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Lecture on Prospect TheoryNational University of Singapore

Prospect Theory

EC4394 L2 1

Outline

Review of expected utility

Prospect theory

Value function

Experimental Evidence

Expected utility vs Prospect theory

Probability weighting function

Experimental Evidence

Expected utility vs Prospect theory

EC4394 L2 2

St. Petersburg Paradox and Expected Utility

How to define the utility/value of a lottery mean?

St. Petersburg paradox: You are faced with a sequence of tosses of a fair coin. The game will end for the first time the coin comes up Head. If this happens on the th trial, you get 2 dollars. What is your willingness to pay to participate in

this game?

EC4394 L2 3

St. Petersburg Paradox and Expected Utility

Mean: U=1

2 2 +

1

4 4 +

1

8 8 + =

1 + 1 + 1 + = + But you will not pay a large amount for the game

Mean is not a good measure of utility

Expected utility: () = , U= 1

2 ln2 +

1

4 ln4 +

1

8 ln8 + =

[ (

2+=1 )] 2 = 22

The utility is finite with a concave

EC4394 L2 4

Expected Utility

A lottery specifies the probability for each prize. = (, ; , 1 ), occurs with probability and occurs with probability 1 .

The expected utility that represents the decision maker - DMs preference would be U = + 1 , where () and () is utility for and .

Note that is the utility function over lottery, while is the utility over outcome/money. Thus, the preference over lottery solely depends on function

EC4394 L2 5

Risk Aversion

DM is risk averse if she prefers the mean of the gamble over the gamble.

DM is risk averse if u is concave. ( + (1 )) () + (1 )() for

any , , and [0,1]; OR (x)0

DM is risk neutral if she is indifferent between the mean of the gamble and the gamble.

DM is risk seeking if she prefers the gamble over the mean of the gamble.

EC4394 L2 6

An Example

= ($100, 0.5; 0, 0.5)

The mean of the lottery is $50

If u() = 0.5,

The expected utility of the lottery is U() =0.5(1000.5)+0.5(00.5)=5

The expected utility of $50 is 50 = 50 = 500.5

50 > U , so DM prefers $50 over the lottery

EC4394 L2 7

Axioms of Expected Utility

Completeness and transitivity

Continuity

Independence: For every lottery , , , and every 0,1 ,

iff + (1 ) + (1 )

EC4394 L2 8

An Example

If 50 100, 0.5; 0, 0.5 , independence implies 50, 0.02; 0, 0.98 100, 0.01; 0, 0.99 .

Why? = 50, = 100, 0.5; 0, 0.5 , R = 0, and = 0.02 + (1 ) = 50, 0.02; 0, 0.98

+ 1 = 100, 0.01; 0, 0.99 .

Is your choice consistent with independence?

EC4394 L2 9

Three Tenets of Expected Utility

Asset Integration: (1, 1; ; , ) is acceptable at asset w iff

( + 1, 1; ; + , ) ()

Risk Aversion: is concave

Expectation: (1, 1; ; , ) = 1(1) + + ()

EC4394 L2 10

Asset Integration

In addition to whatever you own, you have been given 1,000. You are now asked to choose between A1: (1,000, 0.5), and B1: (500)

Most people would choose: 1 1

In addition to whatever you own, you have been given 2,000. You are now asked to choose between A2: (-1,000, 0.5), and B2: (-500)

Most people would choose: 2 2

EC4394 L2 11

Asset Integration

(1) = 0.5( + 1000 + 1000) + 0.5( + 1000) = 0.5 + 2000 + 0.5 + 1000

1 = + 1000 + 500 = + 1500

(2) = 0.5( + 2000 1000) + 0.5( + 2000) = 0.5 + 1000 + 0.5 + 2000

(2) = ( + 2000 500) = ( + 1500)

Expected utility with asset integration could not account for the observation.

EC4394 L2 12

Reference Dependence

1/22/2015 EC4394 L2 13

Reference Dependence

Consider the following example

Ice water in left hand bowl; hot water in right hand bowl; room temperature in the middle bowl.

Immerse left hand in left bowl, and right hand in right bowl.

And then dip both hand in the middle.

EC4394 L2 14

Reference Dependence

DM separates gain and loss relative to a reference point without asset integration

1 = 0.5 1000 + 0.5 0 , and (1) =(500)

2 = 0.5 0 + 0.5 1000 , and 2 = 500

We could have both 1 1 and 2 2 given some function.

