DECISION MAKING AND FORECASTING With Emphasis on Model Building and Policy Analysis Kneale T. Marshall U.S. Naval Postgraduate School Robert M. Oliver )A1 HOC OUOC GIA HA NO! University of California at Berkeley TRUNbTAM THONG TIN.THirVlEN M/Ml 1 r io GIFT OF THE ASIA FOUNDATION NOT FOR RE-SALE McGraw-Hill, Inc. San Francisco Auckland Bogota ^ T N ^ W ^ ' S New York St. Louis London Madrid Mexico City Milan Montreal NewDelb San Juan Singapore Sydney Tokyo Toronto
Preface 1 Basic Concepts 1.1 1.2 1.3 1.4 1.5 The Importance of Models in Decision Making The Nature of a Decision Problem Modeling Uncertainty with Probability A Brief Review of Probability 1.5.1 Outcomes, Events, and Probabilities 1.5.2 Conditional Probability and Independence 1.5.3 Partitions and the Law of Total Probability 1.5.4 Bayes' Rule 1.5.5 Random Variables and Distributions 1.5.6 Expected Values 1.5.7 Marginal, Conditional, and Joint Probabilities 1.5.8 Probabilities and Odds 1.5.9 Conditional Independence 1.5.10 Coherence 1.6 The Role of Forecasting 1.7 Elements of Influence Diagrams and Decision Trees 1.7.1 Directed Arcs in Influence Diagrams 1.7.2 Branches in Decision Trees 1.8 The Decision Sapling 1.8.1 The Value of Perfect Information 1.9 Criteria for Comparing Results 1.10 Clarifying Terminology 1.11 Book Overview 1.12 Summary and Insights XV ^ 2 ^ 4 6 ^ 1^ 12 ^ ]^ 18 20 22 23 24 24 ^5 27 27 28 ^2 ^6 39 4Q IX
2 Using Baseline Forecasts 43 2.1 2.2 2.3 43 4^ A Crop Protection Decision A Decision in Football 2.3.1 The Coach's Decision 2.3.2 Betting on the Football Game 2.3.3 Analysis of the Football Betting Problem 2.3.4 An Alternate Modeling of Outcomes: Point Scores 2.4 A Limited Life Inventory Problem 2.4.1 Marginal Analysis 2.5 The Newsboy Problem 2.6 The Value of Perfect Information 2.7 Airline Seat Allocation Based on Price and Demand 2.7.1 Space Allocation for Two Passenger Classes 2.7.2 Two-Class Space Allocation with Perfect Information 2.7.3 Discount-Seat Allocations in One Passenger Class 2.7.4 A Dynamic Decision Model 2.8 Pricing and Marketing of Hotel Rooms 2.9 The Newsboy Problem with Additional Information 2.10 Sununary and Insights 3 [f^ [f^ 48 48 5Q 52 55 55 58 59 63 65 65 69 71 73 74 77 78 Forecasts for Decision Models 83 3.1 3.2 88 88 89 90 92 95 96 97 99 101 101 103 105 108 J ^ ^ 121 121 ^ 24 ^26 128 130 The Role and Value of Forecasts 3.2.1 Some Examples of Forecasts 3.3 Several Types of Forecasts 3.3.1 Point Forecasts 3.3.2 Probability and Odds Forecasts 3.3.3 Categorical Forecasts 3.4 Decision Probabilities, Likelihoods, and Bayes' Rule 3.4.1 Decision Probabilities and Likelihoods 3.4.2 Bayes' Rule with Probability Forecasts 3.4.3 Bayes' Rule with Categorical Forecasts 3.4.4 Summary of Results in Matrix Notation 3.4.5 Sensitivity of Decision Probabilities 3.4.6 Node and Arc Reversal with Bayes' Rule 3.4.7 Using Bayes' Rule with Odds 3.5 Multiple Likelihoods and Dependent Forecasts 3.5.1 Two Forecasts for a Single Event 3.5.2 Likelihoods for Four Colon Cancer Tests 3.5.3 Forecasts for Sequential Tests 3.6 Optimal Crop Protection 3.7 Credit Scoring Decisions 3.7.1 Notation for Scores and Forecasts 3.7.2 Expected Profit and Risk of an Individual 3.7.3 Expected Profit for the Portfolio 3.8 Summary and Insights
Model Building Xi 4.2 Constructing Influence Diagrams 4.2.1 The Procedure 4.2.2 Arc Reversal and Cycles 4.2.3 No-Forgetting Arcs 4.2.4 Perfect Information 4.3 Examples of Model Formulations 4.3.1 A Decision to Seed Clouds in Hurricanes 4.3.2 Keeping Good Credit Accounts at a Bank 4.3.3 A Navy Mobile Basing Decision Problem 4.3.4 Colon Cancer Diagnosis 4.4 Building and Solving Decision Trees 4.4.1 Node Outcome and Alternative Sets 4.4.2 Drawing Consistent Decision Trees 4.4.3 Perfect Information 4.4.4 Decision Tree Solutions 4.5 The Bank Credit Problem 4.5.1 The Economic Value of a Performance Forecast 4.5.2 Perfect Information about Performance 4.6 Colon Cancer Decision 4.6.1 Sequential Decisions for Colon Cancer Detection 4.7 Irrelevant Decisions and a Game-Show Problem 4.8 A Budget Planning Problem 4.9 Summary and Insights 133 133 135 i- 139 14Q 141 I43 I44 148 152 156 160 161 164 166 167 169 171 172 173 177 179 182 187 190 Model Analysis 192 5.1 5.2 5.3 Betting on the Football Game An Expert Opinion Model 5.3.1 An Aircraft Part Decision Problem 5.3.2 A Crop Protection Problem 5.4 Sensitivity Analysis Using Decision Probabilities 5.4.1 The Economic Value of a Forecast 5.5 Sensitivity Analysis Using Forecast Likelihoods 5.6 with One or More Forecasts 5.6.1 Optimal Policies and Expected Returns 5.6.2 A Numeric Example 5.6.3 A Single Decision with Two Forecasts 5.7 Sequential Decisions Using Sequential Forecasts 5.8 Summary and Insights 192 193 197 199 202 204 207 210 213 215 217 219 222 227 229 Subjective Measures and Utility 232 6.1 6.2 Basics of Utility Theory 6.2.1 Indifference Probabilities and Certainty Equivalents 6.2.2 Assumptions of Utility Theory 6.3 Determination of Utility Functions 232 233 234 235 237 4.1
