giving a probability p of getting (a, b, c), and a probability 1— p of getting {d> e, /). The symbol ~ indicates indifference. Is the decision-maker justified in assuming that her utility function is additive? 3.5 OTHER NORMATIVE APPROACHES

3.5.1 Introduction We have so far described an approach to handling decision problems which is based on reasonable behavioral postulates; we have constantly stressed, however, that there is no obligation for any decision-maker to follow this approach. It is not the case, in our view, that a man is necessarily committing a folly if he fails to follow the precepts of classical decision theory, since he may well not be willing to accept that he can make the preference judgments that the axiomatic development of decision theory requires. Moreover, arguments that show that he will

Decision theory

69

inevitably lose in the long run if he fails to conform to the axioms (such as the Dutch book or money pump arguments for subjective probability see Lindley (1973)) will rarely be supported by practical instances of their occurrence. Most people will modify their judgments, or, more likely, refuse to make them, before unacceptable loss appears. We do not take an imperialist view of decision theory, then, as some advocates do. Rather, we see the framework as a sensible one to adopt when confronting uncertainty and conflicting objectives, and our experience is that many decision-makers agree with us, as the case studies mentioned in Part II of this text testify. We feel it proper, therefore, to describe several alternative normative frameworks that have appeared in the literature of the last thirty years, particularly for attacking the problem of conflicting objectives. Some of these methods are closely related to the decision theory we have described in this chapter, while others stand more on their own. Although there are numerous versions of these techniques, we will outline justfiveof the most prominent of them in this section, in each case giving references where the interested reader can further study the ideas. Some readers may think that the distinctions we make between these methods are not sharp, since there are close links between some of them and with classical decision theory. The principal difference between them, however, is that each procedure is championed by one or more key academics, or by distinct academic traditions. After our brief description of each method, we will discuss, in Section 3.5.7, what aspects of each method are worth considering for use in a particular situation. 3.5.2 Cost-benefit analysis When looking for a technique to compare the positive with the negative effect of possible courses of action, many people will say that what is needed is a cost—benefit analysis. In a general use of this term, this is indeed true; what is the search for a multi-attribute value function if it is not an attempt to balance costs with benefits? Nonetheless, cost—benefit analysis (CBA), as developed in economic theory, is, from our viewpoint, a rather specialized technique, designed specifically for the economic appraisal of social projects, rather than as a general decision-aiding procedure. In carrying out a cost-benefit analysis, however, an implicit construction of a multi-attribute value function takes place, and so in this section we give a very brief summary of CBA, showing how it differs from more general methods. In applying CBA to determine whether a project should be undertaken or not, all the factors of importance are identified, outcomes are measured for each factor, a price is determined for each factor, and all the resulting

jo

Theory

costs are added up. In so doing, an additive multi-attribute value function is constructed. Prices may be interpreted as weights, and then the net cost is simply a weighted sum. At first sight, therefore, there may appear to be little to distinguish CBA from the techniques we have been describing here. There are, however, several important differences in the way numbers are computed for entry into the value function, and the meaning of the answers is interpreted very differently. (See Watson (1981) for a discussion of this.) The most important of these is the philosophical justification of the techniques. The goal of CBA, as developed in economic theory, is to provide an objective measurement of the economic effects of a project. The question to be answered is whether society as a whole will be better off as a result of undertaking the project. Our goal in this chapter, however, has been the very different one of providing a framework for an individual decision-maker to tie together his judgments in a consistent way. As we shall see in Chapter 5, these goals merge to some extent when decision analysis is applied to public policy issues. Despite this, they still remain distinct. Other important distinctions include the handling of equity and the assumption of additivity. Since CBA is predicated on an assumption that net social benefit should be maximized, if one man gains more (in money terms) than another man loses by a transaction involving only the two of them, then the value function of CBA will approve of the transaction, even though equity considerations might render the transaction ethically unacceptable. Multi-attribute value theory is broader in allowing equity to be explicitly considered, by, for example, a measure of inequity featuring as an attribute in the value function. There is a large literature on CBA, and criticisms such as these have not gone unanswered by CBA theorists. Readers interested in studying CBA further are recommended to consult some of the standard texts, such as Layard (1972), Sassone and Schaffer (1978) or Sugden and Williams (1978). Discussions on the limitation of the ideas in practice are also important; a good critical review is given in Fischhoff et al. (1981). In summary, then, CBA has been developed as a tool for the objective analysis of the social desirability of public projects, measuring the relative importance of different factors by prices in the market-place. It was not designed as an aid for an individual decision-maker trying to weigh up his options against the conflicting objectives that he faces. The other four techniques we shall look at, on the contrary, do have this goal. 3.5.3 The multi-criterion decision-making movement A group of researchers, principally in the United States, has developed what has come to be known as the multi-criterion decision-making

Decision theory

71

(MCDM) movement. MCDM is a set of ideas distinct both from classical decision theory and from the main stream of management science. The text by Zeleny (1982), who is one of the key figures in this movement, provides an excellent account of the ideas of MCDM and a wealth of useful references. The intellectual roots of MCDM lie in the study of linear programming (LP). The invention of the simplex algorithm for discovering the optimum of a linear function when there are linear constraints on the variables was a most significant step in the development of management science. For the first time it was recognized that many logistic problems could be considered in this way, and considerable savings were achieved by applying linear programming in a wide range of contexts. Many academics studied developments of LP, and the technique was one of the principal tools of operational research in the 1960s. It was recognized, however, that determining the appropriate objective function in an application was not at all straightforward. Most decision problems involve conflicting objectives and, at least at first sight, the objectives can appear incommensurate. In the LP world, it became natural to study what could be done with linear programs where more than one objective had to be taken into account. Terms such as multi-objective linear programming, goal programming, and compromise programming were used to denote new ways of looking at the problem, usually associated with new algorithms. As Zeleny (1982) points out, the MCDM movement now embraces wider concerns than just the computations (his book contains interesting chapters on the process of decision-making and the generation of decision options). Nonetheless, its methods are still principally centered on the application of concepts from mathematical programming to the multiple criterion problem. The methods are generally suited to problems where there is a continuum of options (e.g. control variables that may take any values in a given range) rather than a set of discrete alternatives. There are, however, some important contributions that the MCDM movement can contribute to the classical decision theory that we have described in this section. The most important of these comes from interactive multiple criterion programming. As we saw in Section 3.1, in applying multi-attribute value theory to a problem, it is necessary to compute weights before the value function can be used to guide choice. It may be, however, that a decision-maker's judgment about weights can only be effectively elicited when real decision options are before him; to ask him questions about what his judgment would be in many hypothetical situations that he is not actually facing involves a degree of mental agility and role-playing that is not widespread among decision-makers. The interactive versions of multiple criterion programming are not subject to this criticism, since they entail the exploration of the set of good options

72

Theory

interactively with the decision-maker. Typically, once the several conflicting objectives have been represented as (usually linear) objective functions, the interactive computer program will explore the Pareto-optimal boundary, allowing the direction of search to be specified at each stage by the decision-maker who is using the computer. Although this still involves asking the decision-maker hypothetical questions about choices he may know that he does not want to consider, it has the potential for being far more useful as a tool for eliciting values than the more straightforward approach of determining weights at the outset. We believe that decision aids which interactively elicit trade-off judgments at the same time that the option space is explored are likely to prove useful in the multiattribute value analysis methods we described in Section 3.1. 3.5.4 Social judgment theory The study of decision-making attracts researchers from many different disciplines. Engineers, mathematicians, statisticians, and philosophers, as well as management scientists, have made many contributions to our understanding of how to improve decision-making. It is to psychologists that we look, however, for an understanding of how decision-making actually takes place, and, as we will describe in Chapter 4, there is a very large literature on this subject. Many of these psychologists have recognized, however, the importance of going beyond the study of how decisions are made, to how decisionmaking could be improved. K. R. Hammond, of the University of Colorado, one of these innovators, is the leadingfigurein social judgment theory, the goals of which theory are perhaps best explained by quoting Hammond directly: Social judgment theorists are interested in creating cognitive aids for human judgment- particularly for those persons who must exercise their judgment in the effort to formulate social policy and who will ordinarilyfindthemselves embroiled in bitter dispute as they do so. Social judgment theorists intend not only to understand human judgment but to create and develop ways of improving it. Hammond et al. 1975, p.276

Social judgment theory is concerned with far more than a technique for determining the best action to take when confronted with conflicting objectives; but it does contain such a methodology. Once attributes have been determined and options laid out (Hammond refers to these as cues and profiles, respectively), the decision-maker is asked for his holistic judgment on the attractiveness of a representative set of options, measured on some suitable scale. A multi-attribute value function, generally of the additive and linear variety, is then fitted to these data by

Decision theory

73

the method of least squares. Since this is a regression analysis, the number of options needed to make an adequate fit goes up as the number of attributes increases. No attempt is made to check whether the appropriate independence conditions are satisfied, although Hammond does allow for non-linear models to be used in certain circumstances. In fact, even the issue of statistical independence is largely ignored, although there are circumstances where the collinearity of data may seriously affect results. In practice, it seems that the method of SJT uses a linear, additive, multi-attribute value function, where the weights are determined by the application of multivariate linear regression to some holistic judgments. We could summarize by saying that a decision-maker's preference structure is elicited by fitting an appropriate functional form to his expressed judgments in a few cases, and then used to guide what his preference ought to be in other cases. There is more to SJT than this, however. Hammond lays stress on the importance of feedback in improving judgment. To this end an interactive computer program has been constructed that presents pictorial information to the decision-maker, and this assists him in refining his judgments (see Hammond et al.y p.6). This program has also been used to assist social judgment when there are many interested parties. Hammond et al. (1980) report several applications of their method where the preference structures of different people involved with the decision were elicited using the program. These were then analyzed using the methods of cluster analysis to identify factions. This information was useful in clarifying the distribution of views among the decision-makers and, more importantly, to separate fact from value. This last property is seen by many users of multi-attribute decision-aiding techniques (not only the users of SJT) as being one of the most important advantages of using an analysis of this kind. In summary, then, SJT has much in common with multi-attribute value theory. SJT differs from other approaches, though; the technique is rooted in a descriptive psychological theory of judgment, and preference structures are elicited by the use of holistic judgments and regression analysis. 3.5.5 Outranking An approach to the problem of decision-making in the face of conflicting objectives which has become widely known in continental Europe (and particularly in France and Belgium) is the set of methods known as "outranking" {surclassement) developed by*B. Roy at the University of Paris. Succinct descriptions of Roy's work in English are not widely available, but extended accounts were given in some conference proceedings during the 1970s (Roy 1973, 1977). We shall now describe the

74

Theory

method recommended by Roy in 1973; we should stress, however, that in recent years there have been developments of these methods. Readers interested in studying the development of outranking are recommended to study the recent literature. Descriptions (in French) of outranking and its relation to other methods can be found in de Montgolfier and Bertier (1978), Scharlig (1985), and Roy (1985). Roy seems to have been motivated in the construction of his methods by an awareness of the false sense of precision which can so easily be associated with numerical representations of the attractiveness of decision options, particularly if they emerge from a computer. He points out that any method which purports to determine (in some objective manner) the best of a set of decision options might well be claiming too much. While it may be possible to determine unequivocally that, say, option A is better than option B, there may be an option C such that we cannot say whether A is better, worse, or on a level with C. Roy therefore seeks a method which results in a clear ordering of options when that is appropriate, but does not produce an ordering if it is inappropriate. He assumes that a well-defined set of options exists, and that each option can be unambiguously defined in terms of a set of attributes with respect to which each option can be measured. (We note that, in common with many other approaches to decision-aiding, little is said about the hard problem of generating options and choosing attributes in thefirstplace (see Chapter 7).) In mathematical terms, we suppose that there are m attributes and n options, and that a score Xij for the / th option with respect to the z'th attribute, is available. We further suppose that a weight W; can be attached to the ith attribute. The writings on outranking pay little attention to where these weights come from or what they mean; as we will see in Section 7.5.4, there are important and subtle difficulties in weight elicitation. Inputs to the method of outranking clearly involve at least as much elicitation as is needed to construct a multi-attribute value function. Now Roy defines two measures to assess the extent to which one option can be said to be better than another. (We should point out that the definitions we are about to give are not the only ones that are used in outranking methods; they are the originals, however.) Roy first defines the concordance measure Crs for the assertion that option r is better than option s by:

crs= where B is the set of indices for the attributes where xir > xis. If the weights are all normalized, so that they sum to unity, the concordance measure is just the sum of the weights on which r is superior to s.

Decision theory

75

The discordance measure for the assertion that r is better than s is defined by: max(Wi(xis - xir)) Drs= ' e max (\wi(xn - xik)\)

where W is the set of indices for which xir < xisy and the maximum in the denominator is over all /, all /, and all k. The discordance is, thus, the biggest weighted deviation of those attributes which score option s higher than option r, normalized by the maximum possible deviation. If Crs = 1, in which case Drs = o, we can say unequivocally that r is better than s; the reverse conclusion holds if Crs = o. If Crs is close to 1, and Drs is close to zero, we might conclude, so it is argued, that r has the edge over s, with these conclusions becoming less strong the smaller C is and the bigger D is. The ELECTRE suite of computer programs has been constructed at the University of Paris to use these ideas. Given the weights and scores, the concordance and discordance measures are computed for each pair of options. Those pairs of options for which the concordance measure is larger than some value coy and the discordance measure smaller than some value do are identified. For such a pair, thefirstoption is said to outrank the second. By this means, a set of options, N, can be found, with the property that no option within N outranks any other with N, but for each option outside N there is at least one option within N that outranks it. As co and do vary, the set N changes; Roy argues that this procedure gives insight to the decision-maker. So long as co is not too low and do is not too high, the exclusion of an option from N probably means that it is not worth any further consideration. By a continuation of this process, a small set of options can be identified which almost surely contains the best option, but the method does not reach afinalconclusion on which is best. Outranking is reported to have been applied to many practical problems, particularly in France and Belgium. If the success of an approach to multi-criterion decision-aiding is to be judged by user satisfaction, then the method is indeed successful. It appears that users find the discipline provided by the method helpful; and the fact that the method serves to rule out many options, but not to make afinalrecommendation about the best option, seems to be a virtue, in that it is clear that the decisionmaker's prerogative to decide is not being abrogated by a computer. There are several criticisms that one can make of the outranking method. First, the mechanisms involved are somewhat arbitrary. The concordance and discordance measures are clearly sensible indicators, but there are many others that would work just as well, and there is no strong reason for adopting these two. Indeed there are versions of outranking which use other, equally arbitrary, formulae. Moreover, this concordance

j6

Theory

measure does not depend on the extent to which one option is better than another on any particular attribute, and the discordance measure depends only on the extreme value of a difference. Thus information contained in the scores and weights is thrown away by the method. Secondly, while a good argument can be made that no analysis should end with a totally unqualified conclusion, the outranking method achieves this end by deliberately confusing a clear situation. The weights and scores that are the input to the method can lead to a definitive conclusion; it seems more reasonable to explore the adequacy of these numbers as descriptors of the problem than to accept them unchallenged as inputs to a complicated algorithm. In summary, then, we see the main contribution of outranking as being the insight that the output of a decision-aiding procedure should not be totally definitive, but rather identify the options that look good. We believe that this is better achieved by a sensitivity analysis on the input numbers for a multi-attribute value analysis than by the outranking algorithms. 3.5.6 The analytic hierarchy process The fourth distinct approach to multi-criterion decision-aiding which we shall consider here is the analytic hierarchy process (AHP) devised by T. L. Saaty of the University of Pittsburgh. His book, entitled The Analytic Hierarchy Process (Saaty 1980), probably gives the best available description of the approach, and interested readers are referred there for a more extensive account than we are able to give here. In advising the US Government, Saaty became aware that in many problems it is natural to think of the factors which are of importance in determining policy as being arranged in a hierarchy. For example, in evaluating policy for managing radioactive waste, we may think of cost, radiological hazard, and environmental degradation as being the chief areas of concern; within the category of radiological hazard we may think of separating hazard to workers from hazard to the general public; within that latter category we may want to distinguish between hazard to the present generation and that to our descendants, and so on. In this way a hierarchy of factors is built up. Saaty's method centers around the idea of a hierarchy. Once such a hierarchy has been constructed, numbers are elicited and computations performed on those numbers to indicate which of the options being considered is best. Once again little attention is paid to how these options are generated in the first place. While we might interpret the process as one way of carrying out a straightforward multi-attribute value analysis, in that weights and scores are computed and the final measure of attractiveness is a weighted score, Saaty and his

Decision theory

77

associates see the method as being very different from MAVT. Indeed the elicitation methods employed in AHP to compute weights and scores are distinctive and unique. First, it is recommended that the inputs from decision-makers should be given in verbal form, although Saaty does allow those who wish to provide numerical inputs to do so. To elicit the weights of attributes at one level in the hierarchy, for example, a decision-maker is asked questions such as: "Consider a pair of attributes. Are they of equal importance, or is one more important than the other? If one is more important, which one and to what extent? Is it weakly more important, strongly more important, very strongly more important, or absolutely more important?" In this case, and this is typical, the decision-maker is allowed nine different possible responses, and he has to choose one of these nine. Then the verbal response is interpreted numerically by attaching one of the numbers 1,3, 5, 7, 9, or 1/3, 1/5, 1/7, 1/9, to the responses listed above and identifying this number with the ratio of the weights of the two attributes being compared. Thus to say that attribute / is weakly more important than attribute ; is interpreted as implying that wjwj = 3. This procedure is repeated for each pair of attributes at the same level on the branch of the hierarchy being studied. It is most likely that the set of ratios elicited in this way will be inconsistent, in the sense that, for some /, / and k: #= (Wi/Wk).

Saaty recommends that the decision-maker should think seriously about inconsistencies and resolve them; that is, the ratios should be altered so that for all / and /, {WiltUj^WjIWk) = (Wi/Wk).

