122421248(1).pdf

8/19/2019 122421248(1).pdf

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DECISION MAKING

UNDER

UNCERTAINTY:

Models and Choices

Charles A. Holloway

Stanford University

TECHNISCHE HOCHSCHULE DARMSTADT

Fachbereich 1

G e s a m t b i b l i o t h e k

B e t r t e b s w i r t s c r t a f t s l e h r e

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Abste

i I-Nr. A 4 3 / 4

Sadigebiele:.

I2L3JL

PRENTICE-HALL INC . Englewood Cliffs New Jersey 07632

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Contents

Preface xix

PART I INTRODUCTION AND

BASIC CONCEPTS

Chapter Introdu ction to the Analysis of Decisions 3

Using Analysis 4

Th e N eed for Some Philosophy 5

Sources of Co mp lexity 5

A Large Num ber of Factors 5

More Than One Decision Maker 6

Multiple Attributes 6

Th e Problems in Choosing Un der Un certainty 7

Evaluating Decisions Under Uncertainty 7

Making Decisions Under Uncertainty 8

Preview 9

Summary 11

Assignm ent M ateria l 11

Selected Reference s on M ultipe rson De cisions 11

Chapter 2 The Ana lytical Approach 13

The Qu ant i ta t ive/Analyt ica l Approach 14

The Mo deling Phase 14

The Choice Phase 15

Decomposit ion 15

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The Use

of

Decomposition

17

Different Ways

to

Decompose

18

The Use

of

Judgment

18

The Role of Managers 19

The Use of Analytical Procedures 19

Analytical Procedures

as

Information Generators

20

Implementation

of

Decisions B ased on Analysis

20

•

*

Steps

in the

Overall Process

21

•

•

Developing Alternatives

21

•

Creating

the

Model: Describing

the

Consequences

22

•

•

Creating

the

Model: Relating Alternatives

to

Consequences

23

•

*

Making

the

D ecision

26

Summary 26

Assignment Material 27

Selected References on Implementation 29

Chapter

3

Modeling Under Uncertainty—

Diagrams and Tables 3

Basic Concepts and Techniques 31

Decision Diagrams 32

Diagramming Conventions

32

Guidelines

and

Rules

for

Diagramming

33

Immediate Decision Alternatives

—

Guideline 1 36

Determine

the

Evaluation Date— Guideline

2 37

Uncertain Events That Affect

the

Consequences

of

the

Initial Alternatives— Guideline

3 37

Future Decisions— Guideline

4 37

Uncertain Events That Provide Information

for Future Decisions— Guideline

5 38

Mutually Exclusive

and

Collectively Exhaustive

Requirements— Guidelines

6 and 7 38

Diagram Events

and

D ecisions

Chronologically— Guideline

8 38

Assignment of Evaluation Units or Measures for

Consequences 40

Payoff Tables 42

The Table Construction

43

Calculation

of

Contribution

43

Decision Diagram Representation

43

'

More on Decision Diagramming 43

•

The

Process

of

Decision Diagramming

45

* What Qualifies

as a

Decision Node?

46

' Alternatives That

Are

Unknown

at the

Decision P oint

47

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• Inferior Alternatives 47

• Evaluation Date 48

' ' Alternatives with Extended Evalua tion Dates 48

• Mutually Exclusive Alternatives 48

• Mutually Exclusive Outcomes 49

• Ordering of Events and Decisions 50

• Exceptions to Chronological Order 51

Ov erall Proce ss 51

Summary 52

Assignment M aterial 53

Sup plem entary References 59

Chapter 4 Introduction to Probability 60

Basic Co nce pts an d Definitions 62

Set 62

Subset 63

Uncertain Event 63

Outcome Space (or Samp le Space) 63

Event 64

Occurrence of an Event 64

Complement 65

Union 66

Intersection 67

. Null (Empty) Set 67

Mutually Exclusive 67

Collectively Exhaustive 68

Tec hnica l Re quire m ents for Probabilities 68

Notation for Probabilities 68

Conditions on Probabilities 68

Notation for Summ ations 69

Limitation of the Technica l Requirements 69

Proba bility Distribu tions 69

Probability Density Function 70

Cum ulative Probability Distribution 71

Sum m ary M easure s for Proba bility Distributions 74

The Mean of a Probability Distribution 74

Expected Values 76

Mean or Expected Value 76

Standard Deviation and Variance 76

Variance 76

Standard Deviation 77

Th e M eanin g of Probabilities 78

Classical View of Pro bab ility 78

Criticism of the Classical View 79

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Limitation of the Classical View 80

