1. 1 Civil Systems Planning Benefit/Cost Analysis Scott Matthews Courses: 12-706 / 19-702/ 73-359 Lecture 12

2. 12-706 and 73-3592 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!  Risk-neutral (what is relation to EMV?)  Risk-averse  Risk-seeking

3. 12-706 and 73-3593 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.

4. 12-706 and 73-3594 (Discrete) Utility Function Dollar ValueUtility Value 15001.00 10000.86 5000.65 2000.52 1000.46 -1000.33 -10000.00

5. 12-706 and 73-3595 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.

6. 12-706 and 73-3596 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.

7. 12-706 and 73-3597 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.

8. 12-706 and 73-3598 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

9. 9 We all need a break. Deal or No Deal http://www.nbc.com/Deal_or_No_Deal/game/

10. 12-706 and 73-35910 Show online game - quickly  Then play it in front of class a few times  With index cards

11. 12-706 and 73-35911 Professor’s Dream  DOND is a constant tradeoff game:  Certainty equivalent (banker’s offer)  Expected value / utility of deal  Attitude towards risk!  Didn’t have this last year as example  To accept a deal, CE must be < offer  How does banker make offers? Not pure EV!

12. 12-706 and 73-35912 Deal or No Deal - Decision Tree  Decision node that has 2 options:  Banker’s offer to stop the game OR  Chance node (1/N equal probabilities) with all remaining case values as possible outcomes

13. 12-706 and 73-35913 Let’s focus on a specific outcome  You’ve been lucky, and have the game down to 2 cases: $1 and $1,000,000  What does your “decision tree” look like?  How much would you have to be offered to stop playing?  What are we asking when we say this?  What if banker offers (offer increasingly bigger from about $100k).

14. 12-706 and 73-35914 And what if your utility looks like.. Utility(Y) Money ($) $0 $1,000,000 1 0 EMV = $500,000.50 0.5 $220k CE - why? Typical risk-averse Risk Prem Risk Prem = EMV - CE

15. 12-706 and 73-35915  The banker offers you $380,000  Who would take the offer? Who wouldn’t?  Would the person on the previous slide take it? Why?

16. 12-706 and 73-35916 And what if your utility looks like.. Utility(Y) Money ($) $0 $1,000,000 1 0 EMV = $500,000.50 0.5 CE - why? Typical risk-averse Risk Prem? Risk Prem = EMV - CE

17. 12-706 and 73-35917 And what if your utility looks like.. Utility(Y) Money ($) $0 $1,000,000 1 0 EMV = $500,000.50 0.5 CE - why? Typical risk-seeking Risk Prem < 0! Risk Prem = EMV - CE ~0.15

18. 12-706 and 73-35918 The banker’s utility function, and decision problem  Minimizing loss!

19. 19 Friedman-Savage Utility Or.. Why Scott doesn’t but lottery tickets until the jackpots get big?

20. 12-706 and 73-35920  http://www.gametheory.net/Mike/applets/Risk/ http://www.gametheory.net/Mike/applets/Risk/  http://www.nbc.com/Deal_or_No_Deal/game/flash.shtml http://www.nbc.com/Deal_or_No_Deal/game/flash.shtml  http://www.srl.gatech.edu/education/ME88 13/Lectures/Lecture22_Multiattribute.pdf