DECISION MAKING UNDER UNCERTAINTY:

DECISION MAKING UNDER

UNCERTAINTY:

Models and Choices

Charles A. Holloway

Stanford University

TECHNISCHE HOCHSCHULE DARMSTADT Fachbereich 1

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Contents

Preface xix

PART I INTRODUCTION AND BASIC CONCEPTS

Chapter 1 Introduction to the Analysis of Decisions 3 Using Analysis 4

The Need for Some Philosophy 5 Sources of Complexity 5

A Large Number of Factors 5 More Than One Decision Maker 6 Multiple Attributes 6

The Problems in Choosing Under Uncertainty 7 Evaluating Decisions Under Uncertainty 7 Making Decisions Under Uncertainty 8 Preview 9

Summary 11

Assignment Material 11

Selected References on Multiperson Decisions 11

Chapter 2 The Analytical Approach 13 The Quantitative/Analytical Approach 14

The Modeling Phase 14 The Choice Phase 15 Decomposition 15

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 Based 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 Decision 26 Summary 26

Assignment Material 27

Selected References on Implementation 29 Chapter 3 Modeling Under Uncertainty—

Diagrams and Tables 30 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 Decisions

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 Point 47

Contents

• Inferior Alternatives 47

• Evaluation Date 48

' ' Alternatives with Extended Evaluation Dates 48

• Mutually Exclusive Alternatives 48

• Mutually Exclusive Outcomes 49

• Ordering of Events and Decisions 50

• Exceptions to Chronological Order 51 Overall Process 51

Summary 52

Assignment Material 53 Supplementary References 59

Chapter 4 Introduction to Probability 60 Basic Concepts and Definitions 62

Set 62 Subset 63

Uncertain Event 63

Outcome Space (or Sample 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

Technical Requirements for Probabilities 68 Notation for Probabilities 68

Conditions on Probabilities 68 Notation for Summations 69

Limitation of the Technical Requirements 69 Probability Distributions 69

Probability Density Function 70 Cumulative Probability Distribution 71

Summary Measures for Probability 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

The Meaning of Probabilities 78 Classical View of Probability 78 Criticism of the Classical View 79

Limitation of the Classical View 80 Relative Frequency 80

Relative Frequency Probability 80

The Conditions and Required Judgment 81 Limitation of the Relative Frequency View 82 Subjective Probabilities 82

Difference Between Objective and Subjective Views 82 Assessment of Subjective Probabilities 83

The Use of Subjective Probabilities 84 Summary 85

Assignment Material 86

Chapter 5 Making Choices Under Uncertainty 90 Direct Choice 92

Outcome Dominance 93 Probabilistic Dominance 94

Direct Choice Using Probability Distributions 98 Direct Choice Using Summary Measures 99 Direct Choice Using Aspiration Level 100 Certainty Equivalents 100

The Insurance Analogy 102

Properties of Certainty Equivalents 103 Assessing Certainty Equivalents 103

Procedures for Assessing Certainty Equivalents 103 Certainty Equivalents for Complex Uncertain Events 105 Using Means or Expected Values 106

Expected Values and Certainty Equivalents 106 Attitudes Toward Risk 107

• Pitfalls in Calculating Expected Values 107 Multistage Problems 108

Sequential Analysis or Rollback 109

• • Rollback Using Direct Choice 110 Rollback Using Certainty Equivalents 114 Rollback Using Expected Values 116

• • Complete Strategies 118

• * Complete Strategy 118

• • Specifying Complete Strategies 119 ' ' Choices with Complete Strategies 121 Summary 121

Direct Choice 121 Certainty Equivalents 122 Means or Expected Values 122 Multistage Problems 122 Assignment Material 123

Contents

Chapter 6 Preferences and Calculation of Certainty

Equivalents 128 Basic Concepts 130

The Reference Gamble 130 Preferences 131

Preference Scale 131 Preference Curve 131

Utility 131

Basic Procedure 131

Assessing a Preference Curve 132 Plotting the Preference Curve 134 Calculating Certainty Equivalents, 135 Summary of the Procedure 136 Basis for the Procedure 136

Substitution of Reference Gambles 136 Reduction to a Single-Stage Gamble 137

* • Justifying the Single-Stage Gamble 138 Choice Between Alternatives 140

Relationship to Expected Preference Procedure 140 Summary 143

Assignment Material 144

PART 2 MODELS AND PROBABILITY

Chapter 7 Calculating Probabilities for Compound

Events 153 Compound Events 155

Examples of Compound Events Formed by Unions 155 The Addition Rule 156

Addition Rule for Mutually Exclusive Events 157 Addition Rule for Non-Mutually 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

Conditional Probabilities 159

The Concept of Conditional Probability 159

Using Tables to Calculate Conditional Probabilities 163 The Multiplication Rule 165

Reversal of Conditioning 166 Independence 170

Multiplication Rule for Independent Events 171

• * Relationship Between Mutually Exclusive Events and Independent Events 172

Summary 174

Assignment Material 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

Assignment Material 201

Chapter 9 Continuous Random Variables, Models,

and Calculations 206 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

Contents.

and Density Functions 212

Calculations Using Continuous Distributions 213 Summary Measures for Continuous Distributions 214 Median 214

Mode 214

Discrete Approximations 215 Procedure for Equally Probable Interval Approximation 215

Procedure for Approximation with Intervals Specified on the Random Variable Axis 218

Using Discrete Approximations to Solve a Problem 219 Summary 221

Assignment Material 224

Chapter 10 Theoretical Probability Distributions 226 Binomial Distribution 228

Illustration of the Binomial Formula 229 The Binomial Distribution 229

Formulation of Problems Using Binomial Distribution 231 Verification of Conditions 231