Relative Frequ ency 80

Relative Frequency Probability 80

The Cond itions and Required Judgment 81

Limitation of the Relative Frequency View 82

Subjective Pro bab ilities 82

Difference Between Objective and Subjective Views 82

Assessm ent of Subjective Probabilities 83

The Use of Subjective Probabilities 84

Summary 85


Chapter 5 Making Choices Under Uncertainty 90

Direc t Ch oice 92

Outcome Dom inance 93

Probabilistic Dom inance 94

Direct Choice Using Probability Distributions 98

Direct Choice Using Summ ary Measures 99

Direct Choice Using Aspiration Level 100

Ce rtainty Eq uivalents 100

The Insurance Analogy 102

Properties of Certainty Equivalents 103

Assessing Certainty Equivalents 103

Procedures for Assessing Certainty Equivalents 103

Certainty Equivalents for Com plex Uncertain Events 105

Usin g M ean s or Exp ected Values 106

Expected Values and Certainty Equivalents 106

Attitudes Toward Risk 107

• Pitfalls in Calculating Expected Values 107

M ultistage Prob lem s 108

Seque ntial Ana lysis or Ro llback 109

• •

Rollback Using Direct Choice 110

Rollback Using Certainty Equivalents 114

Rollback Using Expected Values 116

• • C om plete Strategies 118

• *

Com plete Strategy 118

• • Specifying Com plete Strategies 119

' ' Choices with Com plete Strategies 121

Summary 121

Direct Choice 121

Certainty Equivalents 122

Means or Expected Values 122

Multistage Problems 122

Assignmen t M aterial 123

Contents

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Chapter 6 Preferences and Ca lculation of Certainty

Equivalents 128

Basic Co ncep ts 130

The Reference Gam ble 130

Preferences 131

Preference Scale 131

Preference Curve 131

Utility 131

Basic Pro ced ure 131

Assessing a Preference Curve 132

Plotting the Preference Curve 134

Calcu lating Certainty Equivalents, 135

Summ ary of the Procedure 136

Basis for the Pro ced ure 136

Substitution of Reference Gam bles 136

Reduction to a Single-Stage Gam ble 137

* •

Justifying the Single-Stage Gam ble 138

Choice Between Alternatives 140

Relationship to Expected Preference Procedure 140

Summary 143

Assignm ent M ateria l 144

PART 2 MODELS AND PROBABILITY

Chapter 7 Calculating Probabilities for Com pound

Events 153

Co m po un d Events 155

Exam ples of Co m po und Events Form ed by U nion s 155

Th e A ddition Ru le 156

Addition Rule for Mutually Exclusive Events 157

Addition Rule for Non-M utually Exclusive Events 157

* *

Addition Rule for More Than Two Events 157

Examples of Compound Events Formed by

Intersections 158

Marginal Event 159

Joint Event 159

Co ndition al Proba bilities 159

The Concept of Conditional Probability 159

Using Tables to Calculate Conditional Probabilities 163

Th e M ultipl icat ion Rule 165

Reversal of Conditioning 166

Independence 170

ont nts

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Multiplication Rule for Independent Events 171

•

* Relationship Between Mutually Exclusive Events

and Independent Events 172

Summary 174


Chapter 8 Discrete Random Variables, Outcome Spaces,

and Calculating Probabilities

179

Defining Outcome Space 181

Payoff A dequacy

182

Assessment Adequacy 182

Random Variables 183

Probability Distributions for Random Variables 186

Means

of

Random Variables

186

Standard Deviations and Variances of

Random Variables

186

Independent Random Variables 187

Calculation

of

Expected Values

for

Random Variables

187

•

*

Random Variables as Functions 189

• * Notation

for

Function

189

Probabilities for Compound Random Variables 190

Calculating Probability Distributions for Complicated

Random Variables 193 .

