1. Decision Models Decision Trees

2. Situations In Which Decision Trees Can Be Useful Payoff Tables are fine when a single decision is to be made Sometimes a sequence of decisions must be made Decisions “along the way” will be influenced by events that have occurred to that point Decision Trees can help structure the model so that a series of optimal “what if” decisions can be made.

3. Structure of A Decision Tree A decision tree consists of nodes and arcs Nodes consist of –Start Node –Decision Nodes –States of Nature Nodes –Terminal Nodes Arcs consist of –Decision Arcs give a possible decision and the resulting cost (or profit) of making that decision –States of Nature Arcs give a possible state of nature and the (Bayesian) probability that the state of nature will occur

4. Example -- BGD Developoment Interested in Purchasing Land -- (300,000) To Build/Sell a Shopping Center -- $450,000 A variance must be obtained before building center -- ($30,000) –Variance Approved -- Center Built –Variance Denied -- Center Not Built Can purchase 3-month option to buy before applying for variance -- ($20,000) Can sell the undeveloped land -- $260,000 Can hire variance consultant -- ($5,000)

5. BGD Development Probabilities.4Probability that a variance is approved =.4.6 –Prob variance not approved =.6 Consultant’s Assistance--Consultant’s Assistance-- –P(Consultant Predicts Approval| Approval) =.7 –P(Consultant Predicts Denial| Approval) =.3 –P(Consultant Predicts Denial| Denial) =.8 –P(Consultant Predicts Approval| Denial) =.2

6. Bayesian Probabilities Based on Consultant’s Prediction P(Approval|Predict Approval) =P(Approval|Predict Approval) = P(Pred. Appr.|Approval)P(Approval)/P(Pred. Appr.) =.7 (.7)(.4)/[(.7)(.4)+(.2)(.6)] =.7 P(Denial|Predict Approval) =.3P(Denial|Predict Approval) = 1 -.7 =.3 Note that P(Predict Approval) = (.7)(.4)+(.2)(.6) =.4Note that P(Predict Approval) = (.7)(.4)+(.2)(.6) =.4 P(Denial|Predict Denial) =P(Denial|Predict Denial) = P(Pred. Deny|Deny)P(Deny)/P(Pred. Deny) =.8 (.8)(.6)/[(.8)(.6)+(.3)(.4)] =.8 P(Approval|Predict Denial)=.2P(Approval|Predict Denial) = 1 -.8 =.2 Note that P(Predict Denial) (.8)(.6)+ (.3)(.4) =.6Note that P(Predict Denial) (.8)(.6)+ (.3)(.4) =.6

7. The Decision Tree Start No Consultant $0 Do nothing $0 Buy Land & Variance ($330,000) Approved.4 Build/Sell Center $450,000 $120,000 ($70,000) Buy Option & Variance ($50,000) $100,000 ($50,000) $0 Denied.6 Do nothing Consultant ($5,000) See Next Screen $150,000.4 ApprovedBuy Land/Build/Sell Denied $260,000 Sell Land.6

8. Decision Tree (Cont’d) Start Consultant ($5,000) $115,000 ($5,000) ($75,000) $95,000 ($55,000) $95,000 $115,000 ($5,000) ($55,000) ($75,000) Pred. Approve.4 Buy Land & Variance ($330,000) Buy Option & Variance ($50,000) Buy Land & Variance ($330,000) Buy Option & Variance ($50,000) Pred. Deny.6 $0 Do nothing $0 Do nothing.8 $0 Denied Do nothing.7 Approved Build/Sell Center $450,000 Denied.3$260,000 Sell Land $150,000 Approved.7 Buy Land/Build/Sell $0 Denied.3 Do nothing.2 Approved Build/Sell Center $450,000.8$260,000 Denied Sell Land.2 Approved $150,000 Buy Land/Build/Sell

9. Buy Land & Variance Decision Tree Analysis Start No Consultant $0 Do nothing $0 ($330,000) Approved.4 Build/Sell Center $450,000 $120,000 Denied.6 ($70,000) $260,000 Sell Land Buy Option & Variance ($50,000) Approved.4$150,000 $100,000 Denied.6 ($50,000) $0 Do nothing Consultant ($5,000) See Next Screen Buy Land/Build/Sell $0 (.4)(120,000)+.6(-70,000)$6,000 (.4)(100,00)+.6(-50,000)$10,000 Option/Variance

10. Decision Tree Analysis (Cont’d) Start Consultant ($5,000) Pred. Approve Pred. Deny $115,000 ($5,000) ($75,000) $95,000 ($55,000) $95,000 $115,000 ($5,000) ($55,000) ($75,000).4.6 Buy Land & Variance $0 ($330,000) Approved.7 Build/Sell Center $450,000 Denied.3$260,000 Sell Land Buy Option & Variance ($50,000) Approved.7$150,000 Denied.3 $0 Do nothing Buy Land & Variance $0 ($330,000) Approved.2 Build/Sell Center $450,000 Denied.8$260,000 Sell Land Buy Option & Variance ($50,000) Approved.2$150,000 Denied.8 $0 Do nothing Buy Land/Build/Sell (.7)(115,00)+.3(-75,000)$58,000 ($5,000) (.7)(95,000)+.3(-55,000)$50,000 (.2)(115,000)+.8(-75,000) ($37,000) (.2)(95,000)+.8(-55,000)($25,000) ($5,000) $58,000 Land/Variance ($5,000) Do Nothing.4($58,000)+.6(-$5,000)$20,200

11. Summary Expected Value (No Consultant) = $10,000 Expected Value (Consultant) = $20,200 Hire Consultant! Hire Consultant! and and If consultant predicts approval Buy the land and apply for the variance If consultant predicts denial Do Nothing

12. Review Decision Trees can structure sequences of decisions Nodes are points in time where a decision is to be made or a state of nature will occur Arcs give payoffs or (Bayesian) probabilities Expected Values are calculated for each decision and the best is chosen.