Although perhaps the most obvious way to create consistency would be a least-squares analysis, Saaty argues for a method involving eigenvalue analysis. Once a set of ratios has been derived, a set of weights follows from the requirement that the sum of the weights should be unity. This procedure is repeated to obtain weights at all levels in the hierarchy, and is also used to obtain the scores of the various options at the bottom level, although slight modifications are necessary to the method in this case. Just as a decision tree is folded back, the hierarchy is evaluated by computing the weighted scores for each option at each level in the hierarchy, starting at the bottom. An axiomatization of AHP, defining what rationality means for this process, has recently been given by Saaty (1986). As is the case with the other methods considered in this section, there are many users who are very satisfied with the AHP, and believe it to be a very useful approach to decision-aiding. The aspects which most appeal to users appear to be the ease with which judgments can be elicited and the

jS

Theory

clarity that the formal structuring of the problem provides. The stress on pairwise comparisons as the type of judgment required is most often reported as a strength of the AHP method. While most of the methods we have described here, including MAVT, provide the clarity of formality, there are difficulties over eliciting numerical representations of judgments in some other methods. Some of the advantages of AHP here may by illusory, however. Although it may be easier to state that one attribute is weakly more important than another than to state that a given unit on one scale is three times as significant as a unit on the other scale, AHP makes the untested assumption that, given thefirst,we may infer the second. The problem of making difficult judgments is taken away from the decisionmaker and replaced by standard assignments. We doubt, however, whether this is a virtue; instead of getting a decision-maker to wrestle with his values, the AHP tells him what they ought to be! There are further problems with AHP. For example, the failure to distinguish clearly between options and attributes reduces rather than increases the clarity with which a problem is perceived, in our view. Another difficulty is that the identity of the recommended option can depend on whether or not the analysis includes an option which is clearly not a good one (see Belton and Gear (19 8 3) or Dyer and Wendell (1985)). One aspect of AHP which does seem to us to provide valuable insight is the need to overspecify a model. In using the methods of decision analysis, such as multi-attribute value theory, it makes sense to validate a judgment by getting at it in several ways,. Thus, in obtaining weights, the judgments that one unit of attribute A is worth two of attribute B and that one unit of attribute B is worth three of attribute C imply that one unit of A is worth six of C. This should be tested by direct elicitation, or some other comparable method. In our experience this is often not done in practical decision analysis, usually to conserve analytic resources (i.e. to save time). A virtue of the AHP is that it formalizes the overspecification process. In summary, then, we believe AHP uses a somewhat arbitrary representation of words by numbers which may be implying an underlying judgment structure which the decision-maker is unaware of and may disagree with. It does require more straightforward judgments from the decision-maker than some other methods, however, and the need to overspecify inputs we see as a virtue. 3.5.7 Conclusions Readers who are looking for a formal approach as an aid to decisionmaking on a multi-criterion problem will by now be asking themselves whether all these approaches to decision-aiding compete with each other, and if they do which should be chosen. Some of the approaches have

Decision theory

79

distinct characteristics which set them aside from the others. CBA was originally designed for the economic evaluation of public policy options and, we believe, is best used in this context. A broader version of CBA has, however, been effectively applied elsewhere (see the discussion in sections 6.1, 6.6, and 6.7). Most of the methods in the MCDM set, concerned as they are principally with problems where the decision variables are continuous, are not appropriate to the discrete choice problems with which the other techniques deal. If the problem faced is well modeled by using continuous variables, however, then we recommend investigating the use of MCDM. Similarly, outranking is an approach that stands alone. It could be applied to any procedure that creates a linear additive model, including MAVT, SJT, and AHP. In fact, it seems to have been applied only to the simplest of techniques for creating a linear additive model, where the decision-maker is simply asked what the weights are directly. The other methods we have considered can all be interpreted essentially as procedures for generating weights in a linear additive value function. The question then arises whether any of the methods is better at doing this than the others.To answer such a question we must define what is meant by "better." If the scores produced by the value function could be compared with some objective norm, then goodness could be defined in terms of how well the methods predicted the correct answer. But, if such a norm did in fact exist, there would be little point in constructing a value function at all. An alternative might be to define a procedure to be good if it produces high correlations with directly assessed judgments. Thus decision-makers would first be asked to order, or even to score, options, without any attempt at constructing a decomposed assessment procedure or a value function; the computed orderings produced by any one of the methods would be compared with the holistic orderings. This is the criterion used by Schoemaker and Waid (1982) in an experimental test of MAVT, SJT, and AHP as weight assessment methods. The reader is recommended to consult this paper for a thorough and detailed analysis of a good experiment; they concluded that there was no marked difference between the methods, as measured by the criterion they adopted. We may well ask, however, if this criterion is sensible. The main purpose in constructing a value function is to improve on simple holistic judgments; the intuition is that, by breaking the problem into its component parts and then reassembling them, we may obtain a better representation of our preferences than may be obtained by direct judgment. If we use direct judgment as our criterion for judging the adequacy of an analytic procedure, our argument may fairly be criticized for being circular. On the other hand, if we can validate a methodology on problems where we are confident in the adequacy of direct judgment,

80

Theory

perhaps it is reasonable to argue that the methodology is good for other problems. The alternative to experimental validation of an approach is to investigate each step in the methodology theoretically, and to judge adequacy by whatever principles one is willing to adopt. The theoretical support for the weight elicitation procedures that we give in Section 7.5.4 is strong. That does not entail their adoption, however. So long as a decision-maker is aware of what a method does and is happy to use it, she should choose the one which seems to her to make most sense. This suggests another criterion: how easy is it to use a method? Schoemaker and Waid (1982) addressed this question as well by asking their subjects to rate each of the methods they used according to its mental difficulty and perceived trustworthiness in capturing true preferences. AHP was thought to be the least difficult and SJT the most; perceptions of trustworthiness were about the same for each method. Perhaps the most satisfactory way of eliciting weights is to use insights from each method. The hierarchies and overspecification of AHP seem to be virtues; representing judgments in words (a semantic scale) is quite easy for a decision-maker, but may entail misrepresentation. If some holistic judgments can be made, then a regression analysis may give useful information about weights. The meaning of a weight must be clear to a decision-maker when judgments are being made, however; this is one of the significant virtues of the method we will describe in Section 7.5.

THE PSYCHOLOGY OF DECISION-MAKING

4.1

INTRODUCTION

As we have already remarked, decision-making is a subject which attracts the attention of scholars in numbers of different disciplines. In practical fields where decision-making is a frequent and crucial activity, such as management, medicine, town planning or engineering, much has been written on the subject. The theoretical literature is even larger, however. Mathematicians have studied the implications of the axioms of decision theory; statisticians have been concerned with how to decide in the face of uncertainty; economists have studied how human decision-making causes, and is caused by, economic activity; philosophers have grappled with what it means to make rational decisions. It is to psychologists that we turn, however, for a consideration of some of the central questions of decision-making. These are: (1) What causes us to make a decision? What are the determinants of decision-making ? and, more usefully: (2) How can we make better decisions? Psychologists have given considerable attention to these questions since the 1950s; the purpose of this chapter will be to describe some of this work and its implication for the practice of decision analysis. Good reviews of the literature are available elsewhere (see Edwards 1961, Slovic etal. 1977, Einhorn and Hogarth 1981, Pitz and Sachs 1984 or Slovic et al. 1986). Hogarth (1980) is a good text on this material. Our purpose here is not to repeat these detailed surveys; rather, we shall summarize some of the more important findings and show readers where they may look for a more detailed description of this work. In the remainder of this section we shall discuss what some of the research goals have been in this area of psychology. In the following three sections we shall describe the extent to which these goals have been achieved in three important areas: the perception of uncertainty, subjective expected utility, and judgment. Finally, in Section 4.5, we draw implications from this 81

82

Theory

work for the construction of procedures to help people to make better decisions. What, then, have been the goals of research by psychologists into decision-making? After the normative principles of decision-making, as described in Chapter 3, had been developed, psychologists began asking whether human decision-making conformed to this theory. Although developed as a normative theory (that is, a theory which tells us what to do), the question was asked whether it could be used as a descriptive theory (that is, describing what we actually do). As we have seen, decision theory requires us to express our uncertainties as probabilities and our attitudes towards risk as utilities, and to compute expected utility to guide our actions. There has been considerable attention to the ability of the human mind to express uncertainty in terms of probabilities, and we discuss this in the next section. Similarly, whether we can describe the way we make decisions as though what we are doing is maximizing expected utility has been explored in depth. This work is the subject of Section 4.3. Then, in Section 4.4, we describe what is known about our ability to represent our value judgments in a value function, as required for applying the multi-attribute value theory of Section 3.1. As we will see, there is considerable evidence that the normative theory is not descriptive. It is possible to argue, then, that we should put it to one side and discover better descriptive theories. A different conclusion might be that the normative theories could be modified to conform to the newer descriptive theories. If we were to follow this route, however, it would amount to asserting that good decision-making is identified with what we actually do, since, as we saw in Chapter 2, one of the most satisfactory definitions of good decision-making is whether our processes of decision conform to a norm. Not only would this conclusion clash with our intuition that decision-making could be better (i.e. what we often do is not good decision-making); there is also a considerable body of psychological literature pointing to the inadequacies of human judgment, going beyond the work on the descriptive inadequacy of the normative model of decision theory. Finally, therefore, in Section 4.5, we shall see how the insights from this psychological research should inform our construction of procedures for sound decision-making. 4 . 2 UNCERTAINTY

4.2.1 Is man an intuitive statistician* If we accept that probability theory is the proper way to describe uncertainty, it is of interest to inquire whether, when we use numerical probabilities, we do in fact handle them in conformity with the theory.

The psychology of decision-making

83

One of the key consequences of the axioms of probability theory is Bayes's theorem, which gives the relationships which should be satisfied by conditional probabilities. We may state it as:

where p{B \ A) is the conditional probability of B given A, p(A) the probability of A, and the other terms are similarly defined. If A refers to some uncertain event and B is some other event that relates to it, then Bayes's theorem describes how the probability judgments involved should be related. For example, suppose that A is "Our competitors have begun production of a new product based on material X," and B is "There is a 50% larger than usual demand for material X this month than we predicted three months ago." Bayes's theorem tells us how our original opinion about the likelihood of our competitor's intentions (p(A)) should be modified in the light of the information about material demand. Our new opinion should be p(A | B). The modification factor involves how likely it would be for a new order of X to be placed if the competitor were producing a new product, p(B | A), and how likely this would be anyway, P(B). In the early 1960s, Ward Edwards and some of his co-workers developed a series of experiments to test whether people revised their probability judgments according to Bayes's theorem (see Phillips and Edwards (1966), Phillips et al. (1966), Peterson and Beach (1967), the title of which we have borrowed from for this section, and Edwards (1968)). The conclusion of these, and other similar, experiments is that people seem to revise their opinions less far than the theorem would indicate (see Slovic and Lichtenstein (1971) or Wright (1984) for a clear account of this work). This property of conservatism in updating perceptions of uncertainty seems to have been widespread in laboratory experiments, and as a result a general hypothesis became current in the community of scientists studying this problem that people are generally poor at changing their perceptions of uncertainty in the light of data. Edwards himself is one of many researchers who have questioned this generalized result (Winkler and Murphy 1973, Edwards 1983). They point out that conditions in the real world are very different from those in the laboratory, and that to generalize from one to the other may not be justifiable here. Moreover, they argue, it may not be our perceptions of uncertainty that are at fault; it could just be our ability to represent those perceptions as numerical probabilities that gives us difficulty. It is this issue that was addressed in some very significant work to which we now move.

84

Theory

4.2.2 Heuristics and biases in judgments of probability In 1971 Amos Tversky and Daniel Kahneman published the first of a sequence of papers on biases that they had observed in the judgment of frequency (Tversky and Kahneman 1971, 1973, Kahneman and Tversky 1972, 1973). Their work received considerable public notice, partly through their review article which appeared in the journal Science the following year (Tversky and Kahneman 1974). More recently they have republished these articles, and other contributions to this research field, in a set of readings (Kahneman et al. 1982). In studying whether people are good at properly perceiving the uncertainty they face and then representing those perceptions as probabilities, there is a tricky philosophical problem at the outset. In the subjective interpretation of probability, which, as we argued in Section 3.2, is necessary in decision analysis, there is no such thing as an objectively correct probability; all that is required of a probability assessor is that the numbers he assesses are coherent, in the sense that they should relate with each other according to the rules of the probability calculus. Our intuition that some coherent sets of probability judgments might be better than others at representing our perceived uncertainty has no place in the theory. We are rescued from this problem by recognizing that what we really require is coherence in the large; that is, our judgments should cohere not only among themselves, but also with all other potential probability judgments we might make, and in particular with judgments of probability in true chance situations, such as games of chance. To study perceptions of uncertainty, therefore, Tversky and Kahneman created experimental situations where it was possible to compute probabilities from assumptions of randomness and the application of the probability calculus; their experimental subjects were asked to assess relative likelihoods without carrying out these computations or indeed, in many cases, knowing how to carry them out. Here are short accounts of three of the many experiments they performed; the reader is referred to the original papers, or to Kahneman et al. (1982), if he is interested in more details. One experiment (Kahneman and Tversky 1973) involved giving a group of subjects the following personality sketch: Tom W. is of high intelligence, although lacking in true creativity. He has a need for order and clarity, and for neat and tidy systems in which every detail finds its appropriate place. His writing is rather dull and mechanical, occasionally enlivened by somewhat corny puns and by flashes of imagination of the sci-fi type. He has a strong drive for competence. He seems to have little feel and little sympathy for other people and does not enjoy interacting with others. Selfcentered, he nevertheless has a deep moral sense.

The psychology of decision-making

85

The subjects were then told that Tom W. was a graduate student, and were asked to rank nine possible fields of specialization according to their likelihoods. More than 95% of the subjects judged that Tom W. was more likely to be studying computer science than the humanities or education, even though it was widely perceived at the time that there were many more graduate students study the latter than the former. It appeared to the experimenters that the subjects were ignoring this information about the chances for a randomly selected student to be studying the different subjects when they made their assessments about Tom W. In another experiment (Tversky and Kahneman 1980), subjects were given the following description: A cab was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. You are given the following data: (a) 85% of the cabs in the city are Green and 15% are Blue. (b) a witness identified the cab as Blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colours 80% of the time and failed 20% of the time. What is the probability that the cab involved in the accident was Blue rather than Green? Most subjects gave the answer as 80%. Now, if we are prepared to accept that without the witness's information the probability is 15% that the cab was Blue (the base-rate for Blue cabs), then after getting the witness's information, the probability should be changed to 4 1 % , on application of the rules of probability. Subjects appeared to ignore the base-rate in making their judgments. In yet another experiment (Tversky and Kahneman 1973), subjects were asked whether the letter r was more likely to be in the first or third position in a randomly chosen English word. A clear majority thought that the first position was more likely, yet the reverse is the more frequent occurrence. In trying to explain the consistent failure of their subjects to estimate probabilities in the way that an analysis of underlying frequency would support, Tversky and Kahneman suggested that people typically make these biased judgments as the result of using certain heuristics, or mental rules of thumb. They suggested that their data were explained by the following three heuristics: a. Representativeness. The likelihood that object A belongs to class B is measured by the extent to which A is perceived to be representative of class B. This heuristic explains the results of the first experiment above. The description of Tom W. is perceived to be highly representative of computer scientists and much less so of humanities students; thus the former category is judged more probable than the latter, despite the known distribution of students.

86

Theory

b. Availability. Alternatively, probabilities may be assessed by the ease with which similar occurrences may be called to mind. The results of the third experiment we mention above are explained by observing that it is easier to think of English words beginning with the letter r than words with r in the third place. The availability heuristic leads to a biased judgment of probability. c. Anchoring and adjustment. Sometimes people will estimate a probability by recalling some similar event for which the probability is known and then adjusting that number; the problem is that adjustment is usually insufficient. An example of this phenomenon can be seen in the conservatism which is usually displayed when people are asked for ranges in which an unknown quantity lies. In an interesting series of experiments (see Alpert and Raiffa 1982) people were asked to estimate a series of numerical quantities about which they had some knowledge but which they were not able to specify exactly, such as the tonnage of steel imported into France in 1984 or the population of Australia. They were further asked to give a range of values such that they were 90% sure that the true value fell in that range. We would presumably expect about 90% of such ranges to contain the true value of the number. In fact the proportion was almost always much smaller. Most people appeared to be unable to express properly the true extent of their uncertainty. This phenomenon has been explained by suggesting that subjects first think of their best guess, somewhere in the middle of the range, and then adjust outwards from that anchor. A more subtle bias was investigated by Fischhoff (1975a, b). When making assessments of decision-making it is clearly important to know whether an assessment of uncertainty in the past was a reasonable one. Fischhoff discovered that people tended to be biased in their recollection of how uncertain they were when reconsidering an uncertain event after it had been resolved; he called it the "hindsight" bias. For example, subjects in one experiment were asked, just before President Nixon's visit to China and the USSR in 1972, to make probabilistic predictions about current events, such as: "Will President Nixon meet Mao Tse Tung at least once?" and "Will the USA and the USSR agree to a joint space program?" Some time after the visits, when these events had (or had not) occurred, subjects were asked what their probabilistic predictions had been. There was a clear bias towards what actually occurred; thus subjects' recollections of what their probabilities had been were larger than they actually had been for events that had occurred and smaller for events that had not occurred. This phenomenon of the hindsight bias has the following implications: (i) We must be careful, when judging the decision-making of others, to recognize that what looks almost inevitable in hindsight may have looked quite unlikely at the time; (ii) our memories of past events (on which we

The psychology of decision-making

87

may base our assessments of future events) may be distorted; (iii) overcoming this bias is not easy; and (iv) our ability to learn from experience might not be as good as we sometimes believe. Another good summary of this work is in Fischhoff (1980a). These hypotheses about the heuristics we might use in assessing probability, and the biases or inaccuracies they might lead to, have proved very influential. Many other studies replicated the results of Tversky and Kahneman (see Kahneman et al. 1982); a widespread view appears to have emerged that this work demonstrated a fundamental failing in human judgmental abilities. In 1981, however, in some important papers presented at an international conference in Budapest, considerable criticism was voiced about the meaning of the extensive research work on heuristics and biases which had appeared in the previous ten years (Edwards 1983, Fischhoff 1983, Phillips 1983). As we remarked above, Edwards pointed out that what may be true for artificial situations set up in the laboratory may not be true in real life; moreover, he argued that it is not reasonable to expect subjects ignorant of probability theory to make judgments according to that theory. In practice, he argued, when we need probability assessments, we will make sure that the tools of that theory are available to us. He felt that generalizations from this work had led to a widespread, and unjustified, view that man has severe cognitive limitations. Fischhoff was more cautious in assessing the research work on heuristics and biases; while acknowledging that one could not accept some of the wider generalizations that had been made about man's intellectual incapacities in this regard, the research did show, he felt, that people do need the help of formality in making sensible probabilistic judgments. We will return to discuss further the implications of this body of research in Section 4.5. 4.2.3 Calibration Before we leave our discussion of the research work that psychologists have undertaken on human abilities to judge uncertainty, we will give a brief account of the literature on the property known as "calibration." We mentioned above that the axiomatic approach to subjective probability leads us to accept any assessments of probability as adequate as long as they are coherent. A further property however can be reasonably sought. Suppose a probability appraiser has the task of making a large series of probability judgments: is it not reasonable to hope that, for all propositions assigned a particular probability, the proportion which are true should equal the probability assigned? Thus we would expect 80% of those propositions assigned the probability 0.8 to be true. If a probability-

