5/21/2010. Often you pay for information when you ask for: Investment Advice Management Consultants Market Investigation Palm Reading

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1 Main source: Clemen, R. T. and T. Reilly (2003). Making Hard Decisions With Decision Tools Suite Update 2004 Duxbury. (chapter 12) 1 Often you pay for information when you ask for: Investment Advice Management Consultants Market Investigation Palm Reading Information gathering includes: Consulting experts Conducting surveys Performing mathematical and statistical analysis Doing research Reading books, journals and newspapers You need this information to make a decision in the future: To invest in a particular stock or not To restructure the organization of your company or not To introduce a product or not To marry this person or not INFORMATION TO REDUCE UNCERTAINTY 2 1

2 Given your decision problem, how much should you be willing to pay for this information? To answer this question you might attempt to compute the value (in $$$) of information. We will first discuss a method for determining the value of perfect information, and next for the value of imperfect information and for the value of control. WHICH ONE DO YOU VALUE MORE? 3 Probability and perfect information The expected value of information Expected value of perfect information 4 2

3 Probability and Perfect Information Definition: Clairvoyant Expert on event A If event A is about to occur, the expert says, it will. If event A is not to occur, the expert says, it will not. The expert is NEVER wrong. His information is PERFECT. You are considering investing in a company, but before you want to make sure that the Dow Jones index will go up as this increases your chances of making a good investment. Therefore, you decide to consult a clairvoyant expert on the event A. A = {Dow Jones index goes up} "A" = {Expert Says Dow Jones index goes up} What does it mean to be clairvoyant in probabilistic terms? 5 Pr( { Expert Says Dow Jones } { Dow Jones } ) = =Pr( A A) = 1 Similarly: Pr( A A) = 1- Pr( A A) Pr("A A) = 0 Pr( A A) = 0 1- Pr( A A) = 0 Pr( A A) = 1 Perhaps more importantly, what about? Pr({ Dow Jones } { Expert Says Dow Jones } ) = =Pr(A "A")? 6 3

4 P("A" Pr("A" A)Pr(A) Pr(A "A") = = Pr(" A") Pr("A" A)Pr(A) = = Pr("A" A)Pr(A) + Pr("A" A)Pr(A) 1* Pr(A) = = 1 1* Pr(A) + 0*Pr(A) Conclusion: Pr(A "A") equals 1 no matter what the value of Pr(A) is! 7 What about the probability Pr({Expert Says Dow Jones } )? Pr("A") = Pr( {Expert Says Dow Jones } ) = Pr(" A" A) Pr( A) + Pr(" A" A) Pr( A) = = 1* Pr( A) + 0*Pr( A) = Pr( A) = Pr({ DowJones }) This is true in general: if we consult a clairvoyant expert about an event a with wt possible outcomes { A 1,..., A } then: Pr(" Ai ") = Pr( Ai ), i = 1,..., n { 1 n After consulting the clairvoyant expert about event a, no uncertainty remains about that event. 8 4

5 Expected Value of Information: Stock market example 9 Expected Value of Information: Stock market example Consider first talking to a clairvoyant expert and then making the investment decision What would happen? 10 5

6 Expected Value of Information: Stock market example Of course, the clairvoyant expert will charge a fee, and you would like to know how much you would be willing to pay before using his services. Conclusion: You would be willing to consult the clairvoyant expert if: 1000 X 580 X = 420 (=EVPI) 11 Perfect Information in the Investor s Problem 12 6

7 Expected Value of Perfect Information EVPI = Expected Value of Perfect Information Interpretation: EVPI is the maximum amount of money you would be willing to pay for the services of the clairvoyant expert. If he charges more than $420 you would not consult the expert. (Value of information in a prior sense) A = { Dow Jones index goes up} "A" = {Expert Says Dow Jones index goes up} Consider an expert about event A, who is not clairvoyant, but is considered to be an expert. What does it mean for an expert not to be perfect in his assessment about event A? 13 Pr( {Expert Says Dow Jones } {Dow Jones } ) = Pr("A" A)<1 Hopefully, the probability above is close to 1 (otherwise why consider him/her an Expert?) Pr( {Expert Says Dow Jones } {Dow Jones } ) = Pr("A" nota) > 0 Hopefully, the probability above is close to 0 (otherwise why consider him/her an Expert?) When an expert about an event is not clairvoyant you need to express your trust in his assessment by for example, checking his past performances and interviewing references. 14 7

8 15 Based on your background check of the expert you assess your trust in terms of subjective probabilities & you find: 16 8

9 Actual assessment of the expert Conditional probabilities characterizing economist s forecasting ability 17 Imperfect Information in the Investor s Problem Expected value = 580 Expected value =

