EMGT 269 - Elements of Problem Solving and Decision Making 4. EVALUATING DECISION TREES Objective measured in Dollars Solve Decision Problem using Expected Monetary Value (EMV) Expected Value of discrete random variable Y: n n i 1 i 1 EY [Y ] yi Pr(Y yi ) yi pi Max Profit Trade Ticket EMV= $4.5 EMV= $4 Win (0.20) $25 $24 y $24.00 -$1.00 Pr(Y=y) 0.2 0.8 -$1 EMV= $4.5 Lose (0.80) $0 Win (0.45) $10 =EMV y*Pr(Y=y) $4.50 $0.00 $4.50 =EMV -$1 $10 y $10.00 $0.00 Keep Ticket $0 Lose (0.55) $0 y*Pr(Y=y) $4.80 -$0.80 $4.00 Pr(Y=y) 0.45 0.55 $0 Interpretation: Playing the lottery a lot of times will result in an average payoff equal to the EMV. Instructor: Dr. J. Rene van Dorp Session 4 - Page 52 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen EMGT 269 - Elements of Problem Solving and Decision Making Texaco Pennzoil Case: Max Result Accept $2 Billion 2 Texaco Accepts $5 Billion (0.17) 5 High (0.20) Counteroffer Texaco Refuses (0.50) $5 Billion Counteroffer 10.3 Medium (0.50) Final Court Decision Low (0.30) 0 High (0.20) Refuse 10.3 Medium (0.50) Final Court Decision Low (0.30) Texaco (0.33) Counter offers $3 Billion 5 5 0 Accept $3 Billion 3 Solve tree using EMV by folding back the Tree High (0.20) Final Court Decision Medium (0.50) Low (0.30) Step 1 10.3 5 0 y 10.300 5.000 0.000 Pr(Y=y) 0.2 0.5 0.3 y*Pr(Y=y) $2.06 $2.50 $0.00 $4.56 1. Instructor: Dr. J. Rene van Dorp Session 4 - Page 53 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen =EMV EMGT 269 - Elements of Problem Solving and Decision Making High (0.20) EMV= 4.56 Refuse Medium (0.50) Final Court Decision EMV= 4.56 10.3 Low (0.30) 5 0 Accept $3 Billion 3 2. Max Result Accept $2 Billion 2 Texaco Accepts $5 Billion (0.17) 3. EMV= 4.63 Counteroffer Texaco Refuses (0.50) $5 Billion Counteroffer 5 EMV= 4.56 Texaco Counter offers $3 Billion (0.33) y 5.000 4.560 4.560 Pr(Y=y) 0.17 0.5 0.33 y*Pr(Y=y) $0.85 $2.28 $1.50 $4.63 EMV= 4.56 Max Result Accept $2 Billion 2 EMV= 4.63 Counteroffer $5 Billion 4. Instructor: Dr. J. Rene van Dorp Session 4 - Page 54 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen =EMV EMGT 269 - Elements of Problem Solving and Decision Making Definition Decision Path: A path starting add the left most node up to the values at the end of a branch by selecting one alternative from a decision node or by following one outcome from uncertainty nodes. Definition Decision Strategy: The collection of decision paths connected to a branch of the left most node by selecting one alternative from each decision node along these paths. Optimal Decision Strategy: That Decision Strategy which results in the highest EMV if we maximize profit and the lowest EMV if we minimize cost. Instructor: Dr. J. Rene van Dorp Session 4 - Page 55 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen EMGT 269 - Elements of Problem Solving and Decision Making Max Result Accept $2 Billion 2 EMV= 4.63 Texaco Accepts $5 Billion (0.17) EMV= 4.63 High (0.20) EMV= 4.56 Counteroffer Texaco Refuses (0.50) $5 Billion Counteroffer Final Court Decision Low (0.30) High (0.20) EMV= 4.56 EMV= 4.56 Medium (0.50) Refuse Final Court