Theory

88

0.7

0.8

Subjects' responses

Figure 4.1 A set of empirical graphs describing calibration. Subjects were asked how likely statements were to be true. For all those responses of a subject giving a particular probability, the proportion correct are plotted on the graph. Taken from Lichtenstein et al. 1982, Figure 2.

appraiser displays this property he is said to be well calibrated. Some authors have criticized the notion of calibration on the grounds that the property is not required by subjective probability theory; but our intuition is that it seems a sensible property to require, and, as Dawid (1982) has shown, if assessors are given outcome feedback after each assessment, they can expect to be well calibrated if they are honest and coherent. Experiments to explore whether people are well calibrated have been conducted by many different researchers. A review of this work is given in Lichtenstein et al. (1982). One of the most important practical areas where calibration has been explored is in weather forecasting. In the United States, weather forecasters have been expressing their forecasts of rain in probabilistic terms since 1965, and a record has been kept of their success rate (Murphy and Winkler 1977). These studies report very good

The psychology of decision-making

89

calibration. When predicted probability is plotted against observed frequency for these weather forecasters, an almost straight line is obtained, going through the origin and (1, 1), as would be expected for a perfectly calibrated appraiser (see Murphy and Winkler 1984 for a recent review of this work). The situation is not so satisfactory in laboratory experiments. Numerous studies have used a set of factual questions, the answers to which are known but not to the subject. They have to choose between two possible answers for each question and state the probability that they are correct in their judgments. In these experiments a consistent over-confidence is discovered. Figure 4.1 is taken from Lichtenstein et al. (1982) and shows how, except for judged probabilities of about 0.5, the proportion correct is consistently smaller than the probability estimated. It is not possible to conclude from the results we have mentioned that miscalibration is only discovered in artificial laboratory settings, since other practical studies have also demonstrated poor calibration. It has also been suggested that calibration depends on factors such as the ease of the task and the expertise of the assessor, and there seems to be some substance in these suggestions. The question now arises as to how we can improve our calibration. This problem is discussed in Lichtenstein and Fischhoff (1980). They point out that attempts to correct a bias by adjusting an assessment according to a previously determined calibration curve are unlikely to be satisfactory. First, calibration changes as feedback is provided; and, secondly, even if this is avoided, calibration appears to depend on the difficulty and context of the task, and so a whole set of calibration curves might be needed. They felt that it would be better to devise methods of training people to become better calibrated in the first place. Their paper reports two experiments aimed at training people to be better calibrated probability-appraisers. Most of their subjects inproved their calibration as a result of the training they underwent, which involved feedback following each session. Lichtenstein and Fischhoff were careful not to claim unsubstantiable generalizations of their results, but there can be no doubt that, if calibration is a desirable property, and we believe it to be so, then it is possible to construct training procedures to encourage better calibration. We ought to note, at this stage, that the weather forecasters who exhibit the excellent calibration mentioned above are routinely provided with feedback and, what is more, part of their remuneration depends on how well calibrated they are. The Brier scoring system has been used for many years now to reward well-calibrated assessments of probability (see Lichtenstein et al. 1982 for a discussion of this topic). Put simply, if a forecaster assesses the probability of rain at p, and it rains, then he receives a sum proportional to p2; if it does not rain his prize is proportional to

90

Theory

(i — p)2. It can readily be shown that over the long term the forecaster maximizes his income by being honest and well calibrated. It has been argued that the excellent calibration of weather forecasters can be attributed to the use of the Brier scoring rule for rewarding them. Finally, note that while calibration can be seen to be a desirable characteristic of probability appraisers, the property, even combined with coherence, does not guarantee that a probability-appraiser conforms with our intuitive notions of "goodness." If a weather forecaster always quoted, as his probability of rain, the long-run frequency with which rain occurs, he would be very well calibrated, but of little use to us. Fortunately the Brier scoring system penalizes this kind of behavior! 4 . 3 EXPECTED UTILITY

We discussed in the last section what is known about man's ability to represent his perceptions of uncertainty as probabilities, and to conform to the rules of probability theory when revising his perceptions. We now address the larger question of whether we appear to maximize expected utility when making decisions in the face of uncertainty. Since the evidence suggests that we do not conform to the prescriptions of probability theory, it should be no surprise to the reader to learn that we do not naturally conform to the principle of expected utility either. There has been a considerable amount of interest in this topic; a good review is to be found in Schoemaker (1980), to which the reader should refer for a more detailed account of this research area. We shall limit ourselves here to a brief discussion of some of the experimental work and field studies that are particularly significant. The Allais paradox, which we described in the last chapter, is probably the most famous demonstration of people's failure to choose according to the precepts of expected utility theory. There are many others, however. That due to Ellsberg (1961) is almost as well known. The problem situation presented by Ellsberg to his subjects was as follows, in our words: An urn contains 90 balls, of which 30 are red, and the remaining 60 are black and yellow, but in unknown proportion. A ball is drawn at random and, depending on its identity, you may or may not win a prize of $100. We denote the possible prizes you might win if a red, black or yellow ball is drawn by a triplet (R, B, Y). * Situation A Which of the following gambles do you prefer? AI:($IOO,

or Az: ($0,

$0,

$0)

$100, $0)

The psychology of decision-making Outcome

Utility

100

1

100

1

100

1

100

1

100

1

100

1

Situation A

Situation B

Figure 4.2 Ellsberg's paradox. A preference for Ai over A2 is inconsistent with a preference for Bz over Bi.

92

Theory * Situation B Which of the following gambles do you prefer? Bi: or Bz:

($100, ($o,

$o, $100,

$100) $100)

Ellsberg found that most of his subjects preferred A i to Az, and Bz to B i. These choices can be shown to be inconsistent with expected utility theory, as we shall now demonstrate. We should presumably assess the chance of getting a red ball at 1/3 (although note that there is nothing in subjective probability theory to require us to do so — it is merely that it is reasonable to suppose that a random drawing should be interpreted as one where the chance we give to any particular ball being drawn is the same as that for any other). Since we do not know the proportion of balls that are black, it might be reasonable to give a probability of 1/3 to the ball drawn being black; but we do not need to do this in order to demonstrate a contravention of utility theory. Suppose the proportion of black balls is /?, so that the proportion of yellow balls is 2/3 — p. Then, again taking these proportions as our subjective probabilities, we can represent the situations in decision trees, as in Figure 4.2. Now let us apply expected utility theory. Let us take the utility of $100 to be 1 and that of $0 to be o. Then the expected utility of Ai is 1/3, and of Az is p; a preference of Ai over Az implies that p < 1/3. On the other hand the expected utility of Bi is 1 - /?, and that of Bz is 2/3; hence a preference of Bz over Bi implies that p > 1/3! The usual explanation of this failure to follow the norm or expected utility theory is that subjects prefer to gamble on options for which the probabilities are more clearly established. Slovic and Tversky (1974) report some interesting experiments based both on this problem of Ellsberg and on Allais's paradox. Subjects were presented with the two problems and asked to state their preferences. Then each subject was presented with a counter-argument for his choice on Allais's problem, and then asked to reconsider his decisions on both problems. Not much revision of opinion occurred, and most subjects were happy to maintain a set of choices on both problems that were inconsistent with expected utility theory. Slovic and Tversky end their paper with an intriguing hypothetical dialogue between two decision theorists discussing the implications of these findings. Was people's failure to choose in a manner consistent with expected utility theory simple pig-headedness in the face of rational arguments, or was it that the normative theory was unacceptable for some good reasons? Slovic and Tversky did not give a definitive answer to this question, and indeed none can be given. We reiterate the argument that we made in Chapter 3. There is no moral imperative to follow the prescriptions of expected utility theory. It is, however, an attractive procedure to follow, and, as the mythical Dr S of

The psychology of decision-making

93

Slovic and Tversky's paper points out: "I have observed that, in general, the deeper the understanding of the [theory], the greater the readiness to accept it." We share this observation. The descriptive inadequacy of expected utility theory was demonstrated in another way by Lichtenstein and Slovic (1971), who arranged for subjects to declare a preference between gambles in two different ways. Firstly they were asked to choose between some gambles directly; then they were asked for how much money they would sell the rights to play the gambles. The orderings of gambles produced by these processes were hardly correlated at all. In fact, when simply choosing between gambles, people tended to prefer those with a high chance of winning something; higher prices were asked for those gambles where the prizes for winning were bigger. These inconsistencies of preference were replicated in a study involving gamblers in a Las Vegas casino (Lichtenstein and Slovic 1973). Tversky and Kahneman (1981), in yet another sequence of experiments, supported and generalized these results to show that choices between uncertain options can depend on the way the problem is "framed," to use their term. For example, in one experiment, they gave their subjects the following situations to compare: * Situation A Which of the following options do you prefer? Ai: a sure win of $30. Az: an 80% chance to win $45 (and a 20% chance of winning nothing). * Situation B You have entered a two-stage game. In the first stage there is a 7 5% chance of ending the game without winning anything, and a 25% chance of moving on to the second stage. If you reach the second stage you enter one of two gambles, which must be chosen now. They are: JBi: a sure win of $30. Bz: an 80% chance to win $45 (and a 20% chance of winning nothing). * Situation C Which of the following options do you prefer? Ci: a 25% chance to win $30. Cz: a 20% chance to win $45. These decision options are portayed in Figure 4.3. Now let us analyze these situations using expected utility theory. Since we can arbitrarily define two points on the utility scale, let us set «($45) = 1 and «($o) = o, and write u for u($$o). Then on folding back the tree in situation A, we see that A1 should be preferred to A2 if u > 0.8.

94

Theory

Situation A

0.25

$30

Situation C

Figure 4.3 Choice situations for Tversky and Kahneman's framing experiment. In situation B, the expected utilities of Bi to Bz are, respectively, o.z$u and 0.2, so that we should prefer Bi to Bz if, once again, u > 0.8. Situation C is easily seen to be similar to situation B, in that the expected utilities of Ci and Cz are identical to those of Bi and B2, respectively. Thus consistency with expected utility theory would require that subjects should prefer either the first choice in each situation or the second choice in each situation. The experimenters found that a clear majority of their

The psychology of decision-making

95

subjects preferred Ai to A2 and Bi to B2, but they also preferred Cz to Ci. The contrasts between A and C display a phenomenon similar to that in Allais's paradox. Subjects seemed to ignore the monetary rewards when comparing a certainty with an uncertain option; but when the chances in two options were similar they went for the gamble with the higher prize. A different phenomenon is seen in the inconsistencies between situations B and C. Here it appears that subjects are attracted to the sure outcome in the second part of the two-stage gamble, even though there is, in fact, only a 25% chance of getting it at the outset, just as in situation C. Choice clearly depends on more than just the outcomes and probabilities; it also depends on how the problem is framed. It is not only in laboratory situations that people appear to violate expected utility theory. Schoemaker (1980) describes numbers of field studies which show the failure of expected utility theory to describe how people make choices in practical and uncertain situations. We could now leave the matter here, arguing that while the practice of decision-making does not seem to be described by expected utility theory, this merely shows that the person who is attracted by the normative reasonableness of the theory will need to take care when making decisions if he wants to abide by its precepts. He will need aids to overcome his biases in assessing uncertainty and in evaluating uncertain decision options. It makes sense, however, before taking our discussion further, to mention the possibility of creating alternative descriptive theories, and for two reasons. The more we know of how people actually do make decisions, the easier it will be to know what to do in constructing decision aids to help them conform with the normative theory, if they want to. And, secondly, it may be that adequate descriptive theories will suggest modifications to the standard normative theories, to produce a new normative procedure that is even more compelling, and perhaps nearer to natural decision-making procedures. Chapter 3 of Schoemaker (1980) provides a good review of alternatives, and the volume edited by Wallsten (1980) contains a number of interesting discussions of different descriptive theories of decision-making, particularly in the articles by Pitz, Payne, Wallsten, and Estes. We are not yet in a position, however, to suggest compelling alternatives to the normative theory described in Chapter 3. 4 . 4 JUDGMENT

In the previous two sections we have looked at the evidence that psychologists have produced on the extent to which our perceptions of uncertainty and our decision-making in the face of uncertainty follow the prescriptions of the corresponding normative theory. We now turn to the

96

Theory

issue for which we presented the normative theory first in Chapter 3, namely decision-making when we have no uncertainty about outcomes. Here the problem is solely how to represent our preferences, or our value judgments about conflicting factors. Now we can ask whether the prescriptions of multi-attribute value theory can describe our decisionmaking practice. The first answer to this question is, once again, that it does not. It is, for example, quite easy to find examples of people taking decisions according to the lexicographic model (see Hogarth 1980, p. 57). Here the decisionmaker concentrates on one attribute only, and only brings others into play if options are equally good with respect to that attribute. For example, in choosing an employee he might look first for formal qualifications; if two or more candidates had equal qualifications and these were better than those of all the other candidates, he would eliminate all but those best qualified and then search among those who remained, using a second criterion, such as work experience. It is easy to show that it is not possible to create a value function that is consistent with this choice procedure. Despite this first answer, the second answer to our question above is that in most situations it is quite possible to describe actual decisionmaking as though people were making a decision by maximizing some function of variables describing the attributes of importance. In fact, the linear additive value function (which we discussed in detail in Chapter 3) has demonstrated considerable descriptive power (see Dawes and Corrigan 1974, Dawes 1979, or Fischhoff et al. 1982). At this stage, the interesting question to ask is whether we can learn anything about man's ability to represent his preferences in the form of a value function. Are there, for example, biases in representing judgments of relative importance, just as there appear to be biases in representing perceptions of uncertainty? The literature on these aspects of judgment is large, and it is not possible for us to give a full review of it here. The book by Hogarth (1980) provides an extensive discussion, and the reader is referred there for a fuller account. We shall limit ourselves here to presenting just a few of the ideas and results. One interesting piece of research worth mentioning here is the "bootstrapping" idea (see Dawes and Corrigan 1974). In experiments of this type, experts are observed making judgments in their field of expertise (such as diagnosis in clinical psychiatry [Goldberg 1970] or prediction of success of graduate students [Wiggins and Kohen 1971]), and at the same time the options are measured in terms of a defined set of attributes (personality profiles in psychiatry, and background and aptitude measures for the graduate students). Weights are determined by estimating a linear relationship between the expert judgments (diagnosis as neurotic or psychotic and predicted grade point average, respectively) and

The psychology of decision-making

97

the measured attributes. The computed weights are then used in a linear additive model, giving, of course, different predictions. What is surprising is that these smoothed values are closer to the true values (which could be computed in each case) than are the original judgments. It appears that judgments can be improved by this smoothing process. Dawes and Corrigan (1974) demonstrated an even more remarkable phenomenon; if weights were set equal, once the attribute measurements had been standardized to have the same mean and variance across the options being considered, then the predictive power of the additive model was even greater. Dawes (1979) gives an interesting discussion of the meaning of these results; he sees them as further support of the conclusion of Meehl (1954) that the prediction of numerical descriptor variables is better done by using a linear additive model than by using intuitive judgment. We must be careful not to generalize too freely from these results, to conclude, for example, that in all judgmental situations a simple additive value function will outperform holistic judgments. Nonetheless, there is no evidence here to discourage us from seeking to construct value functions as an aid to decision-making (see Dawes 1977 for a discussion of this point). Another interesting experiment exploring the relationship between holistic judgment and judgment based on a consideration of separate attributes was conducted by Phelps and Shanteau (1978). They ran a series of experiments with livestock judges, whose task is to assess the quality of meat before an animal is slaughtered. Thus any judgment they make can be compared with the actual quality of the meat, which can be objectively determined after slaughtering. One of their experiments was to compare the judgmental ability of their subjects when information about animals was presented as a series of measurements (such as weight, length, etc.) with their performance when information was presented in photographs. They discovered that in the latter case performance was not as good, and could be well modeled by an additive value function using only three or four of the twelve attributes originally identified. On the other hand, they found that, when information was provided in the form of verbal statements, between nine and eleven dimensions were used. We take this as further evidence that, in some circumstances, it is possible to improve on judgments by using the value function approach described in Section 3.1. Shanteau and Phelps (1977) have used these experiments as the basis for an interesting discussion of research work on judgment in general. Another observation from the literature, which is perhaps not surprising, is that the context in which a judgment is made may affect the judgment. We saw this in the last section, in the context of decisionmaking in the face of uncertainty, in Tversky and Kahneman's framing

98

Theory

experiments. The same is true for judging values. For example, Einhorn and Hogarth (1982) showed that the addition of extra choices given to, or supplied by, a subject can change his preferences among the original set of choices. A discussion of some of the other literature on this topic is given in Pitz and Sachs (1984). Fischhoff et al. (1980) have used the word "labile" to describe the difficulty we have in pinning down our values. The Oxford English Dictionary defines labile as "slippery; prone to undergo displacement and change; unstable." In the light of the experimental work described above this is a particularly appropriate adjective. Fischhoff et al. address the question of what might happen when an analyst attempts to measure a subject's labile values. They give a careful account of the ways in which this activity can cause values to move; readers are encouraged to read the paper if they wish to explore this literature further. We just quote some of their conclusions here: Expressed values seem to be highly labile. Subtle changes in elicitation mode can have marked effects on what people express as their preferences.. . . Confronting these effects is unavoidable if we are to elicit values at a l l . . . . If one is interested in what people really feel about a value issue, there may be no substitute for an interactive, dialectical elicitation procedure, one that acknowledges the elicitor's role in helping the respondent to create and enunciate values. Fischhoff et al. 1980, P- 137 4.5 IMPLICATIONS FOR DECISION SYNTHESIS

