Module C 3 Decision Trees Situation In

Module C 3 Decision Trees

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.

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

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

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

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)

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

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. 4 • 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. 6 • P(Approval|Predict Denial) = 1 -. 8 =. 2

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

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

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

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

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

Module C 3 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.