Assessment-Adequate Diagrams 195

Using Tables Instead of Inserting Extra Uncertain Events

into the Diagram 199

Summary 200


Chapter 9 Con tinuous Random Variables, Models,

and C alculations 2 6

Continuous Versus Discrete Models 208

Diagrams for Continuous Models 209

Probability Distributions for Continuous

Random Variables 210

Cumulative Distributions for Continuous

Random Variables .210

Requirements on Probability Density Functions for

Continuous Random Variables

211

Interpretation of Probability Density Functions for

Continuous Random Variables

211

Relationship Between Cumulative Distributions

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and Density Functions 212

Calculat ions Using Con tinuous Distr ibutions 213

Sum ma ry M easures for Co ntinuou s Distr ibutions 214

Median 214

Mode 214

Discrete App roximations 215

Procedure for Equally Probable Interval

Approximation 215

Procedure for Approxima tion with Intervals Specified

on the Random Variable Axis 218

Us ing Disc rete Ap prox ima tions to Solve a Prob lem 219

Summary 221


Chapter 10 Theoretical Probability Distributions 226

Binomial Dis tr ibution 228

Illustration of the Binomial Formu la 229

The Binomial Distribution 229

Formu lation of Problems Using Binomial Distribution 231

Verification of Conditions 231

Using the Tables 232

Poisson D istributio n 234

The Poisson Distribution 237

Formulation of Problems Using the

Poisson Distribution 237

Using the Tables 237

Poisson Approximation to Binomial 238

Th e No rm al Distr ibution 239

Using the Norm al Table 240

Norm al Approximation to the Binomial 243

* * Exp onential Dis tr ibution 244

* * Relationship to Poisson 245

* * The No-M emory Property 245

* * Beta D istributio n 246

Summary 249


App endix 10: Com pact Count ing Techniques

an d the Binom ial Distribu tion 254

Chapter

Em pirical Probability Distributions 257

Discrete R an do m Variables 259

Mechan ics of Obtaining the Distribution 259

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The Problem with a Small Am ount of Data 260

Options in Dealing with a Small Am ount of Data 262

Com bining Em pirical Data with Other Information 263

Comparability 263

Co ntinuous R an do m Variables 263

The Interval-Choice Problem 264

Plotting as a Cum ulative 266

Direct Smoothing 267

Comparability 268

Summary 271

Assignm ent M aterial 271

Ap pend ix 11 A : Acco unting for a Small Am oun t of D ata

in Discrete Distributions 273

Append ix

IB: Improving Comparabil i ty with

a M odel 275

Chapter 12 Sub jective Assessment of Probability

Distributions 280

Subjective Jud gm ents and Probabilities 282

The Technical Requirements 282

The Problem 283

Coherence and Axioms 284

M aintaining Coherence 285

De finition of Subjective Pro bab ility 285

Assessment Lotteries 286

Subjective Probability 287

• *

Relationship to Limiting Relative Frequencies 288

Assessm ent Proced ures 290

Assessment for Specific Events 290

Direct Assessment for a Specific Event 290

Indirect Assessment for a Specific Event 291

Assessment for Continuous Random Variables 295

Extreme Values 295

Cum ulative Plot 296

Filling Ou t the Distribution 296

Finding the Median and Quartiles 296

Visually Fitting Curve 297

Verification 297

Dec om posit ion to Aid Assessment 298

Using Experts 300

• * Decomposition with Continuous Random

Variables 300

Ac cura cy of Subjective Assessments 301

xii Contents

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•

* Modes of Human Judgments 303

•

•

Availability

303

•

*

Adjustment

and

Anchoring

304

•

•

Representativeness

304

•

*

Unstated Assumptions

304

Summary 304


Appendix 12: Axiom Systems for Subjective

Probabilities 306

Notation

306

Conditions

309

Chapter

13

Bayesian Revision

of

Probabilities

311

The Revision Process for Discrete Random

Variables 313

Basic Revision Calculations

313

Interpretation

of

the Revision Process

315

Equal Likelihoods

318

Equal Priors

319

Increasing

the

Amount

of

Evidence

320

Assessment

of

Likelihoods

323

Assessment

of

Likelihoods Using

the

Binomial Distribution

324

*

* Assessment

of

Likelihoods Using

the

Poisson Distribution

324

' • Assessment

of

Likelihoods Using

the

Normal

Distribution

325

*

*

Assessment

of

Likelihoods Using Theoretical Distributions

in General

326

Assessment

of

Likelihoods Using Relative Frequencies

327

*

*

Assessment

of

Likelihoods Using

a

Subjective

Approach

328

*

• The

Revision Process

for

Conjugate

Distr ibutions

330

*

• Normal Prior Distributions with

a

Normal Data- athering

Process

330

*

•

Beta Prior Distributions with

a

Binomial Data-Gathering

Process

331

Some Illustrations of the Use of Bayesian Revision 332

Summary 338


Appendix 13: Formal Notation and Bayes Formula 342

Contents

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Chapter 14 Information and Its Value 343

