Second lecture
Expected utility and prospect theory.
The most sited paper ever published in Econometrica is
Kahneman, D. and A. Tversky (1979). "Prospect Theory: An Analysis of Decision Under
Risk," Econometrica, 47, 263-291.
This paper is also a major reason why Kahneman got the Nobel price. (Unfortunately, Amos
Tversky died a few years earlier.) The paper presents evidence that many choices under
uncertainty cannot be explained by expected utility. An understanding of expected utility is
thus important to understand Kahneman and Tversky’s contribution.
Expected utility builds on a simple idea. Let L1 and L2 be two lotteries, and suppose that you
are indifferent in choosing between them. Now suppose that we put them “inside a lottery”
that is, we run a lottery in two stages. I flip a coin and if head you get 0 and if tail you get to
play either L1 or L2. The key assumption of expected utility, called the independence axiom,
is that if you think L1 and L2 are equally good, then you should also think that the two stage
lotteries are equally good irrespective of whether you get L1 or L2 in the second stage.
To make it simple, suppose all outcomes are some payment between -100 kroner and + 1000
kroner. We also assume that for any payment x in this interval, there is a probability u(x) such
that you will be indifferent between x for sure and the lottery
Lottery u(x):
+1000 kroner (winning)
With probability u(x)
-100 kroner (loosing)
With probability 1-u(x)
Consider another lottery
x
y
With probability p
With probability 1-p
Make this lottery two stage by replacing the payment with x with Lottery u(x) and the
payment y with lottery u(y). What is the probability of winning (getting 1000 kroner in the
end) in this two stage lottery?
Doing this calculation in advance will make it much easier to follow the calculations in class.
You should find that the probability of winning is pu(x)+(1-p)u(y). The argument goes: By
the independence axiom you are indifferent between the one stage and the two stage lottery
(I’ll explain more in class) and suppose you prefer to win with the highest possible probability,
then you should choose lotteries to maximize expected utility, that is:
pu(x)+(1-p)u(y).
Now, when you read Kahneman and Tversky look for two things:
First: May of their results violate expected utility. Try to understand why.
Second: Kahneman and Tversky suggest an alternative theory: Prospect Theory. According to
this theory subjects maximize something similar to expected utility, but with some crucial
differences. In the lottery above it would be
p)v(x-r)+ (1-p)v(y-r).
The key differences are (try to answer the questions while reading the paper):
 While the utility function u(x) takes x as the argument, the function v which they call a
value function takes x-r as argument, where r is a ‘reference point’. Try to find out
what the reference point is, and what evidence they have for the existence of a
reference point.
 The function v has a kink at 0. Is it steeper of flatter above zero (compared to below)?
The existence of the kink is referred to as loss aversion. Why? What evidence do the
present in favor of loss aversion?
 In expected utility u(x) is multiplied by probabilities, while Kahneman and Tversky
suggest using ‘decision weights’ p). How will these weights differ from
probabilities? What is the evidence in favor of weights?
The second paper on the reading list is
Köszegi, B and M. Rabin (2006): A Model of Reference-dependent Preferences, Quarterly
Journal of Economics, CXXI.
The paper is more technical. You do not have to read all details. Before class, look at the
introduction.
 Köszegi and Rabing use a utility function u(c|r)=m(c)+n(c|r). Try to figure out how
this relates to Kahneman and Tversky’s value function.
 What is Köszegi and Rabin’s general idea of how the reference point is formed?
1/21/2013
ECON4260 Behavioral
Economics
2nd lecture
Cumulative Prospect Theory
Suggested approximation
(See Benartzi and Thaler, 1995)
w( p) 
p
p

 (1  p)
1

0,9
1/ 
0,8
Gains
0,7
Losses
0,6
0,5
  0.61 for gains
  0.69 for losses
0,4
0,3
0,2
0,1
0
0
0,2
0,4
0,6
0,8
1
The value function
(see Benartzi and Thaler, 1995)
 xa
v( x)  
b
 l ( x)
10
if x  0
if x  0
5
0
-10
• a = b = 0.88
• l = 2.25
-5
0
5
10
-5
-10
-15
-20
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Evidence; Decision weights
• Problem 3
– A: (4 000, 0.80)
– N=95 [20]
or
B: (3 000)
[80]*
or
D: (3 000, 0.25)
[35]
• Problem 4
– C: (4 000, 0.20)
– N=95 [65]*
• Violates expected utility
– B better than A :
u(3000) > 0.8 u(4000)
– C better than D: 0.25u(3000) > 0.20 u(4000)
• Perception is relative:
– 100% is more different from 95% than 25% is from 20%
Lotto
• 50% of the money that people spend on Lotto is paid
out as winning prices
• Stylized:
– Spend 10 kroner
– Win 1 million kroner with probability 1 to 200 000
• Would a risk avers expected utility maximizer play
Lotto?
