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Decision trees are powerful tools for structuring sequential decision-making processes when events over time impact future choices. They are more effective than payoff tables in situations requiring layered or contingent decisions. A decision tree includes various nodes: start, decision, states of nature, and terminal nodes, interconnected by arcs which represent possible decisions and outcomes. This approach assists in evaluating optimal decisions based on Bayesian probabilities, enabling informed strategies for scenarios like land development and variance approval.
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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) = 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
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
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)
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
($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
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.