Our brief review of some of the psychological research on man's ability to fit in with the prescriptions of decision theory may have led the reader to presume that "man is a cognitive cripple," to use a term identified (and also refuted) by Edwards (1983). Before doing so, we do well to assess the validity of this research as a whole. An important critique of the validity of the kind of decision-making research that we have discussed in this chapter has been provided by Ebbesen and Konecni (1980). Their main point echoes that of Edwards which we referred to in Section 4.2.1: to what extent can results obtained in the laboratory be assumed to hold true in real-world settings? By way of example, Ebbesen and Konecni reported a study of the decisions on bail-setting of a number of judges in a criminal court. First, the judges took part in an experiment as subjects in a laboratory setting that simulated their legal role. They were asked to set bail, in dollars, exactly as they would have done on a real case, but for a number of hypothetical cases. Background information was given in the form of the four cues that previous research had shown were the chief determinants of decision. At a later stage, and unknown to them, the same judges were observed making real bail-setting decisions in court. It was

The psychology of decision-making

99

found that, although the same four cues explained their decision-making behavior, a very different pattern of dependence on these cues emerged. While in the laboratory setting the extent of local ties of the suspect was by far the most significant variable, in real decision-making the recommendations of the district attorney were the most significant. Ebbesen and Konecni argue effectively that, although they had expected to learn about the judges' real decision-making process by the laboratory work it was clearly not possible to do so. This obviously leaves open the whole question of whether we can learn anything from laboratory studies to assist us in aiding real decision-making. This point is followed up by Einhorn and Hogarth (1981). They point out that, lacking hard experimental evidence, we can only give a judgment on the prevalence of judgmental biases in daily life. There are two extreme views. The most optimistic asserts that biases are limited to laboratory situations which are unrepresentative of the natural ecology. . . . The pessimistic viewpoint is that people suffer from cognitive conceit (Dawes 1976); i.e. our limited cognitive capacity is such that it prevents us from being aware of its limited nature. Even in its less pessimistic form, this view is highly disturbing and emphasizes the importance of further research. Einhorn and Hogarth 1981 The issue of generalizability has been recently discussed in greater depth by Hammond (1986) and Hogarth (1986). We can argue, of course, that in Ebbesen and Konecni's case the judges were giving a response that was normatively correct in the artificial cases, but were responding in a pragmatic way to the others. The lack of generalizability can thus be attributed to a mismatch between the experiments and the real world, which could, in principle, be remedied. It does not prove that laboratory results are irrelevant. Where, we may ask, does this rather inconclusive discussion leave us if we wish to apply decision theory to real problems? It should first warn us to be careful in practical attempts to capture values and beliefs in models of the real world, recognizing that the biases demonstrated in the laboratory may be present. Thus, it will be prudent to watch out for the biases of representativeness, availability, and anchoring when assessing probabilities; we will want to recognize how our choices depend on the frame in which the problem is set; and we would do well to address our values in several different ways to check that they are not so labile that they have slipped away from being a representation of our preference structure. But, secondly, we might be tempted to argue that, since we do not conform to the norms of decision theory, we should not use it as a guide for behavior. We argue against this view by reiterating our earlier statement that decision analysis is a normative procedure which we can

ioo

Theory

choose to follow or not as we please; if we do choose to accept its recommendations, then the results of this research will merely show us how necessary it is to attend to the procedures of decision analysis carefully. EXERCISE FOR 4

Identify the bias associated with each of the following examples of erroneous judgment: (a) Subjects repeatedly overestimate the relative frequency of causes of death such as murder and tornadoes, but they underestimate causes such as diabetes and heart disease. (b) In a particular experiment, subjects' estimates of the outcome of a quantitative process were highly correlated with random cues given by the experimenter. (c) Subjects often answer A to the following question: There are two programs in a high school. Boys are in a majority (65%) in program A, and in a minority (45%) in program B. There is an equal number of classes in each of the two programs. You enter a class at random, and observe that eleven out of the twenty students in the class, i.e. 55%, are boys. Is the class more likely to belong to program A or program B?

DECISION ANALYSIS IN ORGANIZATIONS

5.1

INTRODUCTION

In following through the discussion of decision theory that we presented in Chapter 3 the reader will have become aware that it is concerned with how an individual person might reasonably make decisions. Most significant decisions, however, are not made by an individual acting alone but by groups of people. Many examples spring to mind: committee decisions determined by a majority vote; the results of the adversarial process in a law court; or the subtle combination of political pressures that determine events in government. It is also the case that most practical applications of decision analysis are carried out in an organizational setting and focus on organizational decision-making rather than on improving the decision processes of a single decision-maker. It is most important, then, to ask how a normative theory developed to guide individuals can have a role to play in organizational decision-making. It clearly does have such a role in practice; we must ask if that role is valid, or simply bogus in that it does not lead to better decisions. To do this we look first at what is known about how decisions are in fact made in organizations. The literature on this is diverse and enormous, and several different intellectual traditions have contributed to our understanding. Organizational theory is concerned with how groups of people operate when working together; public administration scholars have studied how decisions are arrived at in governmental organizations; social psychologists have investigated how individual decision-making is affected by group pressure. We will not attempt to review all of this literature here; rather we give the reader some entry points to it, and mention what we believe to be the most important findings for a consideration of the use of decision analysis on organizational problems. We will do this in the next section. We have already argued, in earlier chapters, that how people actually make decisions does not necessarily tell us how they ought to make decisions. This is just as true for a group of people as it is for individuals. 101

102

Theory

Much theoretical attention has been paid to normative theories for group decision-making, and we review them below in Section 5.3. Our review will discuss the appropriateness of using these theories for designing real decision-making procedures in organizations. Finally, in Section 5.4, we return to the question of how the normative single-person decision theory of Chapter 3 fits in to organizational decision processes. 5.2 ORGANIZATIONAL THEORY AND DECISION-MAKING

The purpose of this section is to explore two questions: "What do we know about how decision-making is carried out in organizations?" and "What do we know about organizations which will help us in determining whether and, if so, in what way decision theory may be of use for organizational decision-making?" As we observed above, the literature on this question is very large, and it is very diverse. As an introduction to the ideas in organization theory, the reader is recommended to consult the texts by Mintzberg, and in particular Mintzberg (1979). Another way into this literature is the set of readings edited by Pugh (1971), together with the accompanying text (Pugh et ai 1964), which gives useful summaries of the main ideas of some of the principal writers in this field. Finally, the report of a conference held in 1981 in Oregon (Ungson and Braunstein 1982) contain some very useful papers. It was attended by some of the principal scholars in behavioral decision theory and in organizational decision-making, and the connections between the two streams of thought are explored in some detail. Now we shall discuss some of the key ideas from this literature as they impinge on the use of decision analysis in organizational settings. The early social theorists, such as Marx, Weber, and Durkheim, were interested in describing how organizations were formed and how they worked. The concept of a bureaucracy had been discussed as early as the mid-eighteenth century, usually with pejorative overtones (see Albrow (1970) for a history of the idea). Weber, who was a German sociologist at the turn of the century, argued, on the contrary, that a properly run bureaucracy was necessary and desirable. He believed that bureaucratic decision-making could free organizations from the inertia of custom and traditional practices, which usually involved favoritism and the whims of the powerful; instead it could become consistent, impersonal, and objective. His ideas have had considerable influence. In Weber's bureaucracy, there are fixed jurisdictional areas, which are ordered by rules; there is a well-defined hierarchy of positions for people to occupy; and the management, or decision-making, is prescribed by well-defined procedures. This is not the only type of organization that Weber discusses, but it is his

Decision analysis in organizations

103

principal interest (Weber 1947). A most significant investigation of whether Weber's ideal type of bureaucracy actually describes real organizations was carried out by Pugh and his group at Aston University in the sixties (Pugh et al. 1963). They concluded that the characteristics used by Weber to define his bureaucracy were indeed highly correlated in a random sample of organizations, and that it was possible to define a unique measure of the formalization of organizations. If this measure was high, the organization could properly be said to conform to Weber's ideal type of bureaucracy. Critics of Weber's ideal type have argued that, due to its rigidity, decision-making can often fail to lead to what everybody involved can see as a sensible conclusion. While this work gave a framework for studying the nature of organizations, it did not concentrate on decision-making within them. This was the focus of H. A. Simon's seminal work, to which we now turn. Simon has been perhaps the most important thinker on organizational decisionmaking in the last forty years (see, for example, Simon i957a,b, i960). His best-known idea, that of bounded rationality, resulted from his observation that, in administrative decision-making, a manager typically does not maximize some utility in making a choice. His actions are better described by assuming that he just looks for a course of action that is satisfactory rather than optimal. Satisficing behavior has been widely observed, and some economic theories have been founded in the last thirty years on the assumption that administrative man behaves in this way. On the supposition that people make decisions in this way, Simon recommends that organizations should be designed so that this behavior does not lead the organization away from its goals. He introduced the notion of a programmed decision; in a Weberian bureaucracy, with a high degree of formalization, it is possible to prescribe how a decision should be made, prior to its appearance. An example of a set of programmed decisions is in the rules that can be established for operating a warehouse in a factory. The time at which some item should be reordered and the quantity to be reordered can be laid down in advance, as a function of projected demand and the costs involved. The stores clerk does not have to go through a complex decision-making process; he merely carries out a programmed decision. Of course, even in the most formalized bureaucracy, only a small proportion of decisions are highly programmed; but Simon argues that, as computers become more powerful and our understanding of human decision-making becomes more satisfactory, there will be great advantages in attempting to program more and more of the decisions that need to be made within an organization. We shall return to consider the implications of this view for decision analysis in Section 5.4. Simon's contribution to our understanding is not the only one we should mention, however. Clough (1984), who gives an excellent and

104

Theory

entertaining account of both the theory and practice of organizational decision-making, provides a summary of some other theories in his Chapter 6, to which we refer the reader. He does not mention, however, the contribution of Lindblom (1959), which has had an important effect on thinking in this area, and to which we now turn. Lindblom argues that, far from making rational decisions (even of Simon's bounded type), organizations proceed by "muddling through." His work appears to have been performed in response to the over-eager enthusiasm for quantitative approaches to decision-making that were common in the late fifties, particularly in US government circles. A naive application of decision theory to public policy analysis suggests that, in formulating a policy, all possible policies should be explicitly set out, they should each be evaluated in terms of all possible attributes of importance, and then the policy should be adopted which maximizes expected utility. As Lindblom points out, this is a task of enormous proportions for most policy issues; not only are the data requirements of such a program generally infeasible, but most organizations do not have the resource of analysts' time in sufficient quantities to carry this program out. Lindblom argues that, instead of this rather grandiose approach, what people actually do in organizations is to "muddle through"; or, to be more formal, decision-making happens incrementally, with policies being compared with slightly different ones but not with all possible policies. As a description of how policy-making happens, there is much evidence to support Lindblom's view; some writers have gone on to justify this approach as a proper way of reaching decisions in organizations (see, for example, Jannsson and Taylor (1978)). Lindblom's more recent thinking on this topic, after twenty years of discussion of his ideas, is in Lindblom (1979). He still believes that, as a sensible guide for organizational decision-making, incrementalism has much to offer. In his extensive review of the literature in behavioral science that has a bearing on the design of decision support systems, Sage (1981) discusses the various descriptions of decision-making in organizations that have been put forward. As well as the theories we have already mentioned, he discusses the garbage-can model, put forward by Cohen et al. (1972). As Sage says: This . . . model views organizational decision-making as resulting from four variables: problems, solutions, choice opportunities, and people. Decisions result from the interaction of solutions looking for problems, problems looking for solutions, decision opportunities, and participants in the problem solving process. The model allows for these variables to be selected more or less at random from a garbage can. Doubtlessly this is a realistic descriptive model. It is especially able to cope with ambiguity of intention, understanding, history, and organization. Sage 1981, p.666

Decision analysis in organizations

105

As we mentioned in the introduction to this section, it is not only the organizational theorists who have contributed to our understanding of how decision-making happens in organizations. A subject of great interest to political scientists is the way in which the political process results in decisions. Hall and Quinn (1983) have edited a collection of original essays from political scientists which provides a good account of the contribution of public administration theorists to this topic. The readings edited by McGrew and Wilson (1982) are also of great value here, although they range more widely than the political science literature alone. The political science model sees organizational decision-making as a result of the interplay of pressure groups with different views and different power. While most political science literature is concerned with national or local government, its model of decision-making is equally applicable to other kinds of organizations, such as companies, universities or hospital authorities. The outcome of a decision is determined not by rational decision-making, nor by incremental development, but by the confrontation of different groups, negotiation between them, and the exploitation of the influence and power that they have at their command. This is clearly an adequate description of how decisions are arrived at in many contexts; when we come to discuss the role of normative theories of individual decision-making in an organization in Section 5.4, we will need to note that in many contexts it is not possible to think of an organization as being a single actor, as this theory requires. Any application of normative decision theory must recognize the political process that leads to decisions. Finally in this section, we give a brief account of the contribution of social psychology to how decisions are reached in organizations. Janis and Mann (1977) give a good review of the research in this area. One well-known phenomenon identified by Janis is that of "groupthink" (Janis 1972). He argues that if a decision-making group is: (1) highly cohesive; (2) insulated from many external influences; (3) lacking in procedures for systematically evaluating alternatives; (4) subject to a directing leader; and (5) in a condition of high stress; then there is a concurrence-seeking tendency which he calls groupthink. Some of the symptoms of groupthink are: (1) an illusion of the invulnerability of the group to error; (2) collective rationalization of an action; (3) a common belief in the morality of the group; (4) direct pressure on dissenters to conform. Groupthink can lead to defective decision-making in incompletely searching for alternative courses of action or acquiring relevant data, and to biases in processing information at hand, amongst other undesirable effects. Although the concept of groupthink was developed as a result of a consideration of the way decisions were made in major political crises, such as the Cuban missile affair (Allison 1971), it undoubtedly is a phenomenon that occurs in many other contexts.

106

Theory

Social psychologists who study decision-making are interested in how social pressures affect the decisions that an individual makes. Two such pressures, identified by Janis and Mann (1977), are: anticipatory regret, our tendency to worry about how disappointed we, and others, might feel after the event, if we take the wrong decision; and threats, or constraints, imposed by others. These are clearly important influences on decisionmaking, and must be borne in mind when prescribing how to use decision theory in group decision-making. We do not pretend, in this section, to have done any more than touch very briefly on some of the social science literature on decision-making in organizations. Two conclusions can be made, however, which have a bearing on the practice of decision synthesis. The first is that an organization does not make decisions in the way that an individual does; applying decision theory to organizational decision-making will require careful argument. The second is that there are many ways in which organizational decision-making is inadequate; prescriptions for how to make better decisions are necessary. In the next section we discuss some of the normative approaches that have been developed for group decisionmaking, and which are distinct from the decision theory for individuals developed in Chapter 3. Then in Section 5.4 we give our recommendations for how all this theory may be used in organizational contexts. 5.3 NORMATIVE THEORIES FOR GROUP DECISION-MAKING

j.3.1 Collective choice It seems to be easy for us to think of an organization in human terms. For example, when we find that a credit card company, in writing to us apologizing for failing to note our changed address, sends the letter to the old address, we are angry. Our anger, however, may be based on a judgment of the company as though it were an individual. We wonder how they could be so stupid, given that we have informed them, without realizing that the person who wrote the letter is not the same as the person who alters the address list! Similarly, the legal status of a company or a corporation as an individual reflects the convenience of treating a group of people as though they were a single person. A further example of this phenomenon is the tendency of the Press to seek for a single view on some issue of public importance from identifiable groups, such as the Church, the environmental lobby, or the arms industry, even when there is no identifiable common view on the issue among members of the group. It is not surprising, then, that attempts are made to apply decision theory to organizations as though they were individuals. Now, as we have seen, the measurement of both probability and utility depend on it being

Decision analysis in organizations

107

possible to articulate a preference order on options. While it is plausible that an individual acting alone may be able to do this, it becomes problematic to determine what an organization's preference order is, particularly if there are marked differences of opinion. The immediate answer that we might be tempted to make is that the preferences of a group of people can be determined by the traditional method of majority voting. Unfortunately, this does not always lead to a clear-cut group preference, as may be seen from the following example, originally due to Condorcet (1785). Consider a group of three people, Amy, Briony, and Celia. They all wish to go out together, and there are three alternatives before them: they may go fishing, play golf, or go hospital visiting. Amy prefers to fish, but if this is not possible she would rather play golf than go hospital visiting. Briony, the active member of the group, would prefer golfing to the other alternatives. She finds fishing very boring, however, and prefers visiting the hospital to doing that. Celia gains great satisfaction from hospital visiting, however, and prefers this to the other two activities. If she can't exercise her desire, however, she would prefer fishing to playing golf. Their preferences can be displayed in the following table, using an obvious notation. Amy Briony Celia

f > g > h g > h > f h > f > g

Briony suggests that they should determine what to do by voting. She (wisely) suggests that they first determine a group preference between hospital visiting and fishing. Celia joins with her in voting for hospital visiting over fishing, and so fishing is ruled out. Now they vote between golf and hospital visiting; again Briony is in a majority, but this time with Amy, and they vote two to one in favor of golf. Amy is not very happy at this result, and so she suggests, just to confirm matters, that they vote between golfing and fishing. When they do, they find that there is a majority in favor of fishing, which was ruled out at first. Thus by using majority voting the group prefers fishing to golf, golf to hospital visiting, but also hospital visiting to fishing. This is a very simple example; but it shows that, if a group attempts to determine its preferences by simple majority voting, a non-transitive set of preferences may well result; that is, a simple ordering of the options, as is required for utility theory to hold, may not be derived. In passing, note that we often expect a committee to display consistent preferences; Condorcet's paradox shows that, even if each member of a committee displays consistent views, it is perfectly possible for the decisions of the committee to be inconsistent.