Decision Medium (0.50) Low (0.30) Texaco (0.33) Counter offers $3 Billion Accept $3 Billion 5 10.3 5 0 10.3 5 0 3 OPTIMAL DECISION STRATEGY: Counteroffer $5 Billion. Next, if Texaco Counteroffers $3 Billion, refuse the counteroffer. Number of Decision Strategies in Decision Tree above: 3 Instructor: Dr. J. Rene van Dorp Session 4 - Page 56 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen EMGT 269 - Elements of Problem Solving and Decision Making 1. D =“Accept 2 Billion”. Accept $2 Billion 2 2. D =”Counter 5 Billion, Refuse Counter of $3 Billion”. Texaco Accepts $5 Billion (0.17) 5 High (0.20) Counteroffer Texaco Refuses (0.50) $5 Billion Counteroffer Final Court Decision 10.3 Medium (0.50) 5 Low (0.30) 0 High (0.20) Refuse Final Court Decision 10.3 Medium (0.50) 5 Low (0.30) Texaco (0.33) Counter offers $3 Billion 0 3. D =”Counter 5 Billion, Accept $3 Billion”. Texaco Accepts $5 Billion (0.17) High (0.20) Counteroffer Texaco Refuses (0.50) $5 Billion Counteroffer Final Court Decision Medium (0.50) Low (0.30) 5 10.3 5 0 (0.33) Texaco Counter offers $3 Billion Accept $3 Billion 3 Instructor: Dr. J. Rene van Dorp Session 4 - Page 57 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen EMGT 269 - Elements of Problem Solving and Decision Making RISK PROFILES = Graph that shows probabilities for each possible outcome given a decision strategy. Note: Risk Profile is a probability density function for the discrete random variable Y representing the outcomes for the given decision strategy. Accept $2 Billion 2 Outc om e x ($Billion) Pr(Outc ome| D )) 2 1 Pr(Outcome| D ) 1 1 0 .8 0 .6 0 .4 0 .2 0 -1 0 1 2 3 4 5 6 7 O u tc o m e 1. Instructor: Dr. J. Rene van Dorp Session 4 - Page 58 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen 8 9 10 11 EMGT 269 - Elements of Problem Solving and Decision Making C a lc u la t io n Texaco Accepts $5 Billion (0.17) 0.17 0.170 0.50*0.20 0.100 0.50*0.50 0.250 0.50*0.30 0.150 0.33*0.20 0.066 0.33*0.50 0.165 0.33*0.30 0.099 5 High (0.20) Counteroffer Texaco Refuses (0.50) $5 Billion Counteroffer 10.3 Final Medium (0.50) 5 Court Decision Low (0.30) 0 High (0.20) Refuse Final Court Decision Texaco Counter - (0.33) offers $3 Billion Calc ulati on 0.150+0.0 99 0.170+0.250+0.1 65 0.100+0.0 66 10.3 Medium (0.50) 5 Low (0.30) 0 Total Pr (Ou tc ome| D) 0.2 49 0.5 85 0.1 66 1.0 00 1.000 1 P r(Ou tco m e| D ) Outc o me x ($Billion) 0 5 10.3 Prob 0 .8 0 .6 0 .5 8 5 0 .4 0 .2 4 9 0 .2 0 .1 6 6 0 -1 0 1 2 3 4 5 6 7 8 9 10 11 O u tc o m e 2. C a lc u la t io n Texaco Accepts $5 Billion (0.17) Counteroffer $5 Billion Texaco Refuses (0.50) Counteroffer Texaco (0.33) Counter offers $3 Billion 0.17 0.170 0.50*0.20 0.100 0.50*0.50 0.250 0.50*0.30 0.150 0.33 0.330 5 High (0.20) 10.3 Final Medium (0.50) Court 5 Decision Low (0.30) 0 Accept $3 Billion 3 Prob Total 1.000 Calc ulation 0.15 0.33 0.170+0.250 0.1 Pr(Outc ome| D) 0.15 0.33 0.42 0.1 1.000 P r(Ou tco m e| D ) 1 Outc ome x ($Billion) 0 3 5 10.3 0 .8 0 .6 0 .3 3 0 .2 0 .1 5 0 .1 0 -1 3. 