Risk Attitude - Human Decision Making - Lecture Slides, Slides of Human-Computer Interaction Design

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Risk Attitude

2

Introduction

This example illustrates that EMV analysis does not capture risk attitudes of decision makers. Individuals who are afraid of risk or are sensitive to risk are called risk-averse.

Which game would you choose, game 1 or game 2?

Game 1

(0.5)

(0.5)

Payoff $

-$ (0.5)

(0.5)

$2,

-$1,

Game 2

EMV=$14.

EMV=$

If EMV is the basis for the decision, you should choose Game 2. Most of us, however, may consider Game 2 to be too risky and thus choose Game 1.

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Risk Attitude

 Risk-Averse: Afraid or Sensitive to Risk

 Would trade a gamble for a sure amount that is less than the expected value of the gamble  U( x ) is a concave curve

 (^)  

 x

x x

U( ) (continuous) U( x Δ x )U( x )U(Δ x )(discrete)

 (^)  

 x

x x

U( ) (continuous) U( x Δ x )U( x )U(Δ x )(discrete)

 Risk-Seeking: Willing to Accept More Risk

 Would play a state lottery  U( x ) is a convex curve

 Risk-Neutral: An EMV Decision Maker

 Maximizing utility is the same as maximizing EMV  U( x ) is a straight line

x

x 

 U( ) is constant (continuous) U( x Δ x )U( x )U(Δ x ) (discrete)

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Risk Attitude (Cont’d)

Dollars

Utility

Risk-Averse

Risk-Neutral

Risk-Seeking

Shapes of Utility Functions of Three Different Risk Attitudes

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Certainty Equivalent (CE)

EMV

Risk Premium

Utility Curve

Expected Utility (EU)

Utility

Dollar

U(CE) = EU

For a risk-seeking person, CE would be on the right side of EMV on the horizontal axis

Graphical Representation of Expected Utility, Certainty Equivalent, and Risk Premium of Risk-Averse Utility

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Risk Tolerance and Exponential

Utility Function

 Exponential Utility Function

R

x x e

 U( )  1 

R is risk tolerance, showing how risk-adverse the function is. Larger R means less risk-aversion and makes the utility function flatter

0

1

0 1 2 3 4 5 6 7 8 10 11 12 12 14 15

R= R= R=

Exponential Utility Functions with Three Different Risk Tolerances

x ↑ => U( x ) → x =0 => U( x ) = 0

 Find CE of Given Uncertain Event

 First calculate the expected utility (EU) of the uncertain event  Since U(CE)=EU, you can solve the equation to get CE

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Risk Tolerance and Exponential

Utility Function (Cont’d)

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Suppose you face the following gamble: 1) win $2000 with probability

0.4; 2) win $1000 with probability 0.4, or 3) win $500 with probability

0.2, and your utility can be modeled as an exponential function with

R=900. What is your CE of this gamble?

The expected utility of the gamble is:

EU = 0.4∙U($2000)+0.4 ∙U($1000)+ 0.2∙U($500)

= 0.4∙(1-e-2000/900) +0.4 ∙(1-e-1000/900)+ 0.2∙(1-e1-500/900) = 0.

Solve 0.710=1-e-CE/900^ for CE, you can get CE=$1114.

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Exercise

An investor with assets of $10,000 has an opportunity to invest $5,

in a venture that is equally likely to pay either $15,000 or nothing. The

investor’s utility function can be described by the log utility function

U( x ) =ln( x ), where x is the total wealth.

a.What should the investor do?

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Invest

success (0.5)

Don’t Invest

Failure (0.5)

Total Wealth

a.

EU(invest) = 0.5∙U($20,000)+0.5∙U($5,000)=0.5∙ln($20,000)+0.5∙ln($5000) = 9. EU(Don’t invest) = U($10,000) = ln($10,000) = 9.

Therefore, the investor is indifferent between the two alternatives

Invest

Success (0.5)

Don’t Bet

Failure (0.5)

Total Wealth

Bet

Win (0.5)

Lose (0.5)

Don’t Invest 10,000+1,000 = $11,

Invest 10,000-1,000-5,000= $4,

$15,000^ 10,000-1,000-5,000+15,000 =$19,

Don’t Invest 10,000-1,000 = $9,

Invest 10,000-5,000= $5,

$15,000^ 10,000-5,000+15,000 =$20,

Don’t Invest $10,

Success (0.5)

Failure (0.5)

Success (0.5)

Failure (0.5)

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b.

EU(Invest|Win) = 9.

EU(Don’t Invest|Win) = 9.

EU(Invest|Lose) = 9.

EU(Don’t Invest|Lose) = 9.

EU(Bet) = 9.

EU(Don’t Bet) = 9.

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If he wins the bet: EU(Invest) = 0.5∙ln($21,000) + 0.5∙ln($6,000) = 9. EU(Don’t Invest) = ln($11,000) = 9. Therefore, if he wins the bet, he should invest the venture

If he loses the bet: EU(Invest) = 0.5∙ln($19,000) + 0.5∙ln($4,000) = 9. EU(Don’t Invest) = ln($9,000) = 9. Therefore, if he losses the bet, he should not invest the venture

EU(Bet) = 0.5∙EU(Invest|win) + 0.5∙EU(Don’t Invest |lose) = 0.5(9.326)+0.5(9.105) = 9. EU(Don’t Bet) = 9.21 (from part a)

Therefore, he should bet