108

Theory

Since majority voting is not a satisfactory method for determining group preferences, it is natural to ask what method is satisfactory. This is the question that K. J. Arrow asked, and to which he got the startling reply that there is none. Arrow's impossibility theorem (Arrow 1951), as it has come to be known, was one of the most important contributions made by the social welfare economists, in the years after the Second World War. The general question they asked was whether there were general principles for the allocation of resources which were ethically neutral, in some sense, and could therefore become widely accepted. Arrow's result dealt a great blow to that search. What Arrow did was to establish four reasonable conditions that he felt should be fulfilled by a procedure for determining a group's preferences between a set of options, as a function of the preferences of the group members. Arrow's conditions can be stated as follows: (1) Whatever the preferences of the group members, the method must produce a group preference order for the options being compared. (2) If every member of the group prefers option A to option B, then the group must also prefer A to B. (3) The group choice between two options, A and B, depends only on the preferences of the group members between A and B, and not on their preferences between A and B and any other option. (4) There is no dictator; no individual always gets his way. Arrow proved that there is no aggregation procedure which satisfies these four conditions. The reader is referred to Sen (1970) for a proof of this theorem, and also a wider discussion of these issues. Sen refers to Arrow's theorem as a stunning result, which indeed it is. When first published, it created great interest, not only amongst academics. Idealistic hopes that ethical principles could be established for resolving differences of opinion were dashed. A considerable literature was spawned from this result, endeavoring to rescue the hope of creating satisfactory procedures for aggregating group views. Arrow's result is concerned with the combination of preference orders, and does not use the intensity of preferences of individuals, nor, more significantly, interpersonal comparisons of utility. Much has been written on whether it makes sense to consider interpersonal comparisons of intensity of preference in a normative theory of social choice, even though we clearly take such perceptions into account in practice (see Rawls 1971, for example). The problem is one of honesty. In a community of trust, we will often take the minority view if that minority reports an intense opposition to a proposal, while the majority is mildly in favor. If, as is usually the case, we suspect other members of the group of misreporting their feelings in order to get their way, then we are less likely to accommodate them. Harsanyi, amongst other authors, has studied

Decision analysis in organizations

109

these questions in depth, and we recommend readers to consult Harsanyi (1977) if they wish to pursue them. A particular approach to constructing a group utility function, as a function of the utility functions of individuals, has been put forward by Keeney and Raiffa, and is well described in Chapter 10 of Keeney and Raiffa (1976). They imagine a supradecision-maker, whose job it is to construct a utility function for a group of people, maximizing the well-being of its members. They explore the conditions under which the group utility function can be a weighted sum of the individual utility functions. As Brock (1980) points out, however, in an interesting contribution to this debate, there is no consensus on whether this procedure or others that have been suggested make sense even at the theoretical level. What is more, the enormous practical difficulties that would be associated with any attempt to apply these approaches to real group decision problems suggest that this line of approach will be unproductive as the basis for a practical decision aid. We conclude that there is still no satisfactory quantitative method for determining group preferences by some mathematical operation on the expressed preferences of individuals. So attempts to apply decision theory to an organization by the mathematical aggregation of individual beliefs will encounter difficulties. As we shall demonstrate in Section 5.4, however, this does not mean that decision synthesis for an organization is a lost cause! 5.3.2 Game theory An alternative way of prescribing decision-making behavior in organizations is the theory of games. This has more in common with the political science view of decision-making as the interplay of power between different operators than it does with other views of organizational decision-making. Game theory probably originated in the work of Borel (1938), but received far more attention after the publication of von Neumann and Morgenstern's book in 1944. Much of the literature of the forties and fifties is excellently summarized in Luce and Raiffa (1957); after that time, advances in the theory were slow. Game theory is concerned not only with the decisions that an individual might make but also with those of his adversaries in a social situation. To illustrate the concept of a game, consider perhaps the most famous example, that of the prisoners' dilemma. Luce and Raiffa give a good description of this problem, which we reproduce here: Two suspects are taken into custody and separated. The district attorney is certain that they are guilty of a specific crime, but he does not have adequate evidence to convict them at a trial. He points out to each prisoner that each has two alternatives: to confess to the crime the police are sure they have done, or not to

Theory

no

Prisoner B's choices Not confess

Not confess Prisoner A's choices Confess

1 year each 3 months for A 10 years for B

Confess 10 years for-A 3 months for B 8 years each

Figure 5.1 The prisoners' dilemma.

confess. If they both do not confess, then the district attorney states he will book them on some very minor trumped-up charge such as petty larceny and illegal possession of a weapon, and they will both receive minor punishment; if they both confess they will be prosecuted, but he will recommend less than the most severe sentence; but if one confesses and the other does not, then the confessor will receive lenient treatment for turning state's evidence, whereas the latter will get "the book" slapped at him. Luce and Raiffa 1957, P-95 The situation can be described by the pay-off table in Figure 5.1. The problem for either prisoner lies in guessing what the other prisoner might do. If he confesses, he does better than he would if he did not confess, whatever the other decides to do. He could argue that it would be rational, therefore, to confess. But, if they both argue this way and both confess, they both do worse than if they had agreed not to confess! This is truly a dilemma. The theory of games is set up to analyze situations of this kind. It differs from standard decision theory in modeling the decision behavior of other individuals as well as that of the interested party; it is concerned with occasions when the outcome depends on the decisions made by both parties. In reality social situations are often determined by the decisionmaking of more than two people; game theory recognizes this fact, and the study of w-person games, as they are called, has received a great deal of attention. Thirty years ago, when these ideas were first explored, there was considerable enthusiasm for the potential of game theory for analyzing social decision-making and offering normative procedures for guidance in such situations. Unfortunately, these early hopes have not borne fruit except in certain very limited ways. There are several reasons for this. First, the theory has not led to unequivocal recommendations except in very simple cases. The two-person zero-sum game (that is, a game in which, whatever the outcome, what one person gains is entirely at the

Decision analysis in organizations

111

expense of the other - the sum of their gains is zero) is fully analyzed, and the minimax theorem gives strong support for the player adopting a particular strategy (see Luce and Raiffa 1957, Chapter 4, for a discussion of this theory). Very few other cases are amenable to such analysis, however; in the w-person game, the complexities are such that it is difficult to argue for any particular strategy to be followed. Secondly, since in game theory one has to think about one's opponent's processes of analysis, one has to assume some knowledge of his ability to think rationally and of the information available to him. In practice this is very difficult. Normally it is unsafe to suppose that an opponent will act rationally; as we argue elsewhere in this book, there is much room for improvement in the extent to which people apply rational thought to decision-making. There is also the problem of guessing how much bluffing the opponent will engage in. This is the difficulty of the prisoners' dilemma; game theory offers no solution to it. In summary, then, we conclude that the original enthusiasm for the theory of games as a guide for rational decision-making in social contexts has not been justified. Not only is the theory inadequate to cope with the idealizations that can be made of real problems, but also those idealizations are inadequate representations of the real world. We should not dismiss game theory as a useful normative tool for decision-making in social contexts without mentioning some more recent developments which have led to reports of useful applications. Brams (1978, 1979, 1980) has used game theory to produce some imaginative analyses of historical decisions. Metagame theory (Howard 1971) and hypergame theory (Bennett 1977) both offer an analytical tool for complex multi-actor decision problems. We have not space here to describe either of these ideas, or the connections between them, but both have been used to analyze practical problems. For example, metagames have been used on the Arab-Israeli conflict (Howard 1970) and on an international water dispute (Fraser and Hipel 1980), and hypergames on the fall of France (Bennett and Dando 1979) and soccer hooliganism (Bennett et al. 1980). Neither theory claims to provide clear normative advice to an actor in a conflict; both deal with understanding and description. Thus, while these approaches may well become very useful tools for the description of conflict, they do not yet provide us with prescriptions for action. 5.4 USING DECISION ANALYSIS IN ORGANIZATIONS

So far in this chapter we have discussed what we know of how decisionmaking happens in organizations, and the normative methods that have been explored for such organizational decision-making. As we saw in the

ii2

Theory

last section, these normative theories are not very compelling. Decision theory is used, however, and used successfully, to synthesize decisions in organizational settings. In this section we will explain how this may be done satisfactorily. We will describe how a theory developed for individuals can make sense when applied to organizations. j.4.1 The individual in the organization Although, as we saw in Section 5.2, the instances of important organizational decisions being taken by one person acting alone are rare, it is still the case that an individual may have to decide what line to take in the process that leads to a decision. Thus a manufacturing director might have to decide whether or not to support a proposal from the marketing department that a new product should be launched. An example of this may be found in Section 8.4.2. He may properly use decision analysis to help him to answer this question, although he may want to do so privately. This will allow him to include attributes of importance to him that his colleagues might not share (such as personal promotion prospects, the well-being of the manufacturing department, and so on). Very few executives will be able to afford decision analysts to assist on all problems of this type; only problems of sufficient complexity will warrant the employment of a professional decision analyst (we discuss the question of how much analysis to devote to a decision in Chapter 7). Nonetheless, an increasing proportion of managers are being trained in the ideas of decision theory, and microcomputer programs for structuring and evaluating multi-attribute value functions and decision trees are now becoming available. We expect that increasingly decision analysis will be used by managers to help them think through their own actions in the corporate power struggle. 5.4.2 Designing programmed decisions As we saw in Section 5.2, the distinguishing marks of a bureaucracy include programmed decisions, that is, the rules and procedures that are applied to determine many day-to-day decisions. At some stage, of course, somebody has to determine what these rules should be: decisions have to be made about the design of the programs themselves. In a sense these design decisions are like any other organizational decision; so an attempt to use decision analysis will involve the problem of whose values to insert into the analysis. Despite this, some success has been achieved in helping rule-makers to think through the complicated problems of the effect of different rules on the goals of the organization. For example, in health care management, Alemi et al. (1981) report the use of the framework of

Decision analysis in organizations

113

decision theory to set rules for the allocation of nursing care. They constructed a multi-attribute utility function which was used to compute a score for each patient which determined by rule how much nursing care he was allocated. The weightings in the model were determined by a committee and published. The procedure had the following advantages: because decisions were programmed, time was saved; consistency in the allocation of scarce resources improved fairness; the reasons for the decision, as explicated in the model, were clear for all to see. Such an approach to allocating care may alarm some readers: is not the take-over by the machine or the bureaucracy a short step away from this programmed decision process? We too have doubts about such procedures; if the rule goes against common sense or compassion, may it not lead to unintended evil? It is logically impossible for rule-makers to think of all conceivable eventualities. Despite these doubts, we believe that, if used intelligently, this procedure has a lot to be said for it, since its advantages, which we have listed above, are well worth having. There are close similarities between rule-making in a bureaucracy and designing regulations or laws for public actions. As Keeney (1981) points out, we have to regulate activities such as the disposal of toxic waste or the operation of a nuclear power station in order that the decisions of some actors in society (e.g. public utilities) do not have an unacceptable effect on other actors (e.g. residents near a waste disposal facility). The problem for the regulator is how to determine what appropriate regulations should be. Keeney gives a valuable discussion of this problem and shows how decision analysis may help in the design of regulations. The alternative regulations are the decision variables, and a schematic decision tree can be used to scope out the consequences of these different possible regulations. Utility functions for the actors in the process can then be used to explore the desirability of the different regulations from different points of view, namely those of the utility companies, the affected populations, and so on. Keeney does not recommend any formal method for aggregating these different value judgments; in our view this must be a matter for the political process. We know of no attempts actually to determine regulations by an application of the procedures that Keeney discusses; however, it is clear that, as a framework for clarifying the issues in the formation of regulations, decision analysis has something to offer. 5.4.3 Management decisions in a participative setting We discuss the method of decision conferencing in Section 6.5, and we shall not therefore do more here than give a brief description of the method and reasons for its success. (The literature on the decision conference is not yet very large; one of the few papers which describes the

ii4

Theory

approach is Kraemer and King 1983.) The idea is to take a group of people who are responsible for a decision (such as the board of a company) away from their usual place of work for a two- or three-day period, and to devote the time to solving their problem. An analyst, well-versed in the ideas of decision theory, acts as facilitator to the group and, by exploring with them options, goals, and uncertainties, he is able to help them create what Phillips (1984) refers to as a "shared social reality." This is expressed in a model, involving perhaps a multi-attribute value or utility function or a decision tree, which is produced with the aid of appropriate computer software. We recommend the reader to consult Phillips's article for a thoughtful discussion of this idea. We here quote some of his observations about this approach to organizational decision-making. Experience with decision conferences suggests that the adversarial process helps participants to broaden their individual perspectives on the problem, to change their views, to invent new options acceptable to everyone, in short, to create a model that fairly represents all perspectives. But not even the adversarial process can prevent biases entering the model through the influence of an ever-present variable, the climate of the organization. A pervading complex of norms, values, expectations, the acceptable and the unacceptable, influences decisions throughout the organization, often in helpful ways, but sometimes in unhelpful ways. It is here that the decision analyst's role can provide some correction to the unhelpful effects of climate. First, the experienced analyst who has worked with different organizations has both the perspective and detachment to detect and reflect back many of the biases that climate can introduce. Secondly, an analyst who has minimal investment in the consequences of the decision and who works with problem owners away from their company creates a reasonably neutral territory, thus minimizing the effects of the company's climate. This neutral climate also helps the group to maintain a more balanced view of the problem, thus making them more impervious to manipulation by particularly persuasive individuals who do not have a good case, especially when the analyst, detecting attempts to manipulate the group, asks less vocal participants for their views. Phillips 1984, p.44

Thus in a decision conference, there is no attempt to construct some sort of objectively faithful representation of the problem, and then to apply mathematical techniques to determine the optimum solution, as would be done in some versions of systems analysis. Rather, the constructed model is used as a vehicle for discussing issues between the interested parties. No attempt is made, for example, to determine what the trade-off weights in an additive value function "ought" to be; instead the decision implications of different weights are traced through and, usually, a shared view emerges of what to do (rather than what the weights ought to be). Even if a decision conference does not lead to consensus, it provides all the participants with a clearer understanding of the issues involved and why different members of the group prefer different actions. We see this

Decision analysis in organizations

115

application of decision analysis to organizational decision-making as becoming one of the most important in the future. 5.4.4 Public policy analysis Determining public policy on an issue is a more complicated decisionmaking process, by and large, than is decision-making within a company. Not only are there differences of opinion within the decision-making body (the government and its analysts), but in most countries at least some notice must be taken of public opinion on the issues. While a decision conference might help a group of ministers to clarify issues and to reach consensus (although it must be admitted that we do not know of any attempt to run a decision conference at Cabinet level), the need for analyses of public policy to be open provides further opportunities for the ideas of decision theory to be of use. The best-known technique for analyzing public policy is cost-benefit analysis. Indeed, the US government's Executive Order 12291 requires that cost—benefit analysis be applied to projects of a certain kind. We gave a brief discussion of the relation between this technique and decision analysis in Section 3.5; it is sufficient here to note that, in many areas of public policy analysis, enthusiasm for the approach is on the wane. There are many reasons for this, but one of the most important concerns the determination of trade-offs. Integral to a cost-benefit analysis is the expression of all attributes of importance in monetary terms. This amounts to creating a linear additive value function, where the weights are determined by pricing each attribute. Some economists argue that, by a suitable application of economic theory, this can be done in an objective fashion. Others recognize that there are limits to the objectivity of this approach. If we accept that the proper value for a good is the price that it commands in the market-place, it may be plausible to use such prices to determine social trade-offs. Unfortunately, however, in many cases markets do not exist (who buys and sells radiological hazards thousands of years hence?) and even when they do they are often subject to severe imperfections. We have argued (Watson 1981) that on this point, as on others, decision analysis may be more use than the traditional forms of cost—benefit analysis. In a recent study of nuclear waste management in the United Kingdom, multi-attribute value analysis was used to elucidate the best practicable environmental option for the disposal of radioactive waste (Watson 1985). For a particular kind of waste several different disposal options were available. The distinguishing characteristics for these options included cost and radiological exposure. Attributes were defined to measure these factors for each disposal option, and the options were

n6

Theory

scored with respect to each attribute. A multi-attribute value function would now have been very useful, to place the disposal options in some order. To do so, of course, required trade-off judgments. On the one hand, the analysts felt that these judgments should be supplied by the political process, and not by the analysts as sometimes happens. On the other hand, it was felt to be unreasonable to expect busy decision-makers to devote the time to determining the many trade-off weights that were required; nor would the specification of numerical trade-offs appeal to many politicians (understandably, since clarity of stance removes future potential for flexibility). As a compromise, four different sets of weights were suggested by the analysts, chosen to span their perceptions of the variety of public opinion on these trade-offs. These weights were applied to the measures and produced different rank-orderings of the options, corresponding to each of the four sets of weights. The advantage of this approach was to signal to readers of the report that questions of value were involved in this problem, and that it was improper for analysts to insert their own values. It also allowed the decision-makers to see the implications of different views of the relative importance of the attributes. It would, perhaps, further enhance the democratic process if the actual values of different interested parties were incorporated in the analysis, rather than an analysts's guess at them. This approach was adopted by Chen et al. (1979) in an application of decision analysis to the decision by a local authority on whether or not to install a solid-waste shredding plant. They created a decision tree representation of the problem which all the members of the council agreed was an adequate structure; they disagreed, however, on the trade-offs. The analysts helped each member of the authority to elicit values and to roll back the tree. Information about the views of different members was fed back to all the other members of the decision-making body. The result was not to determine what the decision ought to have been; rather, as is often the case with decision conferences, a shared understanding of the problem emerged, which led to the final decision being made more easily. There have been other similar, but more Utopian, suggestions that the structure of decision analysis could be used to determine social decisions. Both Howard (1975) and Edwards (1977) have suggested a formalization of the process of analysis by legislating for social decisions to be made according to the framework of decision analysis. We doubt that their suggestions were put forward as realistic schemes. They were more discussions of what an ideal might look like and, as such, they deserve our notice. Howard suggests that there should be three bodies of people established for solving the problems of society. The first should be responsible for defining the problems and establishing alternatives for dealing with them. The second would flesh out the consequences of each