0 .4 2 0 .4 0 1 2 3 4 5 6 7 O u tc o m e Instructor: Dr. J. Rene van Dorp Session 4 - Page 59 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen 8 9 10 11 EMGT 269 - Elements of Problem Solving and Decision Making CUMMULATIVE RISK PROFILES = Graphs that shows cumulative probabilities associated with a risk profie Note: Cumulative Risk Profile is a cumulative distribution function for the discrete random variable Y representing the outcomes for the given decision strategy. Risk Profiles Cumulative Risk Profiles 1 .2 1 .2 1 P r(Y<=y | D ) Pr(Outcome| D ) 1 0 .8 0 .6 0 .4 1 1 0 .6 0 .4 0 .2 0 .2 0 0 -1 1 0 .8 0 1 2 3 4 5 6 7 8 9 10 0 -1 11 0 0 1 2 3 4 5 6 7 8 9 10 11 y O u tc o m e 1. 1 .2 1 .2 1 P r(Y<=y| D ) P r(Outcome| D ) 1 0 .8 0 .6 0 .5 8 5 0 .4 0 .2 4 9 0 .2 0 .4 0 .2 4 9 0 .2 0 .1 6 6 0 0 1 2 3 4 5 6 7 8 9 10 -1 11 1 0 .8 3 4 0 .6 0 -1 1 0 .8 3 4 0 .8 0 .2 4 9 0 0 1 2 3 4 5 6 7 8 9 10 11 y O u tc o m e 1 1 .2 0.8 1 P r(Y<=y | D ) Pr(Outcome| D) 2. 0.6 0.42 0.4 0.33 0.2 0.15 0.1 0 .8 0 .6 0.4 8 0 .4 0 .2 0 -1 0.1 5 0 0 0 1 2 3 4 5 Outcome 6 7 8 9 10 11 1 1 0 .9 0 .9 -1 0 1 0.4 8 0.1 5 2 3 4 5 6 7 y 3. Instructor: Dr. J. Rene van Dorp Session 4 - Page 60 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen 8 9 10 11 EMGT 269 - Elements of Problem Solving and Decision Making Deterministic Dominance Max Result Accept $2 Billion 2 Texaco Accepts $5 Billion (0.17) High (0.20) Counteroffer $5 Billion Texaco Refuses (0.50) Counteroffer Final Court Decision Medium (0.50) Low (0.30) High (0.20) Refuse Final Court Decision Medium (0.50) Low (0.30) Texaco (0.33) Counter offers $3 Billion Accept $3 Billion 5 10.3 5 2.5 10.3 5 2.5 3 Definition: A = largest 0% point in Cumulative Risk Profile B = smallest 100% point in Cumulative Risk Profile Range of a Cumulative Risk Profile = [A,B] Deterministic Dominance: Step 1: Draw cumulative risk profiles in one graph Step 2: Determine range for each risk profile Step 3: Ranges are disjoint Deterministic Dominance Instructor: Dr. J. Rene van Dorp Session 4 - Page 61 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen EMGT 269 - Elements of Problem Solving and Decision Making 1 .2 Accept 2 Billion 1 0 .8 Counteroffer 5 Billion, Refuse Texaco 3 Billion Counteroffer 0 .6 0 .4 0 .2 0 -1 0 1 2 3 4 5 6 7 8 9 10 11 y Maximize Return "Counteroffer" deterministically dominates "Accept" Instructor: Dr. J. Rene van Dorp Session 4 - Page 62 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen EMGT 269 - Elements of Problem Solving and Decision Making Stochastic Dominance Example 1: Max Result Texaco Accepts $5 Billion (0.17) 5 High (0.20) Counteroffer Texaco Refuses (0.50) $5 Billion Counteroffer 10.5 Medium (0.50) Final Court Decision 5.2 Low (0.30) 0 High (0.20) 10.5 Firm A Refuse Texaco Counter - (0.33) offers $3 Billion Medium (0.50) Final Court Decision 5.2 Low (0.30) 0 Accept $3 Billion 3 Texaco Accepts $5 Billion (0.17) 5 Firm B High (0.20) Counteroffer Texaco Refuses (0.50) $5 Billion Counteroffer Final Court Decision Medium (0.50) Low (0.30) High (0.20) Refuse Texaco Counter - (0.33) offers $3 Billion Final Court Decision Medium (0.50) Low (0.30) 10.3 5 0 10.3 5 0 Accept $3 Billion 3 Instructor: Dr. J. Rene van Dorp Session 