1 Imperfect Information / Utility Scott Matthews Courses: /

and 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.

and 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 ?

and 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 ?

and Additivity, cont.  Now look at p,q labels on handout for the decision problem (top values in tree)  Is it additive if instead p=0.3, q = 0.8?  What if p=0.2 and q=0.2?  Should make us think about sensitivity analysis - i.e., how much do answers/outcomes change if we change inputs..

and EVPI - Why Care?  For information to “have value” it has to affect our decision  Just like doing Tornado diagrams showed us which were the most sensitive variables  EVPI analysis shows us which of our uncertainties is the most important, and thus which to focus further effort on  If we can spend some time/money to further understand or reduce the uncertainty, it is worth it when EVPI is relatively high.

and 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

and

9 Assessing the Expert

and 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.

and 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

and EVII Tree Excerpt

and Rolling Back to the Top

and 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

and Sens. Analysis Graph - on board t v 1 01

Friday  How to use Precision Tree software  Makes solving decision trees easier,  Make sensitivity analysis easier and

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and 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”)

and 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?

and 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

and 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.

and

and (Discrete) Utility Function Dollar ValueUtility Value $ $ $ $ $ $ $ Recall: utility function is a “map” from benefit to value - here (0,1) Try this yourselves before we go further..

and EU(high)=0.5*1+0.3* *0 = EU(low)0.652 EU(save)=0.65

and 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.

and 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.

and 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.

and 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

and 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)

30 Next time: Deal or No Deal