A Taxonomy of Decision Models

Transcription

1 Decision Trees and Influence Diagrams Prof. Carlos Bana e Costa Lecture topics: Decision trees and influence diagrams Value of information and control A case study: Drilling for oil 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 A Taxonomy of Decision Models (In Decision Analysis in the 1990s - L.D. Phillips) Problem dominated by Uncertainty Multiple Objectives EXTEND conversation Event tree Fault tree Influence diagram REVISE opinion Bayesian nets SEPARATE into components Credence decomposition Risk analysis CHOOSE option Payoff matrix Decision tree EVALUATE options Multi-criteria decision analysis ALLOCATE resources Multi-criteria commons dilemma NEGOTIATE Multi-criteria bargaining analysis 1

2 Dealing with uncertainty: Key questions? What are the key uncertainties? What are the possible outcomes of these uncertainties? What are the chances of occurrence of each possible outcome? What are the consequences of each outcome? Hammond, Keeney & Raiffa, Smart Choices (Chapter 7) Decision problem: To drill or not to drill? A small and struggling firm has the mineral rights to a tract of land. A consultant geologist estimates there is a small chance of striking oil. It is expensive to drill for oil, and the cost of drilling if there is no oil will nearly drive the firm to bankruptcy. On the other hand, if they strike oil, the firm will make a big killing. There is another alternative: a rival firm has offered to buy the land. 2

3 What are the key uncertainties? 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 Risk profile Uncertainty: To strike oil in the site (by drilling) Outcome Chance Consequences Oil Small Big profit No oil (dry) High Bankruptcy 3

4 Decision table What are the chances of occurrence of each possible outcome? DECISION TABLES (AND DECISION TREES): MOST CLASSIC APPROACH TO MODEL DECISION PROBLEMS INVOLVING SEQUENCIAL DECISIONS UNDER UNCERTAINTY The idea underlying a tabular representation 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. 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.) 4

5 Decision Tree Decision nodes Represent decisions Chance nodes Represent chance (uncertain) events Consequences are specified at the ends of the branches A DECISION TREE represents all of the possible paths that the DM might follow through time, including all possible decision alternatives and outcomes of chance events: The options represented by branches from a decision node must be such that the DM can choose only one option. Each chance node must have branches that correspond to a set of mutually exclusive and collectively exhaustive outcomes When the uncertainty is resolved, one and only one of the outcomes occurs 5

6 If a chance node is to the right of a decision node, the decision must be made in anticipation of the chance event. Conversely, placing a chance event before a decision means that the decision is made conditional on the specific chance outcome having occurred. Imperfect information: DM waits for inf. before making a decision most attractive least attractive Asymmetric tree with sequential decisions The crescent shape indicates that the uncertain event may result in any value between two limits. INFLUENCE DIAGRAMS NODES: Decision nodes (rectangles) - represent decisions (and alternatives) Chance nodes (ovals) represent uncertain events (and outcomes) (chance events) Consequence (and calculation) nodes represent consequences (and calculations) Nodes are put together in a graph, connected by ARCS. Arcs represent relationships (relevance or sequence) between nodes: Predecessor node à successor node (Done with DPL software) 6

7 DPL Software Using PrecisionTree 1.0 for Excel to solve the problem (Trial available at Palisade website Student version available with Clemen & Reilly, 2001) (PrecisionTree distinguishes calculation nodes from pay-off nodes) 7

8 Oil Dry EMV Drill = 0.25(700)+0.75(-100) Sell Prior probability Highest EMV: Choose Drill Indifference point Drill Sell 0.25: a priori probability 8

9 Supposing that the geologist is clairvoyant EVPI = = Case study: Drilling for oil (continuation: Obtaining imperfect information) However, another option prior to making a decision is to follow the geologist s suggestion of conducting a detailed seismic survey of the land, to obtain a better estimate 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 information) 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 information in a strictly a priori sense. The geologist is not a clairvoyant, unfortunately! I.e., the results of the survey can be imperfect. 9

10 The value of information tells you the value of finding out the state of a chance event before you have to make a decision. Chance events with high values for information present the best opportunities to improve your expected value by thinking of creative new alternatives. Chance events with low values for information are probably not worth further efforts at research, testing, or delay. Important things to remember: Information has no value if it doesn t change your actions, its value is limited to the improvement it provides over what you would get without it. 10

11 Case study: Drilling for oil (continuation: Obtaining imperfect information) 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 information that: If the land is dry, the surveys are unfavourable 80% of the time. However, if there is oil, the surveys are favourable only 60% of the time. O que se sabe: 0.25* * * *0.8 What we know: 0.25*0.6 What we want to know: P(O\F)=? P(D\F)=? P(O\U)=? P(D\U)=? 11

12 What we want to know: P(O\F)=1/2 =.15/.30 P(D\F)=1/2 P(O\U)=1/7 P(D\U)=6/7 Posterior probabilities (Bayes rule) backward induction procedure (rollback) EMV for waiting for the survey results = 153 EMV for deciding without survey = 100 Expected value of imperfect information (EVII) = =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. 12

13 VALUE OF CONTROL Some variables, such as weather, have high information value but are hard to think of good sources of information for. For these variables, move on to the value of control to see if you can think of ways to mitigate the impact of these uncertainties, 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. Most favourable outcome Value of control = =

14 Chance Events with High Value of Control - present the greatest opportunity for improving your outcomes by thinking of creative new ways to either gain control over the uncertainty or to mitigate its impact on your outcomes. Common sources of control: Increased staffing, time, money, or other resources PR or advertising Insurance As with information, 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: Control just improves probabilities. Can t Pick Best State. Important thing to remember about the value of control: it can come either from controlling the underlying uncertainty or by insulating yourself from the effects of that uncertainty, the value of control is normally greater than, or equal to, the value of information. 14

15 CHOOSE option decision tree Graham s decision problem Speed Flexibility Accuracy Cost Weights: Precision Tree (PALISADE) Examples 15

16 Developing Influence Diagrams: Some examples Forecast Hits Miami Outcomes Misses Miami Hits Miami Misses Miami Alternatives Evacuate Stay 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 Source: Clemen, R. (1996), Making Hard Decisions: An Introduction Misses to Decision Miami Analysis Low (2nd risk Edition). Duxbury. Low cost But if there is missing information: The case for sequential decisions... Source: Clemen, R. (1996), Making Hard Decisions: An Introduction to Decision Analysis (2nd Edition). Duxbury. 16

17 More on sequential decisions Source: Clemen, R. (1996), Making Hard Decisions: An Introduction to Decision Analysis (2nd Edition). Duxbury. Developing financial models while accounting for uncertainty 1 st version 3 rd version 2 nd version Source: Clemen, R. (1996), Making Hard Decisions: An Introduction to Decision Analysis (2nd Edition). Duxbury. 17