4 - Page 63 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen EMGT 269 - Elements of Problem Solving and Decision Making Stochastic Dominance: Step 1: Draw cumulative risk profiles in one graph Step 2: CRP do not cross Stochastics Dominance 1 0 .8 0 .6 Firm B Firm A 0 .4 0 .2 0 -1 0 1 2 3 4 5 6 7 8 9 10 11 y Max Return Firm A stochastically dominates Firm B Instructor: Dr. J. Rene van Dorp Session 4 - Page 64 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen EMGT 269 - Elements of Problem Solving and Decision Making Example 2: Max Result Texaco Accepts $5 Billion (0.17) 5 High (0.30) Counteroffer $5 Billion Texaco Refuses (0.50) Counteroffer 10.3 Medium (0.60) Final Court Decision 5.0 Low (0.10) 0 High (0.30) 10.5 Firm C Refuse Texaco Counter - (0.33) offers $3 Billion Medium (0.60) Final Court Decision 5.0 Low (0.10) 0 Accept $3 Billion 3 Texaco Accepts $5 Billion (0.17) 5 Firm D High (0.20) Counteroffer $5 Billion Texaco Refuses (0.50) Counteroffer Final Court Decision Medium (0.50) Low (0.30) High (0.20) Refuse Texaco Counter - (0.33) offers $3 Billion Final Court Decision Medium (0.50) Low (0.30) 10.3 5 0 10.3 5 0 Accept $3 Billion 3 Instructor: Dr. J. Rene van Dorp Session 4 - Page 65 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen EMGT 269 - Elements of Problem Solving and Decision Making 1 0 .8 Firm D 0 .6 Firm C 0 .4 0 .2 0 -1 0 1 2 3 4 5 6 7 8 9 10 11 y Max Return Firm C stochastically dominates Firm D Instructor: Dr. J. Rene van Dorp Session 4 - Page 66 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen EMGT 269 - Elements of Problem Solving and Decision Making Making Decisions with Multiple Objectives Influence Diagram Amount of Fun Fun Job Decision Overall Satisfaction Salary Amount of Work Two Objectives: Making Money (Measured in $) Having Fun (Constructed attribute scale, see page 128) Best(5), Good(4), Middle(3), Bad(2), Worst (1) Instructor: Dr. J. Rene van Dorp Session 4 - Page 67 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen EMGT 269 - Elements of Problem Solving and Decision Making Decision Tree Consequences Salary Fun level Forest Job 5 (0.10) $2600.00 5 4 (0.25) $2600.00 4 $2600.00 3 $2600.00 2 $2600.00 1 3 (0.40) 2 (0.20) 1 (0.05) # hours per week In-Town Job Fun Level 40 hours (0.35) 34 hours (0.50) 30 hours (0.15) $2730.00 3 $2320.50 3 $2047.50 3 Instructor: Dr. J. Rene van Dorp Session 4 - Page 68 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen EMGT 269 - Elements of Problem Solving and Decision Making Analysis based on Salary Objective Salary $2,047.50 $2,320.50 $2,600.00 $2,730.00 Forest Job Prob Prob*Salary 1.00 E[Salary]= In-Town Job Prob Prob*Salary 0.15 $307.13 0.50 $1,160.25 $2,600.00 $2,600.00 0.35 E[Salary]= 1.00 0.80 $955.50 $2,422.88 1.00 0.60 0.40 0.50 0.35 0.20 0.15 0.00 $2,000.00 $2,200.00 $2,400.00 $2,600.00 $2,800.00 Salary Forest Job In-Town Job Optimal Decision Strategy: Forest-Job Instructor: Dr. J. Rene van Dorp Session 4 - Page 69 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen EMGT 269 - Elements of Problem Solving and Decision Making Analysis based on Fun Level