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Utility Examples

Utility Examples. Scott Matthews Courses: 12-706 / 19-702. 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.

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Utility Examples

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

  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. 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

  4. 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

  5. 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

  6. 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

  7. 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 12-706 and 73-359

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

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

  10. Appeal of the Game • DOND is a constant tradeoff game: • Certainty equivalent (banker’s offer) • Expected value / utility of deal • Attitude towards risk! • Recent example from pop culture • To accept deal (for risk neutral), CE < offer • How does banker make offers? Not pure EV! 12-706 and 73-359

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. The banker’s utility function, and decision problem • Minimizing loss! • Banker however “is” playing repeated games with many chances to recover loss 12-706 and 73-359

  18. Play the Game Twice 12-706 and 73-359

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

  20. Is Risk Aversion constant? • Doesn’t seem to be from trials of game • Seems to vary by situation (and timing) • Assumptions of expected value or utility miss the context of the decisions! 12-706 and 73-359

  21. 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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