10 Extra branch on whether to consult an expert is added & you consider first talking to an "Imperfect expert and then making the original investment decision: 20 10

11 Suppose the Imperfect expert said Dow Jones will go UP 21 Suppose the Imperfect expert said Dow Jones will go FLAT 22 11

12 Suppose the Imperfect expert said Dow Jones will go DOWN 23 Note that: After consulting the expert the uncertainty remains. After consulting an imperfect expert, the original decision problem still remains. The only difference is that probabilities of the original decision problem have changed to reflect the additional information, i.e. the expert's advise. To calculate the EMV of the decision problem after consulting the imperfect expert, we have to solve for the probabilities in the decision tree with an added branch. Calculating these probabilities is equivalent to FLIPPING the order of the uncertainty nodes

13 How can we solve for these probabilities? Via Bayes theorem using a probability table 25 Which equals to doing this: 26 13

14 STEP 1: Construct a probability table Posterior probability 27 STEP 2: Insert the probabilities in the probability tree 28 14

15 STEP 3: Calculate EMV after consulting the expert STEP 3A STEP 3B STEP 3C 29 STEP 3D: Calculate EMV after consulting the expert 30 15

16 STEP 4: Calculate EVII for consulting the expert Conclusion: You would be willing to consult the clairvoyant li expert if: 822 X 580 X = 242 (=EVII) EVII = Expected Value of Imperfect Information 31 Interpretation of the Expected Value of Imperfect Information EVII definition EVII is the maximum amount of money you would be willing to pay for the services of the imperfect expert. If he charges more than $242 you would not consult the expert. Note: EVPI EVII Interpretation: Perfect Information is always btt better than imperfect information. When performing sensitivity analysis, EVPI calculation of every uncertain event should be considered (upper bound)! When EVPI is high for a particular uncertain event, investment to reduce uncertainty may be warranted

17 Completed Decision Tree for the Investment Example 33 Value of Information in Complex Problems Market example: one uncertain event, behaviour of the market, uncertainty tit modelled dlld with a simple discrete distribution ib ti How can we handle continuous probability distributions? Same principle, but... Some difficulties Discrete approximation Construct a Monte Carlo simulation model Find theoretical probability models What happens if there are multiple uncertain events and information is available for some or all of them? Same principle... i Decision trees: move chance nodes for which information is to be obtained so that they precede the decision node Influence diagrams: include an imperfect information node that provides information to the decision maker What if there are nonmonetary objectives? Attempt to apply the same logic

18 Value of Information, Sensitivity Analysis, and Structuring Motivation of the chapter: examine situations in which information i is available and to show how decisions i can be made systematically regarding what source of information to select and how much an expert s information might be worth. The value of information is also useful... Elegant and subtle role in the structuring of decisions and in the entire decision analysis process of developing a requisite model. Indicating subsequent focus in the decision modelling process Low EVPI? High EVPI? Large payoffs and guidance for the development of the decision analysis model)

19 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. 37 Most favourable outcome Value of control = =

20 Events with high value of control Opport unity Thinking creative new ways to either gain control over the uncertainty or to mitigate its impact on your outcomes Cost can be modelled in a decision tree or in an influence diagram But control comes with a cost TO REMEMBER: 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 perfect information. Common sources of control: Increased staffing, time, money, or other resources PR or advertising Insurance Possible imperfect control: Control just improves probabilities. Can t Pick Best State

21 This chapter has claimed that information always has a positive value. What do you think of this? Can you imagine any situation in which you would prefer not to have some unknown information revealed? a) Suppose you have just visited your physician because of a pain in your abdomen. The doctor has indicated some concern and has ordered some tests whose results the two of you are expecting in a few days. A positive test result will suggest that you may have a life threatening t i disease, but even if the test t is positive, the doctor would want to confirm it with further tests. Would you want the doctor to tell you the outcome of the test? Why or why not? 41 b) Suppose you are selling your house. Real estate transaction laws require that you disclose what you know about significant structural defects. Although you know of no such defects, a buyer insists on having a qualified engineer inspect the house. Would you want to know the outcome of the inspection? Why or why not? 42 21

22 c) Suppose you are negotiating to purchase an office building. You have a lot of experience negotiating commercial real estate deals, and your agent has explained this to the seller, who is relatively new to the business. As a result, you expect to do very well in this negotiation. Because of the unique circumstance of the building, your agent has suggested obtaining an appraisal of the property by a local expert. You know exactly how you would use this information; it would provide an upper bound on what ht you are willing to pay. This fact is also clear to the seller, who will know whether you obtained the appraisal but will only find out the appraised value if you elect to reveal it. Would you obtain the appraisal? Why or why not? 43 22