Con cept of Inform ation 344

Sources of Info rm atio n 346

Empirical Data 346

Subjective Opinion— Types of Information

from Experts 346

Processing Expert Judgments 347

Value of Inform ation 348

Expected Value of Perfect Information (EVP I) 349

Other Ways to Calculate EVP I 352

Expected Value of Imperfect or Sample Information 353

•

EVSI W ithout Bayes' Theorem 355

• The Relationship Between Value of Information

and Amount of Uncertainty 357

Sensitivity A naly sis 358

• * Value of Information with Different

Risk A ttitude s 359

Summary 361


Chapter 15 Monte Carlo Methods 368

Sam pling from Discrete Probab ili ty Distributions 370

Random Numbers 370

Monte Carlo Sampling— Coin Example 371

Monte Carlo Sampling— Die Example 371

Use of Cum ulative Distributions 372

Summ ary of Monte Carlo Sampling Procedure 373

• Calculating a Probabili ty Distribution Using

M onte Carlo 374

• • Ev ent-O riented Queuing) Problem s 377

Monte Carlo Sampling from Continuous Probabili ty

Distributions 380

• • Comparison of Discrete Approximations

and M onte Carlo 381

Summary 382


PA RT 3 C HO ICE S AN D PREFERENCES

Chapter 16 Attitudes Tow ard Risk and the

Choice Process 389

Contents

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Review of O ption s for Cho osing 391

Direct Choice 391

Certainty Equivalents 391

Risk Av ersion 391

• * Decreasing Risk Aversion 393

Constant Risk Aversion 394

Choices U nd er Risk Aversion 395

• Using Direct Choice with Com plete Strategies 396

' Using Certainty Equivalents 402

Minimizing Variance for Risk-Averse Decision Makers 402

Separability with Constant Risk Aversion 405

• • More Properties with Constant Risk Aversion 407

Risk N eutral i ty 408

Choices U nd er Risk Neu tral ity 409

Separability with Risk Neu trality 409

Risk Seeking 409

Em pirical Eviden ce 411

Summary 414


Chapter 17 Preference Assessment Procedures 419

Th e Preference Assessment Prob lem in G ene ral 420

Choice of the Range of Payoff Values

(Evaluation Units) 421

Preference Assessment Using the Basic

Reference G am ble 422

The Basic Reference Gam ble 422

The Basic Reference Gam ble Assessment Procedure 422

• *

A Variation on the Reference Gam ble Assessment

Procedure 423

Preference Assessm ent Using 50-50 G am bles 425

Comparisons of Methods for Assessing Preference

Curves 427

Assessment for Special Risk A ttitudes 429

Risk Neu trality 429

Risk Aversion 429

• * Constant Risk Aversion 429

Risk-Seeking 431

Re solution of Inconsistencies 431

• • Scale Va lues for Preferences or Utilities 431

Summary 432


Contents

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Chapter 18 Behavioral Assumptions and Limitations

of Decision Analysis

436

The Basic Ideas 438

The Behavioral Assumptions or Axioms for Choice 438

Implications of the Assumptions 441

Limitations Imposed by the Behavioral Assumptions 445

Transitivity

for

Individuals

445

Existence of Preferences for Groups 446

Continuity Assumption with Extreme Outcom es

446

Monotonicity Assumption with Differences in the Time

at Which Uncertainty

Is

Resolved

447

Assumptions and Limitations on the Model 448

Defining Possible Outcomes

448

Subjective Assessment of Probability for Independent Uncertain

Events

450

Assigning Evaluation Units When Payoffs Occur

Over

an

Extended Time Horizon

451

Summary 453


Chapter 19 Risk Sharing and Incentives

456

Risk Sharing 457

Diversification 462

Diversification with Independent Investments 462

Diversification with Dependent Investments 464

Diversification

and

Financial Markets

465

Risk Sharing with Differential Information 465

Agreements with

the

Same Preferences

and

Beliefs

465

Agreement with Different Preferences and Beliefs 466

Incentive Systems 467

Summary 472


Chapter 20 Cho ices with Multiple Attributes 475

The Problem 476

Descriptive Procedures 477

Dominance

478

Sat isficing 478

Lexicographic Procedure

479

Combination Procedure 480

Trade-off Procedures 480

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The Trade-off Procedure 481

Indifference Curves 482

More Than Two Dimensions 483

Multiple Attribute Problems with Uncertainty

Summary 487


484

APPENDICES

Appendix A Binomial Distribution— Individual Terms 493

Appendix B Binomial Distribution—C umulative Terms 500

Appendix C Poisson Distribution— Individual Terms 507

Appendix D Poisson Distribution—C umulative Terms 510

Appendix E Areas Under the Normal Curve

51 3

Appendix F Fractiles of the Beta Distribution

515

Appendix G Random Numbers

Index

517

519

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