Utility Examples
Scott Matthews
Courses: 12-706 / 19-702
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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
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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
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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.
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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
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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
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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
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We all need a break.
Deal or No Deal
http://www.nbc.com/Deal_or_No_Deal/game/
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Show online game - quickly
Then play it in front of class a few times
With index cards
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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
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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
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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).
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Typical risk-averse
And what if your utility looks like..
Utility(Y)
1
0.5
Risk Prem
0
$0
$220k
CE - why?
EMV =
$500,000.50
Money ($)
$1,000,000
Risk Prem = EMV - CE
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The banker offers you $380,000
Who would take the offer? Who wouldn’t?
Would the person on the previous slide take
it? Why?
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Typical risk-averse
And what if your utility looks like..
Utility(Y)
Risk Prem = EMV - CE
Risk Prem?
1
0.5
0
$0
EMV =
$500,000.50
CE - why?
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Money ($)
$1,000,000
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Typical risk-seeking
And what if your utility looks like..
Utility(Y)
Risk Prem = EMV - CE
1
Risk Prem < 0!
0.5
~0.15
0
$0
Money ($)
EMV =
$500,000.50
$1,000,000
CE - why?
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The banker’s utility function, and
decision problem
Minimizing loss!
Banker however “is” playing repeated
games with many chances to recover loss
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Play the Game Twice
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Friedman-Savage Utility
Or.. Why Scott doesn’t buy
lottery tickets until the
jackpots get big?
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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!
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http://www.gametheory.net/Mike/applets/Risk/
 http://www.nbc.com/Deal_or_No_Deal/game/flash.shtml
http://www.srl.gatech.edu/education/ME88
13/Lectures/Lecture22_Multiattribute.pdf
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