Module C3 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 P(Denial|Predict Approval) = =.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) = =.2.4.6

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

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

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

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

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

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.