XII 6.3.1 Utihties as Indifference Probabilities 6.3.2 Utihties from Certainty Equivalents 6.3.3 Cautionary Comments 6.4 Examples of Utihty Functions 6.4.1 An Exponential Utility Function 6.4.2 A Logarithmic Utility Function 6.5 Measures of Risk 6.5.1 Risk Premium 6.5.2 A Risk Aversion Function 6.6 Some Properties of Utility Functions 6.7 Summary and Insights 7 Multiattribute 7.1 7.2 7.3 7.4 A Decision Saphng with Two Attributes The Crop Protection Problem with Two Attributes Car Ranking and Replacement 7.4.1 Ranking Cars by Preference 7.4.2 Car Replacement 7.5 The Added Cost of Conflict Resolution 7.5.1 Car Ranking Revisited 7.6 Assessment of Trade-Offs through Preferences 7.6.1 Two Attributes 7.6.2 Many Attributes and Alternatives 7.6.3 Ranking Cars Using Three Attributes 7.6.4 The Car Replacement Problem Revisited 7.7 A Hierarchical Multiattribute Model 7.7.1 A Hierarchical Cost-Benefit Model 7.7.2* The Two-Hierarchy Multigroup Model 7.7.3* Tradeoff Weights through Indifference Probabihties 7.8 The Analytic Hierarchy Process 7.8.1 Ranking Alternatives with AHP 7.8.2 Avoiding Rank Reversal in AHP 7.8.3 Finding the Weights in AHP 7.9 A Budget Planning Example with Three Attributes 7.10* Multiattribute Utility 7.10.1 An Example with Two Attributes 7.11 Summary and Insights 8 Forecast Performance 8.1 8.2 1 ^ Forecast Calibration 8.2.1 Calibration of Categorical Forecasts 8.2.2 Calibration of Probability Forecasts 8.2.3 Calibration in Expectation 8.3 Forecast Discrimination 8.3.1 Discrimination in Probability Forecasts 8.3.2 Discriminating Categorical Forecasts 8.4 Comparing Discrimination and Calibration 8.5 Forecast Correlation " 238 239 240 241 242 243 244 245 245 248 249 250 252 252 253 256 258 258 261 263 265 266 267 268 269 270 271 272 275 276 278 280 284 286 288 291 295 298 300 303 303 304 304 305 307 g 3Q9 2
8.6 Measuring Forecast Performance with Brier Scores 8.6.1 An Example 8.7 Calibration Effects in Decision Models 8.7.1 Effect of Cahbration on Crop Protection PoUcies 8.7.2 Uncahbrated Forecasts in a Credit Portfolio 8.7.3 Stable Likehhoods 8.7.4 Numeric Example 8.8 Coherent Categorical and Probabihty Forecasts 8.8.1 The Protect Decision with a Categorical Forecast 8.9 Coherent Aggregation of Categorical Forecasts 8.10 Forecast Aggregation and Optimal Decisions 8.11 Summary and Insights Advanced Concepts 9.1 9.2 Classifying Influence Diagrams 9.2.1 A Proper Influence Diagram 9.2.2 Influence Diagrams in Extensive Form 9.2.3 Irrelevant Decision and Chance Nodes 9.3 Chance Influence Diagrams 9.3.1 Directed Graphs, Predecessor and Successor Sets 9.3.2 Equivalent Chance Influence Diagrams 9.3.3 Bayes' Rule and Arc Reversal 9.3.4 Barren Nodes 9.3.5 Cancer Diagnosis 9.3.6 Arc Reversal and Barren Node Removal 9.4 Path History and Rollback Computations 9.4.1 A History Vector Algorithm 9.4.2 Engine Maintenance 9.4.3 Rollback Using Path History Vectors 9.4.4 A Nuclear Reactor Decision Example 9.5 Multiattribute Rollback with and without Trade-Offs 9.5.1 Calculating Noninferior Points with Two Attributes 9.5.2 Rollback with Linear Trade-Offs 9.5.3 Reactor Decision Revisited 9.5.4 Economic Value per Life Saved 9.5.5 History and Rollback with Nonlinear Utihties 9.5.6 Nonlinear Utilities in the Nuclear Reactor Problem 9.6 Reducing Influence Diagrams 9.6.1 Equivalent Influence Diagrams 9.6.2 An Example of ERD Reduction 9.6.3 Chance Node Removal through Expectation 9.6.4 Node Removal through Maximization 9.6.5 Revisiting the Aircraft Part Problem 9.7 Summary and Insights References Author Index Subject Index xiii ^i o l\^n 320 320 -IOT. Z^ 328 328 330 331 333 336 340 342 342 343 345 347 349 35O 351 353 354 356 357 360 362 363 364 365 366 370 370 371 372 375 376 379 381 381 384 385 387 388 390 3