Module C3

1. Module C3 Decision Trees

2. Situation 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 • States of Nature Arcs

4. Nodes in a Decision Tree • Start Node -- A node designating the beginning of the decision process • Decision Nodes -- Points in time where one of a set of possible decisions must be made • States of Nature Nodes -- Points in time where one of several states of nature will occur • Terminal Node -- Gives the cumulative payoff for the sequence of decisions made along the path from the start node

5. Arcs in a Decision Tree • From decision nodes -- gives a possible decision and the resulting cost (or profit) of making that decision • From states of nature nodes -- gives a possible state of nature and the (Bayesian) probability that the state of nature will occur

6. 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)

7. BGD Development Probabilities • Probability that a variance is approved = .4 • Prob variance not approved = .6 • 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

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

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

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

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

12. ($5,000) Decision Tree Analysis (Cont’d) Do nothing ($5,000) $58,000 Land/Variance $0 (.7)(115,00)+.3(-75,000) $58,000 Build/Sell Center Approved $115,000 Start Buy Land & Variance .7 $450,000 ($330,000) Denied Sell Land ($75,000) .3 $260,000 $50,000 (.7)(95,000)+.3(-55,000) ($5,000) Consultant $95,000 Approved Buy Land/Build/Sell Pred. Approve Buy Option & Variance .7 $150,000 .4 ($50,000) Denied Do nothing ($55,000) .4($58,000)+.6(-$5,000) $20,200 .3 $0 Do nothing ($5,000) ($5,000) $0 ($37,000) (.2)(115,000)+.8(-75,000) Build/Sell Center Approved $115,000 .6 Pred. Deny ($5,000) Do Nothing Buy Land & Variance .2 $450,000 ($330,000) Denied Sell Land ($75,000) .8 $260,000 (.2)(95,000)+.8(-55,000) ($25,000) Buy Land/Build/Sell Approved $95,000 Buy Option & Variance .2 $150,000 ($50,000) Denied Do nothing ($55,000) .8 $0

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

14. Module C3 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.