Objective Forest Job In-Town Job Fun Level Prob Fun Level*Prob Prob Fun Level*Prob 0.00% 0.05 0.0% 25.00% 0.20 5.0% 60.00% 0.40 24.0% 1.00 60.00% 90.00% 0.25 22.5% 100.00% 0.10 10.0% E[Fun Level]= 61.5% E[Fun Level]= 60.00% 1.00 1.00 0.80 0.60 0.80 0.40 0.60 1.00 0.20 0.40 0.00 0.20 0.00% 0.40 20.00% 40.00% 0.20 60.00% 80.00% 0.05 0.00 0 1 2 3 0.25 100.00% 120.00% 0.10 4 5 6 Summer Fun Forest Job In-Town Job Optimal Decision Strategy: Forest-Job Instructor: Dr. J. Rene van Dorp Session 4 - Page 70 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen EMGT 269 - Elements of Problem Solving and Decision Making Analysis based on Two Objectives: 1. Before overall score can be calculated the scales of each alternative need to be the same i.e. Transform to 0-1 scale or 0%-100% scale Set Best=100%, Worst=0% Determine intermediate values Having Fun objective: Best(100%), Good(90%), Middle(60%), Bad(25%), Worst (0%) Making Money Objective: $2730.00=100%, $2047.50=0% X $2047.50 100% Intermediate dollar amount X: $2730 $2047.50 2. Assess Trade-off weights k s = Weight for Salary k f = Weight for Fun Level ks k f 1 Using Expert Judgment: Going from worst to best in salary objective is 1.5 times more important than going from worst to best in having fun objective k s 1.5 k f . With k s k f 1 follows that: k f 0.40; k s 0.60. Instructor: Dr. J. Rene van Dorp Session 4 - Page 71 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen EMGT 269 - Elements of Problem Solving and Decision Making Consequences Salary (0.4) Fun level Forest Job 5 (0.10) 81% 100% 4 (0.25) 81% 90% 81% 60% 81% 25% 81% 0% 100% 60% 40% 60% 0% 60% 3 (0.40) 2 (0.20) 1 (0.05) # hours per week In-Town Job Fun Level (0.6) 40 hours (0.35) 34 hours (0.50) 30 hours (0.15) Total Score Fun level Forest Job 5 (0.10) 88.6% 4 (0.25) 84.6% 3 (0.40) 2 (0.20) 1 (0.05) # hours per week In-Town Job 40 hours (0.35) 34 hours (0.50) 30 hours (0.15) 72.6.% 58.6% 48.6% 84.0% 48.0% 24.0% Instructor: Dr. J. Rene van Dorp Session 4 - Page 72 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen EMGT 269 - Elements of Problem Solving and Decision Making Combined Score 48.6% 58.6% 72.6% 84.6% 88.6% Combined Score 24.0% 48.0% 84.0% Forest Job Prob 0.05 0.20 0.40 0.25 0.10 E[Comb Score]= Combined Score*Prob 2.4% 11.7% 29.0% 21.1% 8.9% 73.2% In-Town Job Prob Combined Score*Prob 0.15 3.6% 0.50 24.0% 0.35 29.4% E[Comb Score]= 57.0% Optimal Decision Strategy: Forest-Job Instructor: Dr. J. Rene van Dorp Session 4 - Page 73 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen EMGT 269 - Elements of Problem Solving and Decision Making 1 .0 0 0 .8 0 0 .6 0 0. 50 0 .4 0 0 .2 0 0 .0 0 0 . 0% 0. 40 0. 20 0. 15 0. 05 2 0 .0 % 4 0 .0 % 6 0 .0 % 0.35 0. 25 0.10 8 0 .0 % 1 0 0. 0% C om bin e d S c ore F ore st Jo b In- T ow n Jo b 1.00 0.80 0.60 0.40 0.20 0.00 0. 0% 20. 0% 40.0% 60. 0% 80.0% 100. 0% C o m bin ed Sco r e F or est J ob I n- Town J ob "Forest-Job" stochastically dominates "In-Town Job" alternative in terms of the overall score Instructor: Dr. J. Rene van Dorp Session 4 - Page 74 Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen