29/10/15 Decision Trees and Influence Diagrams Carlos Bana e Costa and Mónica Oliveira REFERENCES: CLEMEN, R. (1996), MAKING HARD DECISIONS: AN INTRODUCTION TO DECISION ANALYSIS (2 ND EDITION). DUXBURY. CHAPTERS 3, 4, AND 12 GOODWIN, P. AND WRIGHT, G. (1998) DECISION ANALYSIS FOR MANAGEMENT JUDGMENT (2 ND OR 3 RD EDITIONS). WILEY. CHAPTERS 6 AND 8 Lecture topics q Decision trees and influence diagrams q Value of informa4on and control q A case study: Drilling for oil q Payoff matrix q Complementary concepts 2 1
A Taxonomy of Decision Models (In Decision Analysis in the 1990s - L.D. Phillips) Problem dominated by Uncertainty MulRple ObjecRves EXTEND conversaron Event tree Fault tree Influence diagram REVISE opinion Bayesian nets CHOOSE opron Payoff matrix Decision tree EVALUATE oprons MulR-criteria decision analysis ALLOCATE resources MulR-criteria commons dilemma SEPARATE into components Credence decomposiron Risk analysis NEGOTIATE MulR-criteria bargaining analysis 3 Dealing with uncertainty: Key ques]ons? What are the key uncertain)es? What are the possible outcomes of these uncertain4es? What are the chances of occurrence of each possible outcome? What are the consequences of each outcome? Hammond, Keeney & Raiffa, Smart Choices (Chapter 7) 4 2
Decision problem: To drill or not to drill? q A small and struggling firm has the mineral rights to a tract of land. A consultant geologist esrmates there is a small chance of striking oil. q It is expensive to drill for oil, and the cost of drilling if there is no oil will nearly drive the firm to bankruptcy. q On the other hand, if they strike oil, the firm will make a big killing. q There is another alternarve: a rival firm has offered to buy the land. 5 What are the key uncertain)es? Uncertainty: To strike oil (by drilling) What are the possible outcomes of this uncertainty? Oil No oil (Dry) What are the chances of occurrence of each possible outcome? There is a small chance of striking oil (There is a high chance that the soil is dry) What are the consequences of each outcome? If they strike oil, the firm will make a big killing If there is no oil it will nearly drive the firm to bankruptcy 6 3
Risk profile Uncertainty: To strike oil in the site (by drilling) Outcome Chance Consequences Oil Small Big profit No oil (dry) High Bankruptcy 7 Decision table What are the chances of occurrence of each possible outcome? 8 4
Decision tables (and Decision Trees): Most classical approach to model decision problems involving sequencial decisions under uncertainty: q The idea underlying a tabular representaron of a problem is that the consequences of any decision can be determined by a number of external factors, out of the control of the DM. q If the DM knew the state of nature that would actually hold, the true state, he could predict the consequence of his choice with certain. (Note: The true state is unknown, but the DM knows which states are possible.) 9 Decision tree Decision nodes Represent decisions Chance nodes Represent chance (uncertain) events Consequences are specified at the ends of the branches 10 5
A DECISION TREE represents all of the possible paths that the DM might follow through Rme, including all possible decision alternarves and outcomes of chance events: The oprons represented by branches from a decision node must be such that the DM can choose only one op9on. Each chance node must have branches that correspond to a set of mutually exclusive and collec9vely exhaus9ve outcomes When the uncertainty is resolved, one and only one of the outcomes occurs 11 If a chance node is to the right of a decision node, the decision must be made in an9cipa9on of the chance event. Conversely, placing a chance event before a decision means that the decision is made condi9onal on the specific chance outcome having occurred. Imperfect informaron: DM waits for informaron before making a decision most attractive least attractive Asymmetric tree with sequenral decisions The crescent shape indicates that the uncertain event may result in any value between two limits. 12 6
DPL sobware 13 NODES: Influence diagrams Decision nodes (rectangles) - represent decisions (and alternarves) Chance nodes (ovals) represent uncertain events (and outcomes) (chance events) Consequence (and calcula4on) nodes represent consequences (and calcularons) Nodes are put together in a graph, connected by ARCS. Arcs represent rela9onships (relevance or sequence) between nodes: Predecessor node à successor node (Done with DPL sohware) 14 7