Decision analysis in organizations

117

alternative, probabilistically if necessary. The third would determine the preferences of society; members of this body might well be elected, as in a representative democracy. The other two bodies would consist, apparently, of technocrats. We see formidable philosophical and practical objections to the introduction of this scheme; it does not surprise us that we know of no attempt to adopt it in practice. Whatever scheme of decision-making is introduced, there will always be scope for misrepresentation and fixing the system. Nevertheless, the vision of creating a system where the views of the citizens are more easily and more obviously taken into account, and where decision-making is more open and more even-handed, has much to commend it. 5.4.5 Negotiations As we observed in Section 5.2, very often decision-making occurs in organizations as a result of the resolution of conflict. Union-management disputes are an obvious example, but there are many others, such as international disagreements or intra-company arguments. The standard approach to these difficulties is to negotiate, and the art of negotiation is sufficiently refined for some people to devote their livelihoods to it. Recently some interesting applications of decision analysis to enhance the process of negotiation have been reported. Raiffa's very readable book (Raiffa 1982) discusses much more than the role of decision analysis in negotiations, but it does cover these ideas. The reader is recommended to consult some of the original papers for detailed accounts (for example, Barclay and Peterson 1976 or Ulvila and Snider 1980). The idea is quite straightforward, but very useful. A multi-attribute value function is constructed for each side in the negotiations, representing their values. If, as is normal, the analysis is carried out to aid just one of the actors in the negotiations, the values of the opposing side must be guessed; good negotiators are skilled in making such guesses, however. Then the scores of each possible agreement are investigated as determined by these value functions. When plotted on a graph, as in Figure 5.2, various conclusions can be drawn. Some of the options dominate others in the sense that, if both sides agree that the value functions properly capture their value structure, they should agree to exclude the dominated option. In Figure 5.2, for example, agreements E and F are dominated. An analysis of this kind can identify dominated options where this fact is not obvious. Moreover, it can lead to the specification of new potential agreements which are more attractive than the positions being advocated by either party. The method does not suggest which solution on the efficient frontier should be adopted; this must be resolved by negotiation tactics. It would be possible, of course, for either party to undertake a

Theory

n8

Value to second party

Figure 5.2 Decision analysis in negotiations. The value of potential agreements A, 6, C, D, £, F to the different parties. Agreement E is dominated by B, and F is dominated by C. decision analysis of the best tactics to be employed, given a previously defined efficient frontier, but, to our knowledge, this has yet to be attempted. The basic methodology was, however, used by US negotiators in the Panama Canal negotiations of 1974—5 (Raiffa 1982, Chapter 12), by the US side in the Philippine military base negotiations of 1978 (see Ulvila and McDonagh 1978), and in international negotiations on tanker standards (Ulvila and Snider 1980). Somewhat less successful applications have been conducted in the field of industrial relations. The value of this approach to negotiations is as difficult to assess as any application of decision analysis. Users spoke positively, however. It at least helped some of the negotiators to think through the complex web of issues before them; moreover, "inducing staff analysts to formalize their assumptions and tradeoffs helped to generate creativity" (Raiffa 1982, p.185). j.4.6

Conclusions

We have seen that, even though decision theory was constructed as a tool to guide individuals in decision-making, there are contexts in organizational decision-making where the ideas can be of great value. First, individuals still have to decide how they will act, even if the context is

Decision analysis in organizations

119

organizational politics; secondly, the framework of decision theory can provide a useful structure for the construction of programmed decisions; thirdly, the decision conference, based on decision theory, can be very useful for guiding participative management decisions; fourthly, in public policy analysis, the framework of multi-attribute value functions can clarify issues and enhance communication; and, fifthly, decision analysis can be used in negotiations to think through complex positions. We have not given a specific discussion of the role of computers in all these contexts, although the reader will have been aware that this role is important in all of these analytical roles, and vital in some, such as the decision conference or in negotiations. There is, of course, a great deal of discussion at present on the role of the computer in management decision-making. A computer system which not only provides a manager with information about his problem in a way that is of use to him, but also allows the manager to analyze that information effectively, has come to be referred to as a decision support system (DSS). The ideas of decision analysis are, arguably, central to the concept of decision support. Indeed, as we discuss in Section 6.6, one way of doing decision analysis is through a DSS. There is now emerging a literature on how to design a DSS, using not only the insights of decision theory, but also the body of knowledge on how decision-making happens in organizations, which we dipped into in Section 5.1 (see, for example, Sage 1981 or Vari and Vecsenyi 1984). It seems to us that decision analysis will continue to play an important and increasingly central role in organizational decision-making in the future; but it is the interaction of the theory with the abilities of the computer which offers the most exciting prospects.

II PRACTICE

Having addressed the abstract and theoretical issues related to rational individual and organizational decision-making, we now turn to the practice of decision analysis. Chapter 6 addresses the strategic (and stylistic) approaches to implementing decision theory for real problems. In Chapter 7 we present the analytic skills (or tactics) needed to practice decision analysis. Then these skills are illustrated with real applications in Chapter 8. Thefinalchapter discusses the validation of analyses, the ethics involved in practicing decision analysis, and research areas in thefieldthat need attention in the future.

ANALYTIC STRATEGY

6.1

INTRODUCTION

The initial discussion between the decision analyst and the decision-maker is pivotal in determining whether the analyst will get a chance to "practice his wares." During this discussion the analyst must ascertain the nature of the decision, the concerns of the decision-maker, and the problems that he is likely to encounter during the course of the analysis. By the end of the discussion the analyst must have described an analytic process that assures the decision-maker that his concerns will be addressed. During the course of the discussion the analyst must tailor his approach to the decisionmaker and his organization, the nature of the decision, and any potential problems. This tailored process must convince the decision-maker that the process will be worth his while. In a nutshell the analyst must convince the decision-maker that the analytic process can be expected to produce new insights into his decision that will be worth the time and money being spent by the decision-maker. The analyst must always remember that it is these insights (as to which alternative best addresses the decision-maker's uncertainties and values) that the analyst has to offer. And it is the analytic process (in which the decision analysis paradigm is imbedded) that will produce these insights. This chapter is devoted to describing five distinct analytic processes (which we will call strategies) that decision analysts have employed in the pursuit of insight. Before defining and describing these five strategies it will be helpful to examine three distinct modes that typically characterize the nature of decisions. We have observed that decision-makers usually want to do one of the following: (1) CHOOSE one alternative from many, (2) ALLOCATE a scarce resource across many alternative projects, or (3) NEGOTIATE a settlement with one or more other parties. In the CHOOSE mode the decision-maker faces the fewest constraints. His alternatives are finite in number, usually between five and fifteen, 123

124

Practice

some of which may be sequential. This is the mode for which the decision tree was devised and to which decision analysis has been applied most often. The ALLOCATE mode is commonly found in the budgeting environment. This mode can be transformed to the CHOOSE mode when the number of projects is limited in number - that is, all possible combinations of projects satisfying the constraint can be enumerated and analyzed. However, the CHOOSE mode can be easily overwhelmed when even a small number of projects are enumerated, especially when there is some uncertainty as to how scarce the scarce resource really is. Just as the ALLOCATE mode can be transformed into the CHOOSE mode, the reverse transformation (CHOOSE to ALLOCATE) is also possible and is sometimes a wise thing to do. When the decision-maker is in the CHOOSE mode but operating under a constraint, the analyst can often provide tremendous insight by turning the constraint into a scarce resource and analyzing a higher-level decision. In both the CHOOSE and ALLOCATE modes the decision-maker is not in a direct give-and-take situation with other decision-makers. This is not to say that the decisionmaker does not have competitors in the market-place making similar decisions or colleagues fighting for the same scarce resource. The NEGOTIATE mode (introduced in Section 5.4.5) occurs when the decisionmaker is interacting with one (or more) other decision-maker(s) for the purpose of mutually accepting a binding agreement. This mode is the most difficult to analyze rigorously and the one for which decision analysis has contributed the least. Throughout the remainder of this section and sections 6.2-6.5 the CHOOSE mode will be used for illustrative purposes. In Section 6.6 an ALLOCATE application is used for exemplary purposes. Section 6.7 provides^ an overview of the uses of decision analysis to support decision-makers who are in the ALLOCATE and NEGOTIATE modes. The reasons for undertaking a decision analysis are the same as the concerns voiced by a decision-maker as he is explaining why he needs help. So decision analysis helps: (1) when the decision-maker flips between options as he emphasizes different factors (criteria, events) and cannot decide which option provides the best balance across all factors; (2) when different actors in an organization are proponents of different options; (3) when the decision-maker cannot combine recommendations across varying areas of expertise (marketing, production); and (4) when the decision must be justified to superiors or the public. There is no direct correspondence between any of these reasons for analysis and the five strategies about to be defined. Yet each reason (or combination), together with the characteristics of the decision-maker and

Analytic strategy

125

his organization, determines the best strategy for a given situation. In addition, the analyst must always factor his own skills and experience (and those of his colleagues) into the choice of a strategy. Five distinctive strategies (or processes) for conducting a decision analysis can be identified in the literature, and were initiated by different decision analysts: (1) MODELING: constructing a model relating options to values; (2) INTROSPECTING: interrogating the decision-maker thoroughly; (3) RATING: using a simple quantitative, multi-dimensional value model to force thought and discussion; (4) CONFERENCING: directing communication between experts and the decision-maker; and (5) DEVELOPING: building a decision support system (DSS) that guides the decision-maker and enhances his productivity. The above terms (MODELING, INTROSPECTING, etc.) are our inventions for these five strategies and should not be attributed to the initiators of each strategy. We choose to use these terms because they, hopefully, more readily communicate the essence of each strategy than do the names of the initiators. These strategies are not meant to be mutually exclusive; in fact, they can be used in combination with each other. Buede (1979) has earlier referred to the MODELING strategy as an "engineering science" approach to decision analysis, and to the CONFERENCING strategy as a "clinical art" approach. Howard (1980) has divided this strategy space into two schools: modeling and direct assessment. MODELING, INTROSPECTING, and RATING were the first three strategies that were applied. Howard (1968) first described the MODELING strategy by using systems terms: decision and state variables. He was the first to define an iterative process for applying decision analysis. Several decision analysisfirmshave adopted this MODELING strategy as their primary business basis. Keeney and Raiffa (1976) provide several examples of INTROSPECTING, eliciting probability and complex utility functions from the decision-maker and calculating the expected utility of the alternatives. This strategy has also been used extensively by decision analysts. RATING, our abbreviation for the simple multi-attribute rating technique (SMART), was justified by Edwards (1971) on the basis of its simplicity and robustness. SMART was conceived as a response to the frustrations of obtaining answers to complex elicitation questions. Typically additive value models are used to evaluate a well-defined set of alternatives. All three of these strategies date back to the late sixties. As part of a basic research project funded by the US Defense Advanced Research Projects Agency from 1972 to 1978, Cameron Peterson and other analysts at Decisions and Designs Inc. of McLean, Virginia, developed the CONFERENCING strategy (Kuskey 1980, Buede 1982,

126

Practice

Kraemer and King 1983, O'Connor 1985). This strategy emphasizes direct and intense communication between the decision-maker and his appointed experts rather than quantitative models or carefully elicited functions. Decision analysis models, of the type used in the RATING approach, are defined to serve as a platform for dialogue and debate between the conference participants. The goal is to get the group thinking and acting as a unit. Quantitative assessments are made so as to provide a conclusion that the group can challenge, test, and discuss. Thefinalstrategy, DEVELOPING a DSS, was developed independently of decision analysis. Nevertheless, it provides a vehicle for the implementation of decision analysis that is very suitable for some decision-makers and problems. Keen and Scott Morton (1978) defined a DSS as an interactive aid tailored to a decision-maker through an iterative development process. This definition is consistent with the principles of decision analysis as long as the aid utilizes the concepts of probability and value or utility in analyzing options. Now that thefivestrategies have been introduced, the reader can easily see that, while distinct, the strategies still share the basic elements of decision analysis: problem structuring, model development, elicitation of subjective judgments, and analysis. The MODELING strategy emphasizes the second element; INTROSPECTING stresses the elicitation element; and RATING and CONFERENCING provide a balance of all four. INTROSPECTING utilizes a rigorous elicitation session with the analyst and decision-maker interacting by themselves. The RATING and CONFERENCING strategies use a far less rigorous format for elicitation. CONFERENCING operates in a group discussion mode, by definition; while RATING and MODELING strategies usually opt for the one-onone interview format. The DEVELOPING strategy is orthogonal to this discussion since it can use any or all possible formats. While all five of these strategies differ significantly in implementation details, there is one common thread that is the key to success in all five: start simple and iteratively develop a more detailed model and results. Any analyst who forgets this basic principle is certain to have less than satisfactory results: at best, limited insights due to too much detail and complication; at worst, an analysis of the wrong decision. Good advice is never to assume the decision-maker (1) knows exactly what his decision involves and where the boundaries of the decision space are and (2) can specify a set of consistent and complete alternatives. This iterative aproach provides: (1) feedback about which variables of the analysis need more attention and which variables have had too much attention, and (2) opportunities to stop when no further analysis is needed. As we stated at the beginning of this chapter, the value of a decision

Analytic strategy

127

analysis is contained in the new insights provided to the decision-maker. These insights can be a realization that one or two factors lie at the heart of the decision and that values for these factors in a given region favor one option while other values favor another option. Alternatively the insights can be as far-ranging as the identification of a new alternative that had not even been imagined or the acceptance of an alternative that had been considered infeasible. There are two other side benefits that each strategy promotes in its own way: increased and improved communication between the decisionmaker and his experts, and an "audit trail" - that is, a statement of reasons, assumptions, and information on which the decision was based. The next five sections explore each of these strategies in detail and illustrate their methods for providing insights, communication, and an audit trail. 6.2 THE MODELING STRATEGY

6.2.1 Background and definition The MODELING strategy focuses upon the linkage from the alternatives through the uncertainties to the outcomes. Thus the knowledge and information of the decision-maker and his experts are used to quantify not only the uncertainties and the values of the decision-maker but also the process underlying the decision problem that relates the way the alternatives impact these uncertainties and values. As described in Howard (1968) the decision analysis cycle is composed of three analytical phases and an information gathering stage (Figure 6.1). In the first phase a deterministic model of the decision is created so that sensitivity analyses can be used to determine which state variables (those not under the control of the decision-maker) are the most critical in causing one alternative to be preferred to another. Figures 6.2 and 6.3 illustrate the deterministic model and sensitivity analysis, respectively. The decision variables are under the control of the decision-maker and are shown as knobs under the model. The state variables (shown to the left of the model) are set by nature (forces outside the control of the decisionmaker). Figure 6.2 also points out that the value variable(s) can vary with time and that the time preference of the decision-maker may require explicit representation. The application discussed in Section 8.4.2 describes this deterministic phase in detail. The critical state variables from the deterministic sensitivity analysis are then the focus of the second phase. Probability distributions are assessed from experts for these critical state variables and the expected value (utility if risk aversion is present) of each alternative is calculated. This

Practice

128 Prior information

Deterministic phase

Probabilistic phase

New

Informational phase

Decision

Act

Gather new information Information gathering

Figure 6.1 Howard's decision analysis cycle. The first description of iterative analysis in support of a decision-maker. Source: Howard 1968, Figure 3.

0 ,0

o

Value

©

00

0

Decision variables

Figure 6.2 Deterministic model. Prior to introducing uncertainty, the complexity of the decision is modeled. Source: Howard 1968, Figure 4.

computation is depicted in Figure 6.4, in which the decision variables have been collapsed into a vector (d) and the time preference function is part of the model. A value lottery for each alternative setting of d is the output of the model, and the risk preference function produces a single number, the expected value . Howard uses the bracket notation {v \ d, §} to denote a probability distribution of value given the decision variable

Analytic strategy

129

8 c

Model

2? CD

Tim

a

State variable s,

Figure 6.3 Deterministic sensitivity. Does the uncertainty in the value of the state variable, sh change the recommended alternative? Source: Howard 1968, Figure 5-

and the prior state of information, £. These brackets are also used for the probability assignments to the state variables. As a result of this second phase the MODELING strategy produces a preferred alternative. However, more analysis is possible, namely, the value of additional information. The inforr ^tional phase uses the concept of clairvoyance (or perfect information;, as shown in Figure 6.$ and discussed in Section 3.3.5, to determine whether the decision-maker would be better off collecting more information before acting upon his preferred decision. If this analysis suggests that additional information should be acquired, the decisionmaker enters the fourth phase, information-gathering, and then repeats the entire process with the resulting information.

Practice

130 Probability assignments

Value lottery

f\ Utility

z\

{*/!«

0

{*JB

Figure 6.4 Stochastic analysis. Probability distributions on the critical state variables result in a value lottery for each alternative (d), which is then converted to an expected utility by using the risk preference function. Source: Howard 1968, Figure 7.

Of particular relevance to our definition of the MODELING strategy is Howard's use of systems theory to reflect these phases. This systems theoretical approach to decision analysis clearly asserts that the analyst's primary function is the specification of the "model" and the accompanying decision and state variables. Other tasks include assessing probabilities and time and risk preference functions; but without the model everything stops. And, in fact, this model never exists when the decision analyst begins working with the decision-maker but is synthesized as a result of their communication process. It is during the development of this model that the architecture for communication and insight is built. The model is built within the confines of physical and economic theories to the extent that they apply. However, the analyst often synthesizes a great deal of expert judgment in defining the model that is based upon years of experience with the real world rather than theory. This MODELING strategy enjoys continuous successful application and evolution. Howard (1984a) redefines the initial flow diagram of analytic tasks to that shown in Figure 6.6. The basis is defined as the three

Analytic strategy

131

.

If this is the case, then wx vi(x\) = w2

and hence which can be read from the value function. Repeating this procedure with the third ranked attribute, then the fourth and so on, yields n — 1 equations relating the weights to each other. Finally we can use the normalization equation: W1 + W2 + . . . + wn = 1.0 to solve a set of n linear equations. Often, however, in practice it is best to overspecify the weights by asking more than n — 1 questions because trade-offs across disparate attributes are very difficult to synthesize. By

202

Practice

overspecifying the set of equations, inconsistencies in the decisionmaker's judgments can be found. We are not trying to show the decisionmaker how irrational he is because he was inconsistent. Rather we are providing new ways to help the decision-maker synthesize the tough choices before him. This numeraire process depends on the validity of the value functions. By showing the decision-maker inconsistencies in his weight judgments we are providing a check on the value functions, an error in which may be the cause of the inconsistency. To illustrate the numeraire assessment approach let us return to the enjoyment hierarchy in Figure 7.11 and assess the weights for the attributes under "ocean access." The decision-maker says that the swing in value for "walking time" is more important than that for "view." We refer back to the value curve for "walking time" (Figure 7.15) and ask if the "view" swing is as important as a change in walking time from ten to five minutes. He synthesizes: "view" is more important than that. We finally narrow the weight of "view" to be equivalent to a change in walking time from ten to four minutes (0.85 on the value curve). We now have two equations and two unknowns: + ^32 = 1.0 = 0.85 where w$\ refers to the relative weight of "walking time" (relative, that is, to the other node at that level, "view"), w32 to "view," w3 to "ocean access," w2 to "cottage luxury," and W\ to "time spent." Making the substitution we find that the relative weight for "walking time" is 0.54 and the "view" relative weight is 0.46. A second approach, which was derived from the conjoint process described in Kuskey et al. (1981), we will call the balance-beam approach. Here the value functions are not used at all. Rather the vector x (1) of the most important attribute is compared to a composite vector of the second and third most important attributes, x (23) : X(23) = (*?, *£, *3, X%, . . ., X°n). We often use a balance-beam analogy to provide the decision-maker with an intuitive understanding of our questions. Since one cannot always expect to find equalities in this process of comparing one improvement from worst to best with two such improvements, the analyst searches for inequalities that bracket the vector x (1) . For example, suppose the decision-maker replies that he prefers the composite of the second and third most important attributes to x (1) , but prefers x (1) to the composite of the third and fourth most important attributes. We now have a set of inequalities that bracket the value of W\:

Analytic tactics W\ < W\ >

203

Wi + W$ W3 + W4.

Typically, after the decision-maker has synthesized many inequalities of this kind, we can use them to obtain numerical expressions of the weights. This is done by assigning an arbitrary value (such as 10) to the least important weight, and using the inequalities as well as additional quantitative judgments of the decision-maker to determine the weights of the more important attributes. Once this is done the normalization equation can be used to rescale the weights. Section 8.6.3 provides an example of how this approach is applied. This approach has even been used to assess value functions and weights, simultaneously, for discrete value variables. 7.5.5 Constructing hierarchical additive value weights We describe the use of value hierarchies for problem- and model-structuring in Section 7.2.3. Now we must deal with assessing weights for the various levels of the hierarchy. Just as we talked about top-down and bottom-up structuring, there is also top-down and bottom-up assessment of the hierarchical weights. The bottom-up approach more accurately captures the swing weights of the hierarchy. Once the value functions have been assessed for all of the bottom-level attributes, and value weights have been assessed at the bottom levels of the hierarchy, then single attributes from each of the bottom-level groupings are chosen (often the most important) and compared with each other. If the hierarchy is large, then these comparisons may be done in several subsets, and then finally single attributes from each subset sampled to complete the comparisons. Usually the number of levels in the hierarchy determines the number of samples that must be compared before the top-level attributes can be weighted. To illustrate the bottom-up assessment approach let us return to the enjoyment hierarchy in Figure 7.11 and use the weights assessed above with the numeraire approach. First we distinguish between relative weights and cumulative weights in the following manner. The cumulative weight for "walking time," W31, is defined to be the product of the relative weights above it, W3W31. This is illustrated in Figure 7.18. Now to assess the relative weights at the top of the hierarchy, we choose one of the bottom-level nodes of "ocean access" and "cottage luxury." The decision-maker rank-orders the swings in value for these attributes as follows: (1) "time spent," (2) "cottage luxury," and (3) "walking time." Now we use the numeraire approach (picking "time spent" as the

204

Practice enjoyment i

time^i spent

cottage"* luxury

ocean " 3 access

I

31

walking" time W3l = w3w3l

view"' 32 W32 = w3w32

Figure 7.18 Relative and cumulative weights. Cumulative weights are the product of all relative weights from the given level in the tree to the top of the tree. numeraire) to assess the other two weights. The change between the best and worst points on the "cottage luxury" scale is considered equivalent to the change of one to three weeks at the beach (0.7 on the value curve, see Figure 7.17). "Walking time" was measured at 0.6 on the numeraire's value curve by comparing the change from worst to best walking times to a change from two to six weeks of time spent. Now we have four equations and four unknowns: W2 = 0.7 wu W31 = 0.6 wu W31 = W3W31 = 0.54 w3 (cumulative weight equation), W\ + W2 + W3 = 1 (normalization equation). Solving for W\ we find it to equal 0.36, making w2 equal to 0.25, and W3 equal to 0.39, making "ocean access" the most important factor at the top of the hierarchy. Solving for W31, we find it to be 0.21. This high ranking of "ocean access" is due to the fact that "walking time' was considered to be 60% of "time spent" and that "walking time" was only slightly more valuable than "view." This result can be used as a validity check on the results to ensure that the calculated weights make sense to the decisionmaker. 7.5.6 Assessing utility functions As we have discussed before (see Section 3.3) a utility function quantifies the decision-maker's risk preference for a specific decision. The analytic strategy section on INTROSPECTING (6.3) defined an approach for assessing a multi-attributed utility function. In this section we will focus on the assessment of a utility function on one attribute. Many decision analysts would then convert the multi-attributed values of the alternatives into value scores on this one attribute (using it as a numeraire) and then calculate the utilities for the alternatives using the single utility function. We shall use the beach cottage investment decision to illustrate the

Analytic tactics

205

$190,000

$100,000

$190,000 $145,000$100,000

Figure 7.19 Risk preference lotteries. The two most common risk-preference assessment questions are (1) asking for the certainty equivalent (X) of a lottery and (2) asking for the indifference probability (p). 1.0 -

0.8 -

0.0 100

140 160 Wealth ($000)

190

Figure 7.20 Risk preference function. Risk aversion is typified by a concave function as shown above. A straight line would indicate risk neutrality (expected value) and a convex function signifies risk-seeking preferences.

206

Practice

elements associated with the assessment of a utility function. Suppose we had completed a probabilistic analysis of the investment and found that the minimum net present value of the cottage was $20,000 (US) and the maximum was $120,000. This would be all that we needed to know if the decision-maker had accepted the delta property for utility functions (Section 3.3.3). However, let us not presume this. Rather we have to start with the minimum and maximum NPVs of wealth for the decision-maker. If the decision-maker starts with an initial wealth of $100,000 and puts up $20,000 to achieve a $20,000 NPV, then the minimum wealth position is $100,000 (100,000 — 20,000-1-20,000). Alternatively, if he puts up $30,000 in a partnership and achieves the maximum NPV (after splitting with his partner), then this maximum wealth position is $190,000 (100,000 — 30,000+ 120,000). So we need to assess a utility function over NPV of wealth from $100,000 to $190,000. While this investment is almost surely an increase in the decision-maker's wealth, he must consider other uses of his $20,000 to $30,000 over the time frame of the beach cottage investment to be able to justify it. Putting the money in the bank could also increase his wealth position. Since a utility function measures risk preference, the only way to assess one is by asking the decision-maker a series of lottery questions. We start by assigning the following values: u( 100,000) = 0.0 and #(190,000) = 1.0.

Figure 7.19 depicts two lottery questions that we could pose to the decision-maker to help synthesize this utility function. The first lottery asks for the wealth position that would make the decision-maker indifferent to afifty-fiftylottery over the best and worst wealth positions possible. Suppose his answer to this question is $130,000. The second question asks for a probability, p, of $190,000 that is equivalent to $145,000, assuming that the starting wealth is obtained with 1 — p. His answer here is 0.7. These two answers are plotted on the curve in Figure 7.20. If this curve is correct, the decision-maker will say that he is indifferent between a wealth of $150,000 and a fifty-fifty lottery over wealths of $190,000 and $130,000. Likewise, he would be indifferent to $120,000 and a fifty-fifty lottery over $145,000 and $100,000. Can you explain why? EXERCISES FOR 7 . 5

1.

Place yourself in a position where you have to buy a house. Suppose that you have been talking to a real-estate agent and given this agent a price range with which you are comfortable. Unfortunately you are called out of town on an emergency before you can look at any of the

Analytic tactics

207

houses. However, you do have time to develop a value structure for a friend of yours to use in evaluating the houses: House value

I

I

I

I

Yard size

Bedrooms I •

Utility expenses per year

• Number

1 Size of master bedroom

(a) Define your units of measurement for each attribute. (b) Determine your minimum and maximum thresholds for each attribute. (c) Use both the bisection and the value difference ranking methods to assess your value curves for each of the four attributes (one method for each attribute). Which method did you use for assessing the value curve for number of bedrooms? Why? (d) Use both the numeraire and balance-beam methods to assess the value weights for this four-attribute hierarchy. Compare the weights derived from the numeraire assessment to those assessed with the balance-beam method. Are there any conclusions or trends? Which set of weights do you believe most accurately reflects your preferences? (e) Now define at least five houses in terms of these four attributes. Be sure that no house is at the bottom of every value function or at the top of every value function. First, develop a holistic rank order of preference for each of these five houses. (Which house do you favor most, all attributes considered? Second, third, fourth, and fifth?) Next, use the weighted value curves to derive a composite value for each house using both sets of value weights. Now you have a holistic rank order, value scores for the numeraire method, and value scores for the balance-beam method. Do the rank orders implied by both sets of value scores agree with your holistic rank order? If not, which rank order do you have most confidence in at this point? Often decision-makers see results at this point that cause them to want to change their value weights and/or curves. Do you? 2.

Use the following bottom-up weight assessments to compute all of the relative and cumulative weights for the hierarchy shown below:

2O8

Practice

Objectives

i Attributes

411

I

I

01

02

i

r

412

421

422

423

4221

4222

0.6

0.4

Bottom-up weights 1. 2.

0.4

3.

0.7

4.

0.5

0.4

0.2

0.3 0.5

3. Develop the equations needed to compute the relative weights for a value hierarchy from a set of bottom-up weights. 4. Using the following lotteries, assess your own risk preference function for money over the interval [o, 10,000]. Use whatever unit of currency is familiar to you. X\ x2 x3 x4

~ ~ ~ ~

{o, 10,000; 0.5} {o, X\\ 0.5} {xu 10,000; 0.5} {x2,x3; 0.5}

Graph the assessments. Are you happy with the results? Modify any of the assessments and sketch the resulting risk preference function. 5. Repeat the fourth exercise with the following lotteries. 5,000 2,500 7,500 5,000

~ ~ ~ ~

{o, 10,000; pi} {o, 5,000; p2} {5,000,10,000; p3} {2,500, 7,500; p4}

Compare the results from this exercise with those of exercise four. Are they the same? If not, do they support the research results discussed in Section 7.5.1?

Analytic tactics P(u(x))

1.0--

0.0

Figure 7.21 Stochastic dominance. If more of x is better, then A is dominated by both B and C. J.6

SENSITIVITY ANALYSIS

Any discipline that utilizes quantitative analysis can benefit from an examination of variations in the solution as input parameters are varied within a reasonable range. Some disciplines, such as linear programming, have well-defined procedures for conducting sensitivity analyses while others are more ad hoc. Decision analysis has some standardized approaches to sensitivity analysis; but, as yet, no general purpose heuristics, equivalent to slack variables in linear programming, have been developed to aid the analyst. Sensitivity analysis is conducted during early iterations in decision analysis so that the analyst and decision-maker can determine where additional modeling and assessment efforts are needed to improve the quality of the analysis. These sensitivity analyses are usually simple variations of one parameter at a time. Then, late in the analysis the sensitivity analysis helps the decision-maker understand the implications of his judgments and determine whether any alternatives are dominated by the others. We call one alternative dominated if there are no circumstances for a defined set of parameter variations such that the alternative in question is preferred. Often this concept of dominance is used in relation to the final probability distribution over utility (or value) for the alternatives, and is

Practice

2IO

Table

Beach cottage results

7-i

NPV

Initial investment $

$

Value

0 0

190,000 170,000

100

57 85

170,000 155,000 110,000 100,000

Value

• Enjoyment value

Total value

Alternatives i 2

3 4 5 6

30,000 30,000 27,000 24,000 20,000 20,000

100 100

0.3

100 90

70

75

70

61 11 0

70

7* 39

0.3

0.4

78 78

15 0

59

30

called stochastic dominance. Figure 7.21 illustrates this stochastic dominance, in which alternative A is dominated by B but the distributions for alternatives B and C cross over, so neither can be said to dominate the other. An example of stochastic dominance can be found in Section 8.2.2. Stochastic dominance is a more robust examination of the optimum policy than just looking at the expected value. However, it holds all inputs to the analysis fixed; therefore, the conclusion of stochastic dominance may well change if some of the inputs are changed. As a result stochastic dominance is usually only used late in an analysis when the decisionmaker is quite comfortable with his model structure, probabilities, and values. Policy region analysis, on the other hand, systematically varies the values of probabilities, values or both to determine in which cases one alternative is preferred. We classify policy region analyses as one-way, two-way and three-way. (We could expand this to four or more if there were ways of displaying or visualizing more than three dimensions in this three-dimensional world.) A one-way policy region analysis varies one parameter and holds the rest constant; or, if the parameter is a probability or value weight, then its peers are held in constant proportions but raised or lowered as a group to satisfy the constraint that their sum must equal 1.0. In order to illustrate these various forms of policy region analyses, we will return to the beach cottage decision. Suppose an analysis of six alternatives had yielded the results shown in Table 7.1. The three value attributes are listed. Both dollar and value results are listed for the initial investment and NPV attributes. Note that the value scale on NPV is linear and that the value scale on initial investment exhibits decreasing returns to scale. The far right column shows the results for value weights of 0.3 for initial investment, 0.3 for NPV, and 0.4 for enjoyment. Figure 7.22 illustrates a one-way policy region analysis as the weight of initial

Analytic tactics

0.1

0.2

211

"0.3

0.4

0.5

Initial investment

Figure 7.22 One-way policy region analysis (proportional). The weight of "initial investment" is varied with the weights of the other two value variables being kept in the same proportion so that all three weights add to 1.0.

0.7

0.3 0.4 Initial investment 0.7

0.6

0.5

0.4

0.3

0.2

0.1

Enjoyment

Figure j.23 One-way policy region analysis (NPV-constrained). This sensitivity analysis is conducted with the weight of "NPV" held at 0.3.

Practice

212

0.2

0.4 0.6 Initial investment

0.8

Figure 7.24 Two-way policy region analysis. While all three weights are varied in this policy region analysis, it is only considered a two-way analysis since the sum of the three weights must equal 1.0. investment is varied between 0.0 and 0.5. The decision-maker might justify a weight of zero since the initial investment is already included in the NPV calculation. On the other hand this initial outlay may be very disruptive of the decision-maker's lifestyle so he may wish to weight it very heavily. The crossover point between alternatives 1 and 4 occurs at an initial investment weight of 0.28. Note that alternative 3 is very close to this crossover point. Options 2, 5, and 6 are clearly dominated in this one-way policy region analysis. Another approach to the above one-way sensitivity analysis would be to vary the weights of initial investment and enjoyment, while keeping the weight of NPV constant at 0.3. Figure 7.23 depicts this sensitivity analysis. The linear curves for each alternative were found by setting the weight of initial investment equal to w and the weight of enjoyment equal to 0.7 — w. So the equations for the alternatives are: Options 1

Values 100

—

IOOW

2

86.4 -

90W

3 4 5 6

75.9- iSw 67.3 + i$w 13.8+ Ssw IOOW

Analytic tactics

213

The only difference between these two forms of one-way sensitivity analysis was the assumption of how to vary the other two weights in proportion to each other. Figure 7.24 uses a two-dimension x — y plot to depict the results of simultaneously varying the three weights (a two-way policy region analysis because the third weight can be uniquely determined from the values of thefirsttwo). Policy region analysis gets it name from this figure because the areas labeled 1, 3, 4, and 5 depict those areas where alternatives 1, 3, 4, and 5 are preferred, respectively. The weights for initial investment and enjoyment are on the x-axis and y-axis, respectively. Lines drawn parallel to the line connecting the (o, 1) and (1,0) points represent different values of the weight of the third variable, the weight of NPV. If NPV is weighted zero, then the (o, I ) - ( I , o) line is the space of weights for initial investment and enjoyment. For NPV weighted at zero, alternative 1 is preferred when the weight of enjoyment is greater than 0.74 (initial investment is weighted below 0.26); alternative 4 is preferred when the weight of enjoyment is between 0.42 and 0.74; and alternative 5 is preferred when enjoyment is weighted below 0.42. The dashed line corresponds to the baseline weight for NPV, 0.3. The dotted line corresponds to a weight of 0.6 for NPV. As can be seen from this figure, alternatives 1 and 4 account for nearly all of the policy region, and the baseline weights occur reasonably close to the boundary between these two regions. Alternative 3 does obtain a small portion of the policy region. This policy region analysis should be used in conjunction with the previous one-way policy region analysis that found alternatives 1, 3 and 4 to be close contenders near the baseline weights. These policy region analyses suggest that alternatives 1 and 4 are nearly equal in value, and that more weight on initial investment favors alternative 4. Also, NPV and enjoyment are so correlated that, if we hold the weight of initial investment constant, the preferred option may change from 4 to 1 to 3 as we vary the weights of enjoyment and NPV, but their values will be so close that the decision-maker would be equally happy with any of the preferred alternatives. The analyses suggest that alternatives 2 and 6 can be discarded. The above examples only focused on the value weights. However, similar sensitivity analyses (including three-way policy region analyses) are done with probabilities or a combination of probabilities and value weights. Such illustrations are shown in Section 8.6. EXERCISE FOR

J.6

Suppose you are helping a friend buy a house. Using the value structure from the exercises at the end of Section 7.5, your friend has evaluated five equally expensive houses on three attributes; yard, size, bedrooms, and

214

Practice

operating expenses. Your friend's value assignments for the five houses appear below: Houses I 2

3 4 5

Yard

5

40 40 90 3O

Bedrooms

Expenses

50 40

90 60 90

10

40 60

10

Helping your friend assess his value weights yields 0.2 for yard, 0.5 for bedroom, and 0.3 for expenses. The result is that the first house has the highest value, but the fifth house is close and houses four and one seem attractive when you think about either the yard or the expenses. (a)

Plot the two-way policy region analysis for these weights to help your friend determine the sensitivity of choice to the weights assessed.

(b)

Your friend says that any of the above weights could be off by as much as 0.1. Are any of the houses besides the first one preferred in this region?

