1. 1 Imperfect Information / Utility Scott Matthews Courses: 12-706 / 19-702

2. 12-706 and 73-3592 Willingness to Pay = EVPI  We’re interested in knowing our WTP for (perfect) information about our decision.  The book shows this as Bayesian probabilities, but think of it this way..  We consider the advice of “an expert who is always right”.  If they say it will happen, it will.  If they say it will not happen, it will not.  They are never wrong.  Bottom line - receiving their advice means we have eliminated the uncertainty about the event.

3. 12-706 and 73-3593 Is EVPI Additive? Pair group exercise  Let’s look at handout for simple “2 parts uncertainty problem” considering the choice of where to go for a date, and the utility associated with whether it is fun or not, and whether weather is good or not.  What is Expected value in this case?  What is EVPI for “fun?”; EVPI for “weather?”  What do the revised decision trees look like?  What is EVPI for “fun and Weather?”  Is EVPI fun + EVPI weather = EVPI fun+weather ?

4. 12-706 and 73-3594 Similar: EVII  Imperfect, rather than perfect, information (because it is rarely perfect)  Example: expert admits not always right  Use conditional probability (rather than assumption of 100% correct all the time) to solve trees.  Ideally, they are “almost always right” and “almost never wrong”. In our stock example..  e.g.. P(Up Predicted | Up) is less than but close to 1.  P(Up Predicted | Down) is greater than but close to 0

5. 12-706 and 73-3595

6. 6 Assessing the Expert

7. 12-706 and 73-3597 Expert side of EVII tree This is more complicated than EVPI because we do not know whether the expert is right or not. We have to decide whether to believe her.

8. 12-706 and 73-3598 Use Bayes’ Theorem  “Flip” the probabilities with Bayes’ rule  We know P(“Up”|Up) but instead need P(Up | “Up”).  P(Up|”Up”) =  =  =0.825

9. 12-706 and 73-3599 EVII Tree Excerpt

10. 12-706 and 73-35910 Rolling Back to the Top

11. 12-706 and 73-35911 Sens. Analysis for Decision Trees (see Clemen p.189)  Back to “original stock problem”  3 alternatives.. Interesting results visually  Probabilities: market up, down, same  t = Pr(market up), v = P(same)  Thus P(down) = 1 - t - v (must sum to 1!)  Or, (t+v must be less than, equal to 1)  Know we have a line on our graph

12. 12-706 and 73-35912 Sens. Analysis Graph - on board t v 1 01

13. 13

14. 12-706 and 73-35914 Risk Attitudes (Clemen Chap 13)  Our discussions and exercises have focused on EMV (and assumed expected-value maximizing decision makers)  Not always the case.  Some people love the thrill of making tough decisions regardless of the outcome (not me)  A major problem with Expected Value analysis is that it assumes long-term frequency (i.e., over “many plays of the game”)

15. 12-706 and 73-35915 Example from Book Exp. value (playing many times) says we would expect to win $50 by playing game 2 many times. What’s chance to lose $1900 in Game 2?

16. 12-706 and 73-35916 Utility Functions  We might care about utility function for wealth (earning money). Are typically:  Upward sloping - want more.  Concave (opens downward) - preferences for wealth are limited by your concern for risk.  Not constant across all decisions!  Recall “how much beer to drink” example  Risk-neutral (what is relation to EMV?)  Risk-averse  Risk-seeking

17. 12-706 and 73-35917 Individuals  May be risk-neutral across a (limited) range of monetary values  But risk-seeking/averse more broadly  May be generally risk averse, but risk-seeking to play the lottery  Cost $1, Expected Value much less than $1  Decision makers might be risk averse at home but risk-seeking in Las Vegas  Such people are dangerous and should be treated with extreme caution. If you see them, notify the authorities.

18. 12-706 and 73-35918

19. 12-706 and 73-35919 (Discrete) Utility Function Dollar ValueUtility Value $15001.00 $10000.86 $5000.65 $2000.52 $1000.46 $-1000.33 $-10000.00 Recall: utility function is a “map” from benefit to value - here (0,1) Try this yourselves before we go further..

20. 12-706 and 73-35920 EU(high)=0.5*1+0.3*.46+0.2*0 = 0.638 EU(low)0.652 EU(save)=0.65

21. 12-706 and 73-35921 Certainty Equivalent (CE)  Amount of money you would trade equally in exchange for an uncertain lottery  What can we infer in terms of CE about our stock investor?  EU(low-risk) - his most preferred option maps to what on his utility function? Thus his CE must be what?  EU(high-risk) -> what is his CE?  We could use CE to rank his decision orders and get the exact same results.

22. 12-706 and 73-35922 Risk Premium  Is difference between EMV and CE.  The risk premium is the amount you are willing to pay to avoid the risk (like an opportunity cost).  Risk averse: Risk Premium > 0  Risk-seeking: Premium < 0 (would have to pay them to give it up!)  Risk-neutral: = 0.

23. 12-706 and 73-35923 Utility Function Assessment  Basically, requires comparison of lotteries with risk-less payoffs  Different people -> different risk attitudes - > willing to accept different level of risk.  Is a matter of subjective judgment, just like assessing subjective probability.

24. 12-706 and 73-35924 Utility Function Assessment  Two utility-Assessment approaches:  Assessment using Certainty Equivalents  Requires the decision maker to assess several certainty equivalents  Assessment using Probabilities  This approach use the probability-equivalent (PE) for assessment technique  Exponential Utility Function:  U(x) = 1-e -x/R  R is called risk tolerance

25. 12-706 and 73-35925 Exponential Utility - What is R?  Consider the following lottery:  Pr(Win $Y) = 0.5  Pr(Lose $Y/2) = 0.5  R = largest value of $Y where you try the lottery (versus not try it and get $0).  Sample the class - what are your R values?  Again, corporate risk values can/will be higher  Show how to do in PrecisionTree (do: Use Utility Function, Exponential, R, Expected Utility)

26. 12-706 and 73-35926 Discussion on Economic Impacts

27. 12-706 and 73-35927 Summary: Thoughts for Project  Don’t forget we will use the “writing rubric” for grading (see syllabus)  35% of your project grade  Don’t just answer the questions - write a report.

28. 28 Next time: Deal or No Deal http://www.nbc.com/Deal_or_No_Deal/game/