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Quantitative Decision Techniques

Quantitative Decision Techniques. 13/04/2009 Decision Trees and Utility Theory. Chapter Outline. 4.1 Introduction 4.2 Decision Trees 4.3 How Probability Values Are Estimated by Bayesian Analysis 4.4 Utility Theory 4.5 Sensitivity Analysis. Introduction .

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Quantitative Decision Techniques

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  1. Quantitative Decision Techniques 13/04/2009 Decision Trees and Utility Theory

  2. Chapter Outline 4.1 Introduction 4.2 Decision Trees 4.3 How Probability Values Are Estimated by Bayesian Analysis 4.4 Utility Theory 4.5 Sensitivity Analysis

  3. Introduction Decision trees enable one to look at decisions: • with many alternatives and states of nature • which must be made in sequence

  4. Decision Trees A graphical representation where: • a decision node from which one of several alternatives may be chosen • a state-of-nature node out of which one state of nature will occur

  5. Thompson’s Decision Tree Fig. 4.1 A State of Nature Node Favorable Market 1 Unfavorable Market Construct Large Plant A Decision Node Favorable Market Construct Small Plant 2 Unfavorable Market Do Nothing

  6. Five Steps toDecision Tree Analysis • Define the problem • Structure or draw the decision tree • Assign probabilities to the states of nature • Estimate payoffs for each possible combination of alternatives and states of nature • Solve the problem by computing expected monetary values (EMVs) for each state of nature node.

  7. Decision Table for Thompson Lumber

  8. Thompson’s Decision Tree Fig. 4.2 A State of Nature Node Favorable Market (0.5) $200,000 1 Unfavorable Market (0.5) Construct Large Plant EMV =$10,000 -$180,000 A Decision Node Favorable Market (0.5) $100,000 Construct Small Plant 2 Unfavorable Market (0.5) EMV =$40,000 -$20,000 Do Nothing 0

  9. 2nd Decision Table for Thompson Lumber

  10. Thompson’s Decision Tree -Fig. 3

  11. Thompson’s Decision Tree-Fig. 4

  12. Expected Value of Sample Information Expected value of best decision withsample information, assuming no cost to gather it Expected value of best decision withoutsample information EVSI=

  13. Expected Value of Sample Information EVSI = EV of best decision withsample information, assuming no cost to gather it – EV of best decision without sample information = EV with sample info. + cost – EV without sample info. DM could pay up to EVSI for a survey. If the cost of the survey is less than EVSI, it is indeed worthwhile. In the example: EVSI = $49,200 + $10,000 – $40,000 = $19,200

  14. Bayes Theorem Posterior probabilities Prior probabilities New data Estimating Probability Values by Bayesian Analysis • Management experience or intuition • History • Existing data • Need to be able to revise probabilities based upon new data

  15. Bayesian Analysis Example: • Market research specialists have told DM that, statistically, of all new products with a favorable market, market surveys were positive and predicted success correctly 70% of the time. • 30% of the time the surveys falsely predicted negative result • On the other hand, when there was actually an unfavorable market for a new product, 80% of the surveys correctly predicted the negative results. • The surveys incorrectly predicted positive results the remaining 20% of the time.

  16. Actual States of Nature Result of Survey Favorable Unfavorable Market (FM) Market (UM) (survey positive|FM) (survey positive|UM) Positive (predicts P P = 0.70 = 0.20 favorable market for product) (survey (survey negative|UM) Negative (predicts P P negative|FM) = 0.30 = 0.80 unfavorable market for product) Market Survey Reliability

  17. Calculating Posterior Probabilities P(BA) P(A) P(AB) = P(BA) P(A) + P(BA’) P(A’) where A and B are any two events, A’ is the complement of A P(FMsurvey positive) = [P(survey positiveFM)P(FM)] / [P(survey positiveFM)P(FM) + P(survey positiveUM)P(UM)] P(UMsurvey positive) = [P(survey positiveUM)P(UM)] / [P(survey positiveFM)P(FM) + P(survey positiveUM)P(UM)]

  18. Probability Revisions Given a Positive Survey Conditional Probability State P(Survey positive|State of Nature Prior Probability Joint Probability Posterior Probability of Nature 0.35 = 0.78 FM 0.70 * 0.50 0.35 0.45 0.10 = 0.22 0.20 0.10 * 0.50 UM 0.45 1.00 0.45

  19. Probability Revisions Given a Negative Survey Conditional Probability State P(Survey Prior Probability Joint Probability Posterior Probability of negative|State Nature of Nature) 0.15 = 0.27 0.15 0.30 * 0.50 FM 0.55 0.40 = 0.73 0.40 UM 0.80 * 0.50 0.55 0.55 1.00

  20. Utility Theory • Utility assessment assigns the worst outcome a utility of 0, and the bestoutcome, a utility of 1. • A standard gamble is used to determine utility values: When you are indifferent, the utility values are equal. • Choose the alternative with the maximum expected utility EU(ai) = u(ai) = u(vij) P(qj)

  21. Utility Theory $2,000,000 Accept Offer $0 Red (0.5) Reject Offer Blue (0.5) $5,000,000

  22. Utility Assessment • Utility assessment assigns the worst outcome a utility of 0, and the bestoutcome, a utility of 1. • A standard gamble is used to determine utility values. • When you are indifferent, the utility values are equal.

  23. Standard Gamble for Utility Assessment (p) Best outcome Utility = 1 Alternative 1 (1-p) Worst outcome Utility = 0 Alternative 2 Other outcome Utility = ??

  24. Figure 4.7 p= 0.80 $10,000 U($10,000) = 1.0 Invest in Real Estate (1-p)= 0.20 0 U(0)=0 $5,000 U($5,000)=p =0.80 Invest in Bank

  25. (0.5) (0.5) (0.5) x1 u(v*) = 0.5 Best outcome (v*) u(v*) = 1 v* u(v*) = 1 Lottery ticket Lottery ticket Lottery ticket (0.5) (0.5) (0.5) Worst outcome (v–) u(v–) = 0 Worst outcome (v–) u(v–) = 0 x1 u(x1) = 0.5 x3 u(x3) = 0.25 Certain outcome (x1) u(x1) = 0.5 x2 u(x2) = 0.75 Certain money Certain money Certain money Utility Assessment (1st approach) II I In the example: u(-180) = 0 and u(200) = 1 X1= 100  u(100) = 0.5 X2 = 175 u(175) = 0.75 X3 = 5 u(5) = 0.25 III

  26. (p) Best outcome (v*) u(v*) = 1 Lottery ticket (1–p) Worst outcome (v–) u(v–) = 0 Certain outcome (vij) u(vij) = p Certain money Utility Assessment (2nd approach) In the example: u(-180) = 0 and u(200) = 1 For vij=–20, p=%70  u(–20) = 0.7 For vij=0, p=%75 u(0) = 0.75 For vij=100, p=%90  u(100) = 0.9

  27. Sample Utility Curve

  28. Risk Avoider Risk Indifference Utility Risk Seeker Monetary Outcome Preferences for Risk

  29. Example Point up (0.45) $10,000 Alternative 1 Play the game Point down (0.55) -$10,000 Do not play the game Alternative 2 0

  30. Utility Curve for Example

  31. Using Expected Utilities in Decision Making Utility Tack lands point up (0.45) 0.30 Alternative 1 Play the game Tack lands point down (0.55) 0.05 Alternative 2 Don’t play 0.15

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