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Civil Systems Planning Benefit/Cost Analysis

Civil Systems Planning Benefit/Cost Analysis. Scott Matthews Courses: 12-706 / 19-702/ 73-359 Lecture 12. Utility Functions. We might care about utility function for wealth (earning money). Are typically: Upward sloping - want more.

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Civil Systems Planning Benefit/Cost Analysis

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

  2. 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 12-706 and 73-359

  3. 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. 12-706 and 73-359

  4. (Discrete) Utility Function 12-706 and 73-359

  5. 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. 12-706 and 73-359

  6. 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. 12-706 and 73-359

  7. 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. 12-706 and 73-359

  8. 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 12-706 and 73-359

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

  10. Show online game - quickly • Then play it in front of class a few times • With index cards 12-706 and 73-359

  11. 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! 12-706 and 73-359

  12. 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 12-706 and 73-359

  13. 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). 12-706 and 73-359

  14. Typical risk-averse And what if your utility looks like.. Utility(Y) 1 0.5 Risk Prem Money ($) EMV = $500,000.50 0 $0 $220k $1,000,000 CE - why? Risk Prem = EMV - CE 12-706 and 73-359

  15. The banker offers you $380,000 • Who would take the offer? Who wouldn’t? • Would the person on the previous slide take it? Why? 12-706 and 73-359

  16. Typical risk-averse And what if your utility looks like.. Utility(Y) Risk Prem = EMV - CE Risk Prem? 1 0.5 Money ($) EMV = $500,000.50 0 $0 $1,000,000 CE - why? 12-706 and 73-359

  17. Typical risk-seeking And what if your utility looks like.. Utility(Y) Risk Prem = EMV - CE 1 Risk Prem < 0! 0.5 ~0.15 Money ($) EMV = $500,000.50 0 $0 $1,000,000 CE - why? 12-706 and 73-359

  18. The banker’s utility function, and decision problem • Minimizing loss! 12-706 and 73-359

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

  20. http://www.gametheory.net/Mike/applets/Risk/ • http://www.nbc.com/Deal_or_No_Deal/game/flash.shtml • http://www.srl.gatech.edu/education/ME8813/Lectures/Lecture22_Multiattribute.pdf 12-706 and 73-359

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