DECISION MAKING UNDER UNCERTAINTY:

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1 DECISION MAKING UNDER UNCERTAINTY: Models and Choices Charles A. Holloway Stanford University TECHNISCHE HOCHSCHULE DARMSTADT Fachbereich 1 Gesamtbibliothek Betrtebswirtscrtaftslehre tnventar-nr. :...2>2&,...S'.?S7. Abste i I-Nr. A 4 3 / 4 Sadigebiele: İ2L3JL PRENTICE-HALL, INC., Englewood Cliffs, New Jersey 07632

2 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

3 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

4 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

5 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

6 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

7 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 Requirements on Probability Density Functions for Continuous Random Variables 211 Interpretation of Probability Density Functions for Continuous Random Variables 211 Relationship Between Cumulative Distributions.

8 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

9 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

10 * 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

11 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

12 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 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

13 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

14 The Trade-off Procedure 481 Indifference Curves 482 More Than Two Dimensions 483 Multiple Attribute Problems with Uncertainty Summary 487 Assignment Material 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 517 Index 519

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