Using PrecisionTree 1.0 for Excel to solve the problem (Trial available at Palisade website Student version available with Clemen & Reilly, 2001) (PrecisionTree disrnguishes calcularon nodes from pay-off nodes) 15 Oil Dry EMV (Expected Monetary Value) Drill 700-100 100 = 0.25(700)+0.75(-100) Sell 90 90 90 Prior probability 0.25 0.75 Highest (EMV): Choose Drill 16 8
Indifference point Drill Sell 0.25: a priori probability 17 Supposing that the geologist is clairvoyant EVPI = 242.5-100 = 142.5 18 9
Case study: Drilling for oil (con]nua]on: Obtaining imperfect informa]on) However, another opron prior to making a decision is to follow the geologist s suggesron of conducrng a detailed seismic survey of the land, to obtain a bemer esrmate of the probability of finding oil. The cost of the survey is 30,000. Since EVPI=142.5 (the maximum amount that the DM should be willing to pay the clairvoyant for perfect informa4on) far exceeds 30, it may be worthwhile to proceed with the seismic survey and wait for its results before making a decision. NOTE: We are thinking about the value of informaron in a strictly a priori sense. The geologist is not a clairvoyant, unfortunately! I.e., the results of the survey can be imperfect. 19 q The value of informaron tells you the value of finding out the state of a chance event before you have to make a decision. q Chance events with high values for informa4on present the best opportunires to improve your expected value by thinking of crearve new alternarves. q Chance events with low values for informa4on are probably not worth further efforts at research, tesrng, or delay. Important things to remember: InformaRon has no value if it doesn t change your acrons, Its value is limited to the improvement it provides over what you would get without it. 20 10
21 Case study: Drilling for oil (con]nua]on: Obtaining imperfect informa]on) A seismic survey obtains seismic soundings that indicate whether the geological structure is favourable to the presence of oil. Based upon past experience, our DM got the informaron that: If the land is dry, the surveys are unfavourable 80% of the Rme. However, if there is oil, the surveys are favourable only 60% of the Rme. 22 11
Case study: Drilling for oil (con]nua]on: Obtaining imperfect informa]on) What we know 0.25*0.6 0.25*0.4 0.75*0.2 0.75*0.8 23 What we know: 0.25*0.6 What we want to know: P(O\F)=? Oil Underground P(D\F)=? P(O\U)=? Oil Underground P(D\U)=? 24 12
What we want to know: P(O\F)=1/2 =.15/.30 Oil Underground P(D\F)=.15/.30 P(O\U)=.1/.7 Oil Underground P(D\U)=.6/.7 Posterior probabili4es (Bayes rule) 25 backward induc4on procedure (rollback) EMV for wairng for the survey results = 153 EMV for deciding without survey = 100 Expected value of imperfect informaron (EVII) =153-100=53 (>30) (The DM would never want to pay more than 53,000 for the survey) Make the survey. If favourable, drill. If unfavourable, sell. 26 13
Value of control Some variables, such as weather, have high informaron value but are hard to think of good sources of informaron for. For these variables, move on to the value of control to see if you can think of ways to mirgate the impact of these uncertainres, even if you can t predict them. The value of control for an event tells you the value of being able to choose the outcome of the uncertainty rather than taking your chances. The value comes from being able to guarantee the most favourable outcome and prevent less favourable outcomes. 27 Most favourable outcome Value of control = 700-100 = 600 28 14
Chance Events with High Value of Control present the greatest opportunity for improving your outcomes by thinking of crea4ve new ways to either gain control over the uncertainty or to mi4gate its impact on your outcomes. Common sources of control: q Increased staffing, Rme, money, or other resources q PR or adverrsing q Insurance As with informaron, sources of control are rarely free. Those whose cost is less than their benefit should be modelled explicitly in your influence diagram and decision tree. Common types of imperfect control: q Control just improves probabilires. q Can t pick best state. 29 Important things to remember about the value of control: q it can come either from controlling the underlying uncertainty or by insularng yourself from the effects of that uncertainty; q the value of control is normally greater than, or equal to, the value of informaron. 30 15
CHOOSE op]on decision tree Graham s decision problem Speed Flexibility Accuracy Cost 90 100 90 90 25 0 0 20 100 100 100 100 30 0 0 30 60 100 90 80 0 0 0 0 40 0 0 40 Weights: 0.70 0.15 0.05 0.10 31 Assess the Cash Flows and probabili]es using the Precision Tree sobware 32 16