Using the Tables 232 Poisson Distribution 234 The Poisson Distribution 237 Formulation of Problems Using the Poisson Distribution 237 Using the Tables 237

Poisson Approximation to Binomial 238 The Normal Distribution 239

Using the Normal Table 240

Normal Approximation to the Binomial 243

* * Exponential Distribution 244

* * Relationship to Poisson 245

* * The No-Memory Property 245

* * Beta Distribution 246 Summary 249

Assignment Material 250

Appendix 10: Compact Counting Techniques and the Binomial Distribution 254

Chapter 11 Empirical Probability Distributions 257 Discrete Random Variables 259

Mechanics of Obtaining the Distribution 259

The Problem with a Small Amount of Data 260 Options in Dealing with a Small Amount of Data 262 Combining Empirical Data with Other Information 263 Comparability 263

Continuous Random Variables 263 The Interval-Choice Problem 264 Plotting as a Cumulative 266 Direct Smoothing 267 Comparability 268 Summary 271

Assignment Material 271

Appendix 11 A: Accounting for a Small Amount of Data in Discrete Distributions 273

Appendix 1 IB: Improving Comparability with a Model 275

Chapter 12 Subjective Assessment of Probability

Distributions 280 Subjective Judgments and Probabilities 282

The Technical Requirements 282 The Problem 283

Coherence and Axioms 284 Maintaining Coherence 285

Definition of Subjective Probability 285 Assessment Lotteries 286

Subjective Probability 287

• * Relationship to Limiting Relative Frequencies 288 Assessment Procedures 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

Cumulative Plot 296

Filling Out the Distribution 296 Finding the Median and Quartiles 296 Visually Fitting Curve 297

Verification 297

Decomposition to Aid Assessment 298 Using Experts 300

• * Decomposition with Continuous Random Variables 300

Accuracy of Subjective Assessments 301

xii Contents

• * Modes of Human Judgments 303

• • Availability 303

• * Adjustment and Anchoring 304

• • Representativeness 304

• * Unstated Assumptions 304 Summary 304

Assignment Material 305

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 Distributions 330

* • Normal Prior Distributions with a Normal Data- Gathering Process 330

* • Beta Prior Distributions with a Binomial Data-Gathering Process 331

Some Illustrations of the Use of Bayesian Revision 332 Summary 338

Assignment Material 339

Appendix 13: Formal Notation and Bayes Formula 342

Chapter 14 Information and Its Value 343 Concept of Information 344

Sources of Information 346 Empirical Data 346

Subjective Opinion—Types of Information from Experts 346

Processing Expert Judgments 347 Value of Information 348

Expected Value of Perfect Information (EVPI) 349 Other Ways to Calculate EVPI 352

Expected Value of Imperfect or Sample Information 353

• EVSI Without Bayes' Theorem 355

• The Relationship Between Value of Information and Amount of Uncertainty 357

Sensitivity Analysis 358

• * Value of Information with Different Risk Attitudes 359

Summary 361

Assignment Material 362

Chapter 15 Monte Carlo Methods 368 Sampling from Discrete Probability Distributions 370 Random Numbers 370

Monte Carlo Sampling—Coin Example 371 Monte Carlo Sampling—Die Example 371

Use of Cumulative Distributions 372

Summary of Monte Carlo Sampling Procedure 373

• Calculating a Probability Distribution Using Monte Carlo 374

• • Event-Oriented (Queuing) Problems 377 Monte Carlo Sampling from Continuous Probability Distributions 380

• • Comparison of Discrete Approximations and Monte Carlo 381

Summary 382

Assignment Material 383

PART 3 CHOICES AND PREFERENCES Chapter 16 Attitudes Toward Risk and the

Choice Process 389

Contents

Review of Options for Choosing 391 Direct Choice 391

Certainty Equivalents 391 Risk Aversion 391

• * Decreasing Risk Aversion 393 Constant Risk Aversion 394 Choices Under Risk Aversion 395

• Using Direct Choice with Complete 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 Neutrality 408

Choices Under Risk Neutrality 409 Separability with Risk Neutrality 409 Risk Seeking 409

Empirical Evidence 411 Summary 414

Assignment Material 415

Chapter 17 Preference Assessment Procedures 419 The Preference Assessment Problem in General 420 Choice of the Range of Payoff Values

(Evaluation Units) 421

Preference Assessment Using the Basic Reference Gamble 422

The Basic Reference Gamble 422

The Basic Reference Gamble Assessment Procedure 422

• * A Variation on the Reference Gamble Assessment Procedure 423

Preference Assessment Using 50-50 Gambles 425 Comparisons of Methods for Assessing Preference Curves 427

Assessment for Special Risk Attitudes 429 Risk Neutrality 429

Risk Aversion 429

• * Constant Risk Aversion 429 Risk-Seeking 431

Resolution of Inconsistencies 431

• • Scale Values for Preferences or Utilities 431 Summary 432

Assignment Material 432

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 Outcomes 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

Assignment Material 454

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

Assignment Material 473

Chapter 20 Choices 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

Contents

The Trade-off Procedure 481 Indifference Curves 482

More Than Two Dimensions 483

Multiple Attribute Problems with Uncertainty Summary 487

Assignment Material 488

484

APPENDICES

Appendix A Binomial Distribution—Individual Terms 493 Appendix B Binomial Distribution—Cumulative Terms 500 Appendix C Poisson Distribution—Individual Terms 507 Appendix D Poisson Distribution—Cumulative Terms 510 Appendix E Areas Under the Normal Curve 513 Appendix F Fractiles of the Beta Distribution 515

Appendix G Random Numbers Index

517 519