– Is Lotto participation a challenge to expected utility?
• Can prospect theory explain why people participate
in Lotto?
• What is maximum willingness to pay for this winning
prospect, for an
– Risk avers expected utility maximizer?
– A person acting acording to prospect theory?
Suggested answer
• A risk neutral expected utility maximizer will value the
winning prospect to the expected value
– 1 million kroner* (1/200 000) = 5 kroner
– WTP for a risk avers person < 5 kroner
• Prospect theory
–
–
–
–
–
p = w(1/200 000) ≈ 0.0002
v(1 million) = (1 000 000)^0.88 ≈ 190 000
WTP = x where: 2.25(x)^0.88 ≈ 190 000 * 0,0002
Solution: WTP ≈ 27.5 kroner
Would buy Lotto even with only 2 kroner expected value for
each 10 kroner spent. (5 kroner/2.7)
• Some people do NOT buy Lotto tickets
– Is that a challenge to CPT?
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Value function
Reflection effect
• Problem 3
– A: (4 000, 0.80)
– N=95 [20]
or
B: (3 000)
[80]*
• Problem 3’
– A: (-4 000, 0.80)
– N=95 [92]*
or
B: (-3 000)
[8]
• Ranking reverses with different sign (Table 1)
• Concave (risk aversion) for gains and
• Convex (risk lover) for losses
The reference point
• Problem 11: In addition to whatever you own,
you have been given 1 000. You are now
asked to choose between:
– A: (1 000, 0.50)
– N=95 [16]
or
B: (500)
[84]*
• Problem 12: In addition to whatever you own,
you have been given 2 000. You are now
asked to choose between:
– A: (-1 000, 0.50)
– N=95 [69]*
or
B: (-500)
[31]
• Both equivalent according to EU, but the
initial instruction affect the reference point.
Isolation Effect
(recall the independence axiom)
”In order to simplify the choice between alternatives,
people often disregard components that the
alternatives share and focus on the components
that distinguishes them”
•
Problem 10:
Consider the two-stage game. The first stage is (2. stage,
0.25; 0, 0.75) (proceed to stage to with 25% probability. If
you reach the second stage you have the choice between
A: (4000, 0.80) and B (3000) [78%]. Your choice must be
made before the game starts.
•
The choice in 10 is equivalent to.
A’: (4000, 0.20) [65%] and B’: (3000,0.25)
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1/21/2013
The editing phase
Finding the reference point
• Combination
(200,0.25,200,0.25) =(200,0.5)
• Segregation
(300,0.8;200,0.2)=200 + (100,0.8)
• Cancellation
(200,0.2;100,0.5;-50,0.3) vv (200,0.2;150,0.5;-100,0.3)
Can be seen as a choice between
(100,0.5;-50,0.3) vv (150,0.5;-100,0.3)
• Simplifications
– (500,0.2 ; 99,0.49) dominates
(500,0.15; 101,0.51) if the last part is simplified to
(…
; 100,0.50)
Stochastic dominance
Lottery A (58%)
White
red
green
yellow
Probability %
90
6
1
3
Price
0
45
30
-15
Lottery B (42%)
White
red
green
yellow
Probability %
90
7
1
2
Price
0
45
-10
-15
White
red
green
blue
yellow
Prob. %
90
6
1
1
2
Lottery C (A)
0
45
30
-15
-15
Lottery D (B)
0
45
45
-10
-15
The status of cumulative
prospect theory
(See Starmer 2000)
• Explains data much better than alternative theories.
– Rank dependent utility, does a fair job but not as good
• Starmer claim a limited impact on economic theory
– But loss aversion is increasingly referred to
• But, some very interesting application
– We will use it to understand equity return
• See Camerer 2000 for other examples.
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