Precision Tree (PALISADE) Examples 33 Decision Trees vs. Influence Diagrams Influence Diagrams Strengths Compact Good for communicaron, in parrcular in the structuring phase Good overview of large problems Good for understanding the relevance between uncertainty nodes Decision Trees Displays details, being good for in-depth understanding Flexible representaron Best for asymmetric decision problems Adequate for performing sensirvity analysis Weaknesses Details suppressed Becomes very messy for large problems Complementary use of decision trees and influence diagrams! 34 17
Forecast Hits Miami Misses Miami Developing Influence Diagrams: Some examples Outcomes Hits Miami Misses Miami Alterna4ves Evacuate Stay Source: Clemen, R. (1996), Making Hard Decisions: An IntroducRon to Decision Analysis (2nd EdiRon). Duxbury. Choice Outcome Conseq. risk Conseq. cost Evacuate Hits Miami Low risk High cost Misses Miami Low risk High cost Stay Hits Miami High risk High cost Misses Miami Low risk Low cost 35 But if there is missing informa]on: The case for sequen]al decisions... Source: Clemen, R. (1996), Making Hard Decisions: An IntroducRon to Decision Analysis (2nd EdiRon). Duxbury. 36 18
More on sequen]al decisions... Source: Clemen, R. (1996), Making Hard Decisions: An IntroducRon to Decision Analysis (2nd EdiRon). Duxbury. 37 Developing financial models while accoun]ng for uncertainty 1 st version 3 rd version 2 nd version Source: Clemen, R. (1996), Making Hard Decisions: An IntroducRon to Decision Analysis (2nd EdiRon). Duxbury. 38 19
PAYOFF TABLE 39 Depar]ng from a Payoff Table with informa]on for making a choice under uncertainty Some situarons in which there is uncertainty, but no probabilisrc informaron Several strategies Several scenarios Example Table with Net Present Value (NPV) from different investment strategies and scenarios: Strategy Scenario C1 Scenario C2 Scenario C3 S1 500 200 100 S2 300 175 150 S3 200 180 160 40 20
Algorithms to make Choices under Uncertainty A ) Laplace ( principle of insufficient reason ) B ) Wald (Pessimist - Maximin) C ) MaxiMax (OpRmist) D ) Hurwicks (Intermediate) E) Savage ( MiniMax Regret ) 41 A ) LAPLACE principle of insufficient reason Equiprobable E[NPVs,S1] = (500+200+100)/3 = 800/3 E[NPVs,S2] = (300+175+150)/3 = 625/3 E[NPVs,S3] = (200+180+160)/3 = 540/3 B ) WALD Pessimist - Maximin min (S1) = 100 min (S2) = 150 min (S3) = 160 Choose S1 Strategy Scenario C1 Scenario C2 Scenario C3 S1 500 200 100 S2 300 175 150 S3 200 180 160 Maximin = 160 Choose S3 42 21
C ) MaxiMax Op4mist Strategy Scenario C1 Scenario C2 Scenario C3 S1 500 200 100 S2 300 175 150 S3 200 180 160 Max (S1) = 500 Max (S2) = 300 Max (S3) = 200 MaxiMax = 500 Choose S1 D ) Hurwicks Op4mism coefficient - α (between 0 and 1), weighted average of the extremes à Ex: α = 0.7 S1 500 0.7 + 100 0.3 = 380 S2 300 0.7 + 150 0.3 = 255 S3 200 0.7 + 160 0.3 = 188 Choose S1 (380) 43 Strategy Scenario C1 Scenario C2 Scenario C3 S1 500 200 100 S2 300 175 150 S3 200 180 160 E) SAVAGE ( MiniMax Regret - Minimize maximum regret) Regret Matrix Strategy Scenario C1 Scenario C2 Scenario C3 S1 0 0 60 Max = 60 S2 200 25 10 Max = 200 S3 300 20 0 Max = 300 minimax 60 à Choose S1 44 22
Complementary concepts EXPECTED MONETARY VALUE RISK PROFILE 45 The Risk Profile concept A risk profile is a graph that shows the chances associated with possible consequences. Probability Risk Profile For Oil Diagram of oil_infl.xls 0,6 0,5 0,4 0,3 0,2 0,1 0-100000 -50000 0 50000 100000 150000 200000 250000 300000 Value Each risk profile is associated with a strategy, a parrcular immediate alternarve, as well as specific alternarves in future decisions. 46 23
The Cumula]ve Risk Profile concept q In this format, the verrcal axis is the chance that the payoff is less than or equal to the corresponding value on the horizontal axis. q It results from adding up, or accumularng the chances of the individual payoffs à Along the horizontal axis we can read the chance that the payoff will be less than or equal to that specific value. 1,2 Cumulative Probability For Oil Diagram of oil_infl.xls Cumulative Probability 1 0,8 0,6 0,4 0,2 F(y) = P(Y y) = P(Y = i) i:i y 0-100000 -50000 0 50000 100000 150000 200000 250000 300000 Value 47 The Expected Value concept The random variable Y has many possible outcomes! Expected value: BEST GUESS for Y, what number would you give? Ε Y n = y i p i [ ] = y i P(Y = y i ) i=1 i=1 n Interpreta4on: If you were able to observe many outcomes of Y, the calculated average of all the outcomes would be close to E[Y]. 48 24