8 APPLICATIONS

8.1 GENERAL INTRODUCTION

The primary purpose of this chapter is to use real applications of decision analysis to illustrate the analytic skills described in the previous chapter, "Analytic tactics." So this chapter has the same organization of sections as Chapter 7, with the exception that problem- and model-structuring are each allocated a separate section here to emphasize the difference between the two and the importance of structuring in general. Our secondary purpose with this chapter is to provide examples of real decision analyses across the decision-making spectrum: commerce/ industry, public policy, regulations, and the professions (e.g. the medical field). A guide that relates the applications presented in this chapter to these decision-making areas is presented below: Commerce/industry 8.2.2 New product and facilities Hertz and Thomas 1983 planning 8.4.2 Facilities investment and Spetzler and Zamora expansion 1984 8.6.2 Merchandizing Dyer and Lund 1982 Medical Fryback and Keeney 8.3.3 Indexing trauma severity 1983 8.7.2 Cardiac catheterization Macartney et al. 1984 Public policy Project appraisal Howard et al. 1972 8.5.2 Seeding hurricanes Kirkwood 1982 8.7.3 Nuclear power plant siting Evaluation of programs 8.2.3 Selection of trajectories Dyer and Miles 1976 8.4.3 System design Buede and Choisser 1984 215

2i 6

Practice

Resource allocation 8.5.3 Magnetic fusion choices 8.6.3 Marine Corps program development Regulatory 8.3.2 Emission control strategies

8.2

North and Stengel 1982 Kuskey et al. 1981 North and Merkhofer 1976

PROBLEM-STRUCTURING

8.2.1 Introduction This section provides two examples of problem-structuring; that is, the definition of the problem space, including boundaries, decision-makers, and alternatives. The new product and facilities planning analysis described by Hertz and Thomas (1983) illustrates the use of a financial model to help eliminate inferior alternatives. The search for an appropriate analytic methodology is examined in the trajectory selection case study of Dyer and Miles (1976). Dyer and Miles recognized the importance of matching the methodology and its required subjective inputs to the decision-makers and experts available within the organization. Inattention to this point can doom the best formulated model. 8.2.2 Hertz and Thomas (1983) — new product and facilities planning The disguised ABC corporation was considering a change in strategy in the egg carton market, to which "decision and risk" analysis was applied as an aid for the decision-makers. For several years ABC had been developing "Egg 'N Foam," a plastic product which would compete with paper egg cartons; and strategic options ranged from abandoning the product to considerable expansion. Before reasonable alternatives could be specified, the analysts engaged in significant problem definition and investment analysis. The first step was to construct a simple investment flow chart (see Figure 8.1), which sized the egg carton market, ABC share price, ABC revenue, manufacturing costs, overhead, investment, and the resulting profit and return on investment. From this, more detailed flow charts were developed, assumptions (e.g. market share) stated, data (e.g. forecasts of total egg production) gathered, and a deterministic model built. One of the major uses of the model was a sensitivity analysis of plant operating characteristics. Different equipment combinations and plant sizes were examined, resulting in the definition of two primary plant sizes — small (one extruder and two thermoformers) and super (two extruders

Applications

217

Total egg carton market

ABC foam share

X

ABC selling price

ABC revenue ABC manufacturing costs

ABC total overhead \ Profit

I

Return on investment

t

Investment

Figure 8.1. Initial Egg'N Foam flow chart. Investment flow model used to structure alternatives. Source: Hertz and Thomas 1983, Figure 1. and four thermoformers). The analysis also found that the two most sensitive factors in determining investment return were demand and the caliper (or thickness) of the Egg 'N Foam. This analytical approach to problem-structuring provided the options that were then analyzed. These four investment alternatives were (see Figure 8.2): (1) superplant/fast, (2) superplant/slow, (3) small plant/fast, and (4) small plant/slow. The results of the probabilistic decision analysis are shown in Figure 8.3, which demonstrates that the fast superplant alternative stochastically dominates the other three alternatives. 8.2.3 Dyer and Miles (1976) - selection of trajectories In 1973 Dyer and Miles used decision analysis in a group decision context to help a Science Steering Group (SSG) within the US National Aeronautics and Space Administration (NASA) select a pair of trajectories to be used for the launch of two spacecraft in 1977. These spacecraft were to swing by Jupiter in 1979 and Saturn in late 1980 or early 1981 on their

Practice

218

Key: MW = Midwest SE = Southeast SW = Southwest NE = Northeast

Super plant-strategy Number of thermoformers

Total investment = $8 million 1968

1969

1970

1971

1972

1973

1974

1975

1976

Small plant strategy Number of thermoformers Fast ISW

SW SW 1

'

MW

MW

I—

1

r—

J

Slow , __ IMW

IMW

SE

ISE

ISE

INE

Total investment = $8 million

J

1968

1969

1970

ISW I

1971

1972

1973

1974

1975

1976

Figure 8.2. Egg'N Foam investment strategies. Alternatives that were the result of the problem-structuring process. Source: Hertz and Thomas 1983, Figure 3.

Applications

219

Probability /1.0 Strategy 4

Strategy 2 /

Strategy 3/

Strategy 1

Figure 8.3. Graph of cumulative probability density functions for each strategy. Strategy 1, the fast implementation of the superplant, stochastically dominates the other strategies. Source: Hertz and Thomas 1983, Figure 5. way out of the solar system. The Science Steering Group was composed of the team leaders of specialized science teams, which are listed in Table 8.1 along with their primary investigations. The selection process was governed by the following guidelines: (1) The process should focus on obtaining a trajectory pair compatible with both the science requirements and the mission constraints. (2) The process should be compatible with the Project resources allocated for mission analysis. The existing Project management structure and science interfaces should be used. (3) The process should not divert the efforts of the SSG from other Project activities, nor should it create dissension among the SSG members. (4) The process should be conceptually simple, and any documentation presented to the SSG should be essentially self-explanatory. These guidelines are typical of the constraints placed upon most applications for real clients. In discovering how the various science teams valued trajectories, the authors found that it was absolutely necessary to evaluate the trajectories as a pair. For some scientific investigations the critical information was of such value that the two trajectories provided redundancy in the collection process that was vital. However, for other investigations it was not possible for one trajectory to obtain all of the information; thus, each

220

Practice Table 8.1 The MJSyy science investigations'1

Science team

Abbreviation

Radio science

RSS

Infrared radiation

IRIS

Imaging science

ISS

Photopolarimetry

PPS

Ultraviolet spectroscopy

UVS

Cosmic ray particles

CRS

Low energy charged particles

LECP

Magnetic fields Plasma particles

MAG PLS

Planetary radio astronomy Plasma waves b

PRA PWS

Primary measurements Physical properties of atmospheres and ionospheres. Planet and satellite masses, densities, and gravity fields. Structure of Saturn rings. Energy balance of planets. Atmospheric composition and temperature fields. Composition and physical characteristics of satellite surfaces and Saturn rings. Imaging of planets and satellites at resolutions and phase angles not possible from Earth. Atmospheric dynamics and surface structure. Methane, ammonia, molecular hydrogen, and aerosols in atmospheres. Composition and physical characteristics of satellite surfaces and Saturn rings. Atmospheric composition including the hydrogen-to-helium ratio. Thermal structure of upper atmospheres. Hydrogen and helium in interplanetary and interstellar space. Energy spectra and isotopic composition of cosmic ray particles and trapped planetary energetic particles. Energy spectra and isotopic composition of low energy charged particles in planetary magnetospheres and interplanatary space. Planetary and interplanetary magnetic fields. Energy spectra of solar-wind electrons and ions, low energy charged particles in planetary environments, and ionized interstellar hydrogen. Planetary radio emissions and plasma resonances in planetary magnetospheres. Electron densities and local plasma wave-charged particle interactions in planetary magnetospheres.

a The broad spectrum of complex scientific phenomena requires an evaluation methodology that systematically captures inputs from each science team. b Not selected by NASA at the time of this study. Source: Table 1 of Dyer and Miles 1976.

trajectory provided unique results. Naturally, detailed modeling could capture all of these valuation considerations, but not within the above guidelines. An additive multi-attribute utility model was considered for the selection methodology and rejected for the following reasons: In the first place, it is not an easy task to identify the appropriate set of utility-independent attributes of a trajectory pair. Even if such sets of attributes exist, they may be different for the different science teams. Further, the level of effort required to identify these attributes for each science team and to construct the utility functions seemed inconsistent with the guideline to develop a concep-

Applications

221

tually simple process that could be presented to the SSG in a self-explanatory document. Also, at this early stage in the Project the scientists might be reluctant or unable to provide such detail concerning their preferences. They might feel that the disclosure of this information, even if correctly stated, could prove to be disadvantageous in future negotiations with the Project for resources and in decisions affecting the science investigations These reasons for rejecting a multi-attribute analysis apply more to the INTROSPECTING strategy than to the RATING strategy. However, the final reasons concerning the disclosure of information, if truly held by the SSG and science teams, would be adequate to rule out any multi-attribute formulation for the selection of trajectory pairs. Thirty-two trajectory pairs were identified for consideration through an iterative process with the individual science teams. These trajectory pairs spanned the range of feasible alternatives, met at least the minimum requirements of each science team, and were of interest to at least one science team. The evaluation scheme that the authors devised for this selection process consisted of a comparison of the trajectory pairs by the science teams on an individual basis, followed by a collective choice analysis. As part of the science team comparison, each team rank-ordered and then scaled the thirty-two trajectory pairs. The scaling process used lotteries consistent with von Neumann—Morgenstern utility theory to achieve a cardinal (interval) scale. Each trajectory pair was compared by each science team to a lottery between the most and least favored pairs. Then, in order to measure the value difference between the least and most favored trajectory pairs across science teams, each team was asked to assign a probability that made the team indifferent between its least favored pair and a lottery between its most favored pair and a "no-data" trajectory pair (the Atlantic Ocean Special). The larger this final probability is, the smaller the value difference is between the least and most favored trajectory pair. The authors openly stated that this last judgment did not achieve the desired information. The analysts attempted to get the science teams to revise these judgments, but were "only partially successful." It is recognized that the science teams can game the process by assigning a probability that is lower than they truly believe, thus overstating the value of the difference between the least and most favored pairs. The authors state that it was not appropriate to elicit this information from the science teams, and ultimately relied upon the project scientist for the information. Table 8.2 provides the rank and utility data for the ten science teams for the thirty-two trajectory pairs. Numerous collective choice rules were examined with the information collected from the science teams, ranging from rank sum, addition of

Table 8.2 Ordinal rankings and cardinal utility function values

load* I 2

3 4 5 7 8 9 10 11

13 15 17 18

19 20 21 22

23 24 25 26 27 28

29 30 31 32 33 34 35 36

IRIS

CSS

Trajectory Rank

Utility

Rank

Utility

O.87I O.875 O.925 6.O O.948 O.6OO 9.O O.943 O.6OO O.6OO 3 2 . O O.75O O.6OO 2O.O O.886 O.6OO I 2 . O O.93O O.887 I I.O O.935 O.6OO 13.O O.928 O.652 18.O O.915 O.887 29-5 O.869 O.6OO 4.0 O.966 8.0 O.944 O.6OO O.6OO 26.5 O.873 O.772 10.0 O.937 O.6OO 14.5 O.925 O.6OO 29.5 O.869

Rank

!3-5

6.O O.772 28.O 6.O O.772 24.5 2O.5 0.6OO 14-5

22.O

2O.5

26.O

20.5 20.5 2O.5 20.5 3-o 20.5 8.O 3.0 2O.5 20.5 2O.5 6.O 20.5 20.5 20.5 O.6OO 24-5 3 . 0 O.887 17.0 2O.5 O.6OO 7.0 20.5 O.6OO 2.0 1.0 20.5 O.66O I.O I.OOO 23.0 2O.5 O.6OO 5.0 2O.5 O . 6 0 0 22.0 2O.5 O . 6 0 0 3-o 2O.5 O.6OO 19.0 2O.5 O.6OO 26.5 2O.5 O.6OO 16.0 2O.5 O.6OO 21.0 2O.5 O.6OO 31.0

4.0 28.5 32.O 10.0 16.0 24.0 7.0

22.0 16.0 J 3-5 10.0 10.0 31.0 3-o 10.0 O.875 16.0 O.919 26.0 O.947 28.5 O.996 10.0 I.OOO 30.0 0.876 26.0 0.957 19.0 I.O 0.878 5.0 0.987 0.890 19.0 6.0 0.873 0.921 19.0 2.0 0.883 0.813 22.0

Utility

uvs

PPS

ISS Rank

O.85O 2 2 . O O.75O 25.O O.96O 28.5 O.7OO 2O.O O.67O 17.O O.6OO 32.O 5.O O.9OO O.8OO I3.O O.72O 8.5 O.92O 3-5 O.75O 28.5 O.8OO 18.5 O.85O 6-5 O.9OO 15.0 O.9OO 28.5 O.62O 22.0 O.97O 31.0 O.9OO 18.5 O.8OO 25.0 O.7OO 28.5 O.67O 25.0 O.9OO 11.0 O.65O 16.0 O.7OO 14.0 O.77O 6.5 I.OOO 12.0 O.94O 8.5 O.770 22.0 O.93O 10.0 O.77O 3-5 2.0 O.98O 1.0 O.75O

Utility

O.78O O.65O O.72O O.8lO O.52O O.58O O.75O O.63O O.52O

O.55O O.5IO

O.58O O.53O

Rank

Utility

Rank

Utility

O.82O O.79O O.9IO O.970 4.O O.92O I8.O O.8lO I3.5 O.830 26.0 0.760 19-5 O.8OO 28.5 O.72O J 3-5 0.830 9 . 0 O.84O 13.5 O.83O 22.0 O.79O 26.0 O.76O 32.0 O.63O 22.0 O.79O 31.0 O.640 30.0 O.67O

24.5

O.85O

14.O

O.7OO

15.O

O.5OO

5-5 28.5 28.5 I3.O 32.O 2I.O

O.9IO

7.0

O.83O I9.O O.45O *7-5 O.48O 26.0 O.9OO 6.0 O.43O 25.0 O.74O 7.o O.6IO 16.0 O.6OO 24.0 O.49O 31.0 O.32O 30.0 O.65O 20.0 O.77O 14.0 O.560 17-5 O.5OO 29.0 O.58O 27.5 O.44O II.O O.63O 23.0 O.87O 22.0 O.3OO 32.0 O.96O 5.0 O.98O 1.0 I.OOO 4.0

O.48O O.35O O.35I

1.0

I.OOO

O.84O O.85O O.93O O.83O O.84O O.76O O.86O

O.84O 2 8 . O O.84O 2 7 . O O.88O 6.O O.8OO

3O.O

O.86O 9-5 O.89O 2 . 0 O.96O 5-5 O.9IO 31.0 O.8lO 17.5 O.87O 24.5 O.85O 13.0 O.88O 13.0 O.88O 21.0 O.860 28.5 O.84O 5-5 O . 9 I O 5-5 O . 9 I O 28.5 O.84O 1 . 0 O.97O 17-5 O.87O I 7-5 O.87O 24.5 O.85O 3.0 O.93O 17-5 O.87O

II.O I9.O 2O.O 26.O 3I.O l6.O

8.0

O.9OO

13.0 O.88O 9-5 O.89O O.83O 21.0 O.86O O.83O 13.0 O.88O O.77O 14.5 O.85O O.72O O.8OO

9.O

24.O 25.O 22.0 29.O I7.O 8.O 32.O 5.O 3.O I.O

13.O O.72O

PIS

MAG

Utility

9.0 O.53O O.7OO 7.0 3.0 O.6lO O.64O 13-5 9.0 O.75O O.69O 26.0 6.0 O.72O O.55O 28.5 O.7IO 19-5 O.8lO 13-5 O.82O 13.5 I.OOO 24.0

O.52O

LECP

Rank

O.55O I7.O O.53O 22.0 5.O O.52O 2.0 O.57O O.6OO O.5OO

CRS

Utility

Rank

17.5

Rank

Utility

PRA Rank

26.5 O.6OO 19.O II.O O.75O I8.O 7.0 O.9OO 24.O 7.O O.9OO 25.O 6.0 O.8O0 i-5 I.OOO O.36O 26.5 O.6OO 26.0

O.546 0-547

9.0

O.556 0.552 O.558

O.7OO

17.5 17.5 O.4OO 17-5 O.26O 32.0 O.28O 26.5 O.43O II.O O.49O

O.5IO

O.485 O.3OO

O.35O O-575 0.405 0.408 O.25O

O.95O I.OOO O.97O

O.485

17-5 26.5 26.5 26.5 7.0 26.5 26.5 26.5 II.O i-5 4.0 II.O

O.7OO O.7OO O.7OO O.5OO O.6OO O.75O O.7OO O.6OO O.6OO O.6OO O.9OO O.6OO O.6OO O.6OO

O.75O I.OOO

O.95O O.75O

13.0 7.0 27.0 28.0 22.0 8.0 29.0 30.0 16.0 31.0 23.0 20.0 32.0 1.0 2.0

3.0 II.O

2.0

O.99O

4.0

O.95O

4.0

10.0 O.75O 21.0 4.0 0-975 3.0 18.0 O.62O 13.0 21.0 O.59O 12.0 15.0 O.67O 8.0 12.0 O.78O 9.0 23.0 O.57O 10.0

O.42O

17-5 4.0

O.7OO

12.0 5.0 14.0 15.0 17.0 10.0 21.0

2.O

O.99O

Utility

O.98O O.57O O.573

17-5 17.5

O.95O O.7OO O.7OO

II.O

O.75O

O.595 17-5 O.594 26.5

O.7OO O.6OO

O.6OO

O.IO9 O.IO8 O.6OO O.IO7

O.IO6 O.IO5

O.543 O.557 O.IO4 O.IO3

O.549 O.IO2

O.542 O-545 0.101 1.000 0.990 0.980 0.554 0-970 0.553 0.960 0.551 0.550 0.548 0.555 0.544

Note: "In order to reach a recommendation the results from each science team must be aggregated across the 12 trajectory pairs. Note that 6 , 1 2 , 1 4 , and 16 are missing from this list of 36. Source: Table II of Dyer and Miles 1976.

Table 8.3 Collective choice rankings and valuesa Form

Rank sum

Additive

Nash 0.6 or 0.8

P'