1/22/2015 EC4394 L2 15

Risk Aversion

You are asked to choose between A1: (3,000) and B1:(4,000,0.8). Most people choose A1 B1, risk averse in the

gain domain

You are asked to choose between A2: (-3,000) and B2: (-4,000,0.8). Most people choose B2 A2, risk seeking in the

loss domain

This is not consistent with the usual assumption of risk aversion

EC4394 L2 16

Diminishing Sensitivity

Principle of diminishing sensitivity applies to sensory dimensions (Weber-Fechner law).

Turning on a weak light in a dark room versus turning on a weak light in a bright room

Diminishing sensitivity in gain, u(x)0; diminishing sensitivity in loss, u(x)>0 for x

Loss Aversion

You are asked to choose between A1: 0 and B1: 100, 0.5; 100, 0.5

Most people choose A1 B1, risk averse across gain and loss domain

0.5(100) + 0.5(100) < (0) = 0 => (100) < (100)

EC4394 L2 18

Loss Aversion

0.5 + 0.5 < 0 => () < ()

Loss looms larger than gain.

EC4394 L2 19

Valuation Function

Gain/loss relative to reference dependence

Diminishing sensitivity towards gain and loss

Loss aversion (Loss looms larger than gain)

EC4394 L2 20

Reference

point

Sensitivity to probability

Russian roulette: a potentially lethal game of chance in which a player places n bullets in a revolver (full with 6 bullets), spins the cylinder, places the muzzle against his or her head, and pulls the trigger. What is your willingness to pay for each of the reduction of bullets as follows.

6 to 5 (From sure to 5/6 chance)

3 to 2

1 to 0 (From 1/6 chance to impossible)

People like to pay more for 6 to 5, and for 1 to 0.

EC4394 L2 21

Allais Paradox: Common Ratio

It is commonly observed that

(3000, 0.90) (6000, 0.45)

(6000, 0.001) (3000, 0.002)

The expected utility of the choice:

0.9u(3000)> 0.45u(6000)

0.001u(6000)>0.002u(3000)

Contradiction!

EC4394 L2 22

Allais Paradox: Common Ratio

Here we show that it violates the independence

:(3000, 0.90) , :(6000, 0.45), R=0, =2/900

+ 1 =(3000, 0.002).

+ 1 =(6000, 0.001).

Hence (3000, 0.90) (6000, 0.45) and (6000, 0.001) (3000, 0.002) violates independence.

EC4394 L2 23

Allais Paradox: Common Ratio

When expected utility, U = +1 , could not account for the choice

behavior, we need to extend the theory further.

We introduce probability weighting function to have U = () + 1

We examine the properties of

EC4394 L2 24

Allais Paradox: Common Ratio

We derive the condition for , under which it can exhibit the choice patterns.

0.90 (3000) > 0.45 (6000)

0.001 (6000) > 0.002 (3000)

0.45

0.90< 3000

6000< 0.001

0.002

Subproportionality:

<

EC4394 L2 25

Allais Paradox: Common Consequence

It is commonly observed that

2400 (2500, 0.33; 2400, 0.66)

(2500, 0.33) (2400, 0.34)

The expected utility of the choice:

2400 > 0.33 2500 + 0.66 2400

0.33 2500 > 0.34 2400

Contradiction!

Check violation of independence axiom!

EC4394 L2 26

Allais Paradox: Common Consequence

EC4394 L2 27

1-33 (0.33) 34 (0.01) 35-100 (0.66)

Gamble A1 2500 0 2400

Gamble B1 2400 2400 2400

Gamble A2 2500 0 0

Gamble B2 2400 2400 0

Suppose there are 100 balls numbered from 1 to 100. You randomly pick a ball, the number drawn determines your earning as follows.

Allais Paradox: Common Consequence

Similarly, we derive the condition for

0.33 2500 > 0.34 2400

2400 > 0.66 2400 + 0.33 2500

(1 0.66)) 2400 > 0.33 2500 > (0.34)(2400

0.66 + 0.34 < 1

Subcertainty: + 1 < 1

EC4394 L2 28

Gambling and Insurance

Gambling: (5000, 0.001) (5)

Expected utility: 0.001 5000 > 5 5

5000< 0.001

If () > 0, 5

5000> 0.001,

contradiction!

Insurance: (-5) (-5000, 0.0001) Check whether it contradicts the expected utility!

EC4394 L2 29

Gambling and Insurance

Similarly, we derive the condition for

0.001 5000 > 5

0.001 > 5

5000> 0.001

Overweighting small p: >

Check for insurance: (-5) (-5000, 0.0001)!

EC4394 L2 30

Subadditivity for small p

It is commonly observed that (0.001,6000) (0.002, 3000)

Expected utility: 0.001 6000 >0.002 3000 (3000)/(6000) < 0.5

If () > 0, 3000

6000> 0.5,

contradiction!

EC4394 